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A context for belief revision: forward chaining-normal nonmonotomic rule systems
Annals of Pure and Applied North-Holland
269
Logic 67 (1994) 269-323
A context for belief revision: forward chaining-normal nonmonotonic rule systems V.W. Marek” Department
qf Computer Science, University qf Kentucky,
Lexington.
KY 40.506-0027. USA
A. Nerode** Mathematical
Sciences Institute,
Cornell University,
Ithaca, NY 14853, USA
J.B. Remmel*** Department
of Mathematics.
University
of California
at San Diego, La Jolla, CA 92903, USA
Communicated by D. van Dalen Received 1 December 1992 Revised 15 June 1993
Abstract Marek, V.W., A. Nerode and J.B. Remmel, A context for belief revision: forward chaining-normal nonmonotonic rule systems, Annals of Pure and Applied Logic 67 (1994) 269-323. A number of nonmonotonic reasoning formalisms have been introduced to model the set of beliefs of an agent. These include the extensions of a default logic, the stable models of a general logic program, and the extensions of a truth maintenance system among others. In [13] and [16], the authors introduced nonmonotonic rule systems as a nonlogical generalization of all essential features of such formalisms so that theorems applying to all could be proven once and for all. In this paper, we extend Rieter’s normal default theories, which have a number of the nice properties which make them a desirable context for belief revision, to the setting of nonmonotonic rule systems. Reiter defined a default theory to be normal if all the rules of the default theory satisfied a simple syntatic condition. However, this simple syntatic condition has no obvious analogue in the setting of nonmonotonic rule systems. Nevertheless, an analysis of the proofs of the main results on normal default theories reveals that the proofs do not rely on the particular syntactic form of the rules but rather on the fact that all rules have a certain consistency property. This led us to extend the notion of normal default theories with respect to a general consistency property. This extended notion of normal default theories, which we call Forward Chaining normal (FC-normal), is easily lifted to nonmonotonic rule systems and hence applies to general logic programs and truth maintenance
Correspondence to: V.W. Marek, Department of Computer Science, University of Kentucky, Lexington, KY 40506-0027, USA. Email:[email protected] * Work partially supported by NSF grant IRI-9012902. ** Work partially supported by NSF grant DMS-8902797 and AR0 contract DAAG629-85-C-0018. *** Work partially supported by NSF grant DMS-9006413. 016%0072/94/$07.00 0 1994- Elsevier Science B.V. All rights reserved SSDI 0168-0072(93)E0053-Q
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et al.
systems as well. We prove that FC-normal nonmonotonic rule systems have all the desirable properties possessed by normal default theories and a number of other important properties as well. We also give an analysis of the complexity of the set of extensions of FC-normal nonmonotonic rule systems. For example, we prove that every recursive FC-normal nonomontonic rule system has an extension which is r.e. in the halting set. Similarly any finite FC-normal nonomonotonic rule system S has an extension which can be computed in polynomial time in the size of S. Again these results automatically apply to general logic programs, truth maintenance systems, and default theories. Moreover, even in the case of normal default theories, our complexity results are new.
1. Introduction and motivation A number of nonmonotonic reasoning formalisms have been introduced to model the set of beliefs of an agent. For example, Reiter [26] introduced default logic where the set of beliefs of an agent reasoning with incomplete information corresponded to an extension of a default theory (D, W). In the realm of logic programming, the stable models of a general logic program, as introduced by Gelfond and Lifschitz [9], can be used to model the set of beliefs of an agent. Similarly, extensions of truth maintenance systems as defined by Doyle [‘i’] and De Kleer [6], with subsequent contributions of Reinfrank, Dressler, and Brewka [25] can be used to model the set of beliefs of an agent. We introduced nonmonotonic rule systems as a non-logical generalization of all essential features of default theories, truth maintenance systems, and logic programs so that theorems applying to all could be proven once and for all, see [13] and [16]. We put forth the hypothesis that any of the above systems can be used to effectively model the beliefs of an agent as long as we restrict ourselves to a rather wide class of default theories, general logic programs, or truth maintenance systems which correspond to the forward chaining-normal nonmonotonic rule systems, or FC-normal systems, introduced in this paper. For example, suppose that belief sets are identified with stable models of an FC-normal logic program. To explain the role of FCnormality, we employ the paradigm of a blocks world for a robot with a hardwired motion planner. We think of the robot as making moves based on facts about where blocks are and based on rules of thumb such as “as long as such and such configuration is not observed, move as follows” based on logic programming. We choose as our current belief set a stable model, if any, incorporating known facts and rules. First difficulty: a general logic program may have no stable model. A second difficulty can arise even if our logic program has a stable model. That is, we want to deduce robot moves in this stable model. But if the robot observes a new fact, a block position not in the stable model, we have to abandon the previous stable model of the old program and find a new stable model of the logic program extended by the new facts. Again, for arbitrary logic programs, there may be no such stable model. So if stable models are to model beliefs and the robot is to have a belief set no matter what facts arise, with a hardwired underlying logic program, we have to find a useful and general condition on the logic program which guarantees that the program, extended by new facts,
A context
for
belief
always has a stable model. Such a condition
271
revision
is the irreducible
model belief sets as stable models and to specify belief revision
minimum operations
in order to as program-
ming operations on stable models. In default logic, Reiter [26] (see also Etherington [S]) defined a simple class of default rules, called normal, for which one never gets stuck in finding extensions. His normality
condition is syntactic and unduly restrictive, i.e., in a normal default theory It turns out that this syntactic condition does not all rules must be of the form y. generalize directly to nonmonotonic rule systems. However, an analysis of the proofs of the main results of Reiter on normal default theories reveals that his proofs do not rely on the particular the form 7 generalization
syntactic
form of his rules but rather on the fact that all rules of
have a certain consistency property. This led us to define a far reaching of normal default theories with respect to a general consistency prop-
erty. Moreover, this generalization is easily extendible to nonmonotonic rule systems and hence applies to general logic programs and truth maintenance systems as well. Thus we introduce in this paper the definition of what we call FC-normal nonmonotonic rule systems. We shall see that when we translate FC-normal nonmonotonic rule systems back into default theories, we will define a large class of default theories which we call FC-normal default theories. We shall see that the class of FC-normal default theories strictly contains the class of normal default theories and that FC-normal default theories have all the desirable properties of normal default theories. Finally, there are natural analogues of FC-normal default theories or FC-normal nonmonotonic rule systems in the formalisms of logic programming and truth maintenance systems as well. To repeat, we claim that extensions of nonmonotonic rule systems can be used to model beliefs of an agent, provided that we restrict ourselves to the class of FC-normal nonmonotonic rule systems introduced in this paper. Here an extension of a nonmonotonic rule system is the common generalization of extensions of default theories, stable models of logic programs, and extensions of truth maintenance systems. Restricting ourselves to FC-normal nonmonotonic rule systems avoids our ever getting stuck in finding new extensions in the face of new facts and there is no syntactic limitation. Rather, we employ an axiomatic “consistency property”, different for each application, which covers all areas of intended application we have examined. Our notion of consistency property can be thought of as a version of Scott’s “information systems” [27] tailored to extensions of nonmonotonic rule systems. In future papers this will allow us to use, for example, metaprogramming as a belief revision programming language operating on “FC-normal” logic programs and representations of stable models. Now the problem of whether there exists a stable model of a finite propositional program is known to be NP-complete [20]. The same result applies to the problem of whether there exists an extension of a nonmonotonic rule system. So the problem of finding a consistency property under which a logic program P or a nonmonotonic rule system (U, N ) is FC-normal is, at least, NP-hard. But in all applications, we can see directly what consistency notion to introduce, dictated by natural considerations of consistency for the semantics of the application.
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V. W. Marek et al.
We shall not only prove that FC-normal nonmonotonic rule system have the desirable properties possessed by normal default theories but we shall also prove that FC-normal nonmonotonic rule systems have a number of other important properties as well. For example, given an FC-normal nonmonotonic rule system 9’ = (U, N), we shall show that every extension of Y can be constructed via a simple forward chaining
algorithm
which is based on a well-ordering
< of the strictly nonmonotonic
rules of Y which we denote by nmon(Y). That is, each ordering mines an extension of Y and every extension of 9’ is determined However,
many orderings
ings themselves
may determine
the same extension.
may be taken as computational
surrogates
of nmon(Y) deterby an ordering <.
This means
for extensions.
the orderIn a com-
panion paper [lS], we introduce a more general forward chaining process based on a well-ordering of the nonmonotonic rules which can be applied to arbitrary nonmonotonic rule systems 9’. For general nonmonotonic rule systems, our forward chaining process also yields an extension, but for a possibly smaller rule system than our original rule system. It turns out that one can view our definition of FC-normal nonmonotonic rule systems as a sufficient condition which guarantees that the general forward chaining algorithm always produces an extension of the original rule system. This is important because we will show that the forward chaining algorithm, when applied to finite FC-normal nonmonotonic rule systems, produces an extension in polynomial time in the sum of the lengths of the rules of the system. More precisely one can construct an extension of an FC-normal nonmonotonic rule system in time which is of order the square of the sum of the lengths of the rules of system. As usual, this same result also applies to constructing extensions of FC-normal default theories, stable models of “FC-normal” logic programs, and extensions of “FC-normal” truth maintenance systems. Thus FC-normal nonmonotonic rules system have the property that one can construct an extension or go from one extension to another in a highly efficient manner. Finally we explore the complexity of the sets of extensions of arbitrary recursive FC-normal nonmonotonic rule systems. Briefly, a nonmonotonic rule system 9 = (U, N) is called recursive if the universe U is a recursive set of integers and the set of rules N is a recursive set. We showed in [15] that for every countably branching recursive tree, there is a recursive nonmonotonic rule system 9’ = (U, N) such that there is a recursive one-to-one correspondence between maximal branches of the tree and the extensions of the 9’. Conversely, the set of all extensions of a recursive nonmonotonic rule system always so arises. Since there are recursive trees without hyperarithmetic maximal branches, but with a continuum of maximal branches, it follows there are recursive NRS with a continuum of extensions but no hyperarithmetic extensions. This does not happen with FC-normal recursive NRS. They always have extensions recursive in 0”. However, recursive FC-normal nonmonotonic rule systems are very expressive. We show that given any highly recursive tree, i.e. a recursive tree which is finitely branching and which has the property that we can effectively find all the successors of any node in the tree, there is an FC-normal nonmonotonic rule system Y = (U,N) such that there is a recursive one-to-one
A context for belief reoision
correspondence
between
the maximal
branches
273
of the tree and the extensions
9’. This is shown via coding trees into suitably constructed a simultaneous construction of a suitable consistency property. for example,
that there are recursive
FC-normal
nonmonotonic
of the
recursive NRS with This will imply that, rules systems which
have no recursive extensions. The outline of this paper is as follows. In Section 2, we shall briefly review Reiter’s [26] definition of normal default theories and state the main properties of such theories. Then in Section 3 we introduce the basic definitions of nonmonotonic rule systems.
In Section
define FC-normal
4, we shall introduce nonmonotonic
our abstract
consistency
properties
rules systems. We shall also introduce
and
our forward
chaining construction and state our major results about FC-normal nonmonotonic rules systems. The proofs of these theorems will be postponed until Section 7. In Section 5, we shall show how to translate the definitions and results of Section 4 back into logic programming, default logic, and truth maintenance systems. In Section 6, we shall provide a background on the complexity of general recursive nonmonotonic rule systems and state our basic complexity results for recursive FC-normal nonmonotonic rule systems. The proofs of the complexity results in Section 6 will be postponed until Section 8.
2. Normal default
default rule is a rule of proof of the form c~:M$i,...,
M$,,,
(1)
Y
where q, $ 1, . . . , Grn, y are formulas of a propositional language 9. A default theory is a pair (0, W) where D is a set of default rules and W G 9. For any subset of formulas S c 9, we let Cn(S) denote the set of all logical consequences of S. Also if D is a set of default rules, let
v:M$,,..., Y
Given a subset S c 9, define T(S) as the least set T(under conditions: 1. WLT; 2. Cn(T) = T; 3. Whenever rED is a default lIc/j$S, then YET.
inclusion)
satisfying
these
rule of the form (1) and VET and for all j d m,
214
V.W. Marek
It is easy to see that T(S) always
et al.
exists. We say that S E _Y’ is an extension of
(D, W) if T(S) = S. A default rule of the form (1) is called generating for S if cp~S, i$,#S. Let S c 9’. Then we define GD(D, S) as the set of all generating 1$1,..., rules for S in D and c(GD(D, S)) is the set of their conclusions. A rule r is normal if it is of the form
q:Mti II/ A default proved
theory
(2)
. (D, W) is normal if every rED is a normal
the following
theorems
about
normal
default
default
rule. Reiter [26]
theories.
Theorem 2.1. Every normal default theory possesses an extension. Theorem 2.2 with D’ E D. A = (D, W). 1. E’ G E
(Semi-monotonicity). Suppose that D and D’ are sets of normal defaults Let E’ be an extension of the normal default theory A’ =
2. GD(E’, A’) G GD(E, A). Theorem 2.3 (Orthogonality of Extensions). Zf a normal default theory (D, W) has distinct extensions E and F, then E u F is inconsistent. Theorem 2.4. Suppose that A = (D, W) is a normal default theory and WV c(D) is consistent. Then A has a unique extension. Theorem 2.5. Suppose that A = (D, W> is a normal default theory and that D’ c D. Suppose further that E; and E; are distinct extensions of (D’, W). Then A has distinct extensions E, and Ez such that El c E, and E; c E,.
3. Monotonic and nonmonotonic rule systems In this section we recall the basic definitions of monotonic and nonmonotonic rule systems from [13, 163. Tarski [28] characterized monotonic formal systems by means of monotonic rules of inference. Such systems include intuitionistic logics, classical logics, modal logics, and many others. Suppose that a nonempty set U is given. In a particular application U may be the collection of all formulas of a propositional or first-order logic, of all legal strings of a formal system, or of all atomic statements as in logic programming. A monotonic rule of inference is a tuple r = (P, cp) where P = (aI, . . . , a,) is a finite (possibly empty) list of objects from U and cpis an element of U. Such a rule r is usually
215
A context for belie1 revision
written
in the suggestive al,
form
a”
.‘.,
(3)
Y=
.
cp
We call fxl , . . . , a, the premises
of r and cp the conclusion
of r.
Definition 3.1. (a) A monotonic formal system is a pair (U, M), where U is a nonempty set and M is a collection of monotonic rules. (b) A subset S c U is called deductively closed over (U, M) if for all rules r=
Ml,
. ..>
%
EM,
al,
....
(x,ES implies
cpeS.
cp Inspired
by Reiter [26], and Apt [3], we introduced
the notion
of a non-monotonic
system (U, N) in [13, 161. A nonmonotonic rule of inference is a triple (p, R, cp>, where P = {ul, . . . . an}, R = {PI, . . . . Pm} are finite lists of objects from U and cp~U. Each such rule is written in a more suggestive form as
formal
r=
al> . . . . u,: 81, .“> Pm
(4)
cp
Here c1i, . . . . CI, are called the premises of rule r, PI, . . . , /3,,,are called the constraints of rule r, and cp is called the conclusion of rule r. For any rule r as in (4), we shall write prem(r) = (a,, . . . . a,}, cons(r) = {j3,, . . . . pm}, and c(r) = cp. Either prem(r), or cons(r), or both may be empty. If prem(r) = cons(r) = 0, then the rule r is called an axiom. A nonmonotonic rule system is a pair (U, N ), where U is a nonempty set and N is a set of nonmonotonic rules such that prem(r), cons(r), and {c(r)} are subsets of U for all rEN. Each monotonic formal system can be identified with the nonmonotonic which every monotonic rule has an empty set of constraints. A subset S s U is called deductively closed if for all r=
El, .‘., a,:
Bl,
cp
...,
Pm
system in
EN,
whenever all the premises cx1, , a, of r are in S and all the constraints r are not in S, then the conclusion cp of r belongs to S. In nonmonotonic systems, deductively closed sets are not generally
/11, . . . . /?,,,of closed under
arbitrary intersections as in the monotone case. But deductively closed sets are closed under intersections of descending chains. Since U is deductively closed, by the Kuratowski-Zorn Lemma, for any I E U, there is at least one minimal deductively closed superset of I. Given sets S E U and I s U, an S-deduction of cp from I in (U, N) is a finite sequence (cp 1, . . . , qk) such that (Pk = cp and, for all i < k, each Cpiis in I, or is an axiom, or is the conclusion of a rule rEN such that all the premises of r are included
276
V. W. Marek et al.
in (cpl,.... vi-
1 } and
all constraints
S-consequence of I is an element
of r are in U - S (see [19],
of U occurring
in some S-deduction
also [25]).
An
from I. Let C,(Z)
be the set of all S-consequences of I in (U, N ). Clearly I is a subset of Cs( I). However, note that S enters solely as a restraint on the use of the rules imposed by the constraints in the rules. A single constraint in a rule in N may be in S and thus prevent the rule from ever being applied in an S-deduction from I, even though all the premises
of that rule occur earlier
in a deduction.
Thus S contributes
no members
directly to C,(Z), although members of S may turn up in C,(Z) by an application of a rule which happens to have its conclusion in S. For a fixed S, the operator Cs( .) is monotonic.
That is, if I c J, then C,(Z) c C,(J).
Also, C,(C,(I))
= C,(I).
However,
for fixed I, the operator C,(Z) is anti-monotonic in the argument S. That is if S’ E S, then C,(Z) s C,*(Z). Generally, C,(I) is not deductively closed in (U, N ). It is perfectly possible that all the premises of a rule be in C,(I), the constraints of that rule are outside C,(I), but a constraint of that rule be in S, preventing the conclusion from being put into C,(Z). Example
3.1. U = {CI,8, y}, N = ($, y>,
S = {/I}. Then
C,(O) = (cz> is not deduct-
ively closed. However, Proposition
the following 3.2.
holds:
If S G C,(I),
then C,(I)
is deductively closed.
We say that S c U is grounded in I if S E C,(I). We say that S s U is an extension of I if C,(Z) = S. The notion of groundedness is related to the phenomenon of “reconstruction”. S is grounded in I if all elements of S are S-deducible from I (remember that S influences only the negative sides of rules). S is an extension of I if two things happen. First, every element of S is deducible from I, that is, S is grounded in I (this is an analogue of the adequacy property in logical calculi). Second, the converse holds: all the S-consequences of I belong to S (this is the analogue of completeness). Thus extensions are analogues for a nonmonotonic systems of the set of all consequences for monotonic systems. Both properties (adequacy and completeness) need to be satisfied-if we want S to be an extension. The notion of an extension is related to that of a minimal deductively closed set. Indeed, the following propositions are proved in [13]. Proposition 3.3. Zf S is an extension of I, then: (1) S is a minimal deductively closed superset of I. (2) For every Z‘ such that Z E Z‘ G S, Cs( I ‘) = S. Proposition 3.4. The set of extensions of I forms an antichain. That is, if S,, Sz are extensions of I and S1 E S2, then S1 = Sz.
211
A context for belief rel;ision
With each rule r of form (4), we associate
obtained
from r by dropping
a monotonic
all the constraints.
rule of form (3)
Rule r’ is called the projection
of rule
r. Let NG (S, Y) be the of all S-applicable rules. That is, a rule r belongs to NG (S, 9) if all the premises of r belong to S and all constraints of r are outside of S. We write M(S) for the collection of all projections of all rules from NG(S, 9’). The projection (U, N ) Is is the monotone system (U, M(S)). Thus (U, N ) Is is obtained as follows: First, non-S-applicable rules are eliminated. Then, the constraints are dropped altogether. We have the following characterization theorem. Theorem 3.5. A subset S c U is an extension deductive closure of I in (U, N )is.
of I in (U, N)
if and only if S is the
For the rest of this paper, we shall only consider extensions of 8 unless explicitly stated otherwise. We say that T is an extension of Y if T is an extension of 0 in 9. We shall end this section by giving yet another characterization of extensions. For this we need the concept of a proof scheme. A proof scheme for cp is a finite sequence P = (
ro,
GO),
. . . . (cp,,
rm,
G,>>
(6)
such that (P,,, = cp and (1) If m = 0 then: (a) cpo is an axiom Go = 0, or (b) cp is a conclusion
(that is, there exists a rule reN of a rule r =
(2) If m > 0, ((Cpi, ri, Gi))yCol
‘b:..“br,
such that r = $), r. = r, and
r0 = r, and Go = cons(r).
is a proof scheme of length m and (P,,,is a conclusion
of
where io, . . . . i, < m, r, = r, and G, = G,_ 1 u cons(r). The formula qrn is called the conclusion of p and is written cln(p). The set G, is called the support of p and is written supp(p). The idea behind this concept is as follows. An S-derivation in the system (U, N ), say p, uses some negative information about S to ensure that the constraints of rules that were used are outside of S. But this negative information is finite, that is, it involves a finite subset of the complement of S. Thus, there exists a finite subset G of the complement of S such that as long as G n Si = 0, p is an S1 -derivation as well. Our notion of proof scheme captures this finitary character of S-derivation. We can then characterize extensions of (U, N) as follows.
V. W. Marek et al.
278
Theorem 3.6. Let Y = (U, N > be a nonmonotonic rule system and let S c U. Then S is an extension of 9 if and only if (i) for each cp~S, there is a proof scheme p such that cln(p) = cpand supp(p) A S = $79, and
(ii) for
each
cp$ S, there
is no proof
scheme
p such
that cln(p) = cp and
SUPP(P) c-l s = 0.
There is a natural utilize.
Given
preordering)
preordering
a proof
scheme
proof scheme
is always
a minimal
with the same conclusion.
recursion-theoretic considerations. more detail in [13, 161
4. FC-normal nonmonotonic
of proof schemes according p, there
The concept
to the set of rules they (with respect
to that
This fact will be used in our
of minimal
proof scheme is treated
in
rule systems
In this section we shall define forward chaining-normal nonmonotonic rule systems (FC-normal nonmonotonic rule systems) and state the analogues of all the theorems of the previous section. We shall postpone the proof of all theorems in this section until Section 7. Let 9 = (U, N ) be a nonmonotonic rule system. Let man(Y) be the set of all rules rEN such that r has no constraints. Thus man(Y) = {rEN :cons(r) = 0}. We let nmon(Y) = N - mon(9’). We shall refer to man(Y) as the monotonic part of Y and nmon(Y) as the nonmonotonic part of 9’. We say a set W c U is monotonically closed if whenever r = -Emon and CI~,. . . . CI,,EW, then YEW. Given any set A G U, the monotonic-closure of A, written cl,,,(A), is defined to be the intersection of all monotonically closed sets containing A. It is easy to see that such a set is itself monotonically closed. So far our investigations of nonmonotonic rule systems did not get beyond the information established already in [13, 161. Next we introduce the notion of consistency property over (U, N ), which leads us to the main subject of this paper. We say that a subset Con E 9(U) (where Y(U) is the power set of U) is a consistency property over 9’ = (U, N ) if 1. @Con; 2. VA, B E U(A E B A Con(B) * Con(A)); 3. VA z U (Con(A) * Con(cl,,,(A))); 4. whenever A! c Con has the property
that
then Con( U JZ?). Condition 1 says that the empty set is consistent. Condition 2 requires that a subset of a consistent set is also consistent. Condition 3 postulates that the closure of a consistent set under monotonic rules is consistent. Finally, the last condition says
A context for b&f
of a directed family of consistent
that the union
219
revision
sets is also consistent.
We note that
conditions 1, 2, and 4 are Scott’s conditions for information systems. Condition 3 connects “consistent” sets to the monotonic part of the rule system; if A is consistent then adding
elements
derivable
from A via monotonic
ency property r=
is FC-normal
rules preserves
“consistency”.
rule system and let Con be a consist-
Now suppose ,Y = (U, N ) is a nonmonotonic over (U, N ), Then we say a rule @l>
‘..>
/31,. .. . Bk
&I:
E nmon(Y)
Y
(with respect
to Con) if Con(Vu
{r})
and not Con(Vu
{y, pi}) for all
i d k whenever V G U is such that Con(V), clmO,,(V) = V, c(i, . . . . C(,EV, and 1,. . . , pk 4 V. We say that Y = (U, N ) is a FC-normal (with respect to Con) if all Y>a rEnmon(.Y) are FC-normal with respect to Con. Finally, we say that (U, N) is FC-normal nonmonotonic rule system if for some consistency property Con s P(U), (U, N ) is FC-normal with respect to Con. Example 4.1. Let U = {a, b, c, d, e, j>. Let the consistency following condition: A $ Con
if and only if
either
(2) ;,
(3) ;,
be defined by the
{c, d } c A or {e, f } c A.
Thus {u,b,c,e}, {u,b,c,f), {u,b,4e},and {u,b,Lfj which are in Con. Now consider the following set of rules, N: (1) ;,
property
(4) q,
are the maximal
subsets of Y(U)
(5) 5
Then for the nonmonotonic rule system Y = (U, N ), rules (l), (2), and (3) form the monotonic part of Y and rules (4) and (5) form the nonmonotonic part of 9. First it is easy to check that Con is a consistency property over 9. The monotonically closed subsets of P(U) which are in Con are {a}, (a, d}, {a, e}, {a,f}, {a, b, c}, {a, d, e}, {a, d,f}, {a, b, c, e}, and {a, b, c,f}. It is then easy to check that Y is FC-normal with respect to Con. Moreover, one can easily check that Y has a unique extension A4 = {a,b,c,e}. If we add to N the rule : to get a set of rules N’, then Con is no longer a consistency property over 9” = (U, N’) because {c}ECon but the monotonic closure of {c} relative to Y’ = (U, N ‘) which equals {a, b, c, d} is not in Con. If we add the rule y to N to form a new NRS 9” = (U, N”), Con will still be a consistency property over 9” = (U, N”) because the property of being a consistency property depends only on the monotonic part of the rule system. However 9” = (U, N”) is not FC-normal with respect to Con because r = 7 is not FC-normal with respect to Con. That is, for the monotonically closed set {a, b, c, e}, we have prem(r) E {a, b, c, e}, cons(r) n {a, b, c, e} = @, but cI,,,( {c(r)} u {a, b, c, e}) = {a, b, c, d, e} is not in Con.
280
V. W. Marek et al.
Finally if we add to N the rule 7 9”’ = (U, N”‘) is still FC-normal extensions,
M1 = {a, 6, c,e) and M2 = {a, b, c,f).
We have the following Theorem
to get a set of rules N”‘, then the resulting NRS with respect to Con but now there are two
analogue
4.1. Let 9 = (U, N)
of Theorem
0
2.1.
be an FC-normal
nonmonotonic rule system with
respect to consistency property Con. Then there exists an extension of 9. It is easy to adopt
the proof of Theorem
4.1 to get the following
result.
Theorem 4.2. Let Y = (U, N ) be a normal nonmonotonic rule system with respect to consistency property Con and let I be a subset of U such that IECon. Then there exists an extension I’ of I in 9. In fact, we will show that there is a uniform construction of extensions of FCnormal NRS Y = (U, N > which depends on a well-ordering < of the nonmonotonic rules of 9, i.e. the rules in nmon(Y). We shall call this construction the forward chaining construction with respect to < (and this is the reason why we call our systems, for which this construction always succeeds in producing an extension, forward chaining-normal). To this end, fix some well-ordering < of nmon(Y). That is, the well-ordering < determines some listing of the rules of nmon(Y), {rn : LXE~}where y is some ordinal. Let 0, be the least cardinal such that y < 0,. In what follows, we shall assume that the ordering among ordinals is given by E. Our forward chaining construction will define an increasing sequence of sets {Eg<}aE~,. We will then define E< = uorso. Ed and show that ES is always an extension of Y. Moreover we shall show that all extensions
of Y arise from this construction.
The forward chaining construction of E < Case 0: CI= 0. Let E$ = cl,,,(O). Case 1: ~11 = y + 1 is a successor ordinal. Given that r1 = where E&I
a,,
.'., a,:B1,
E,<, let {(cc) be the least %EY such
.'.> Pk
*
CI~, ., CY,EE,~ and pl, . . . . Pk, Ic/$ E,+. = Ed = E,,,. Otherwise, let E6 1 = Eh = &,,(E~
If there
u (c(rc(,))>).
Case 2: CYis a limit ordinal. Then let Ed = U pEaED<. This given, we have the following.
is no
such
C(a), then
let
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Theorem 4.3. If Y = (U, N ) is an FC-normal well-ordering of nmon(Y), then 1. E X is an extension of Y. 2. (Completeness of the construction) a suitably
chosen ordering
It is quite straightforward with respect to consistency Thus the following
nonmonotonic
Every extension
rule system and < is any
of 9
is of the form E X for
< of nmon(9’). to prove by induction property
is an immediate
that if 9’ = (U, N ) is FC-normal
Con, then E,< icon consequence
for all CIand hence E -Xicon.
of Theorem
4.3(2).
Corollary 4.4. Let Y = (U, N) be an FC-normal nonmonotonic rule system respect to consistency property Con, then every extension of Y is in Con.
with
Example 4.2. If we consider the final extended program of Example 4.1, it is easy to check that any ordering <1 in which the rule r1 = $ precedes the rule r2 = “4 will have E ),,a
in stages. This given, we will then define E-X = u,,,
E,<.
The countable forward chaining construction of E < Stage 0. Let E$ = cl,,,(@). Stage n + 1. Let L(n + 1) be the least SEO such that
where c(i) . ., a&E,< and pi, . . . . Pk, $4 E,+. If there is no such /(n + l), then let E,? 1 = E,<. Otherwise, let
This given, we then have the following. Theorem 4.5. If 9 = (U, N ) is a countable FC-normal nonmonotonic rule system, then 1. Es is an extension of Y if E-X is constructed via the countable forward chaining algorithm with respect to < where < is any well-ordering of nmon(Y) of order type w.
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2. (Completeness
Every extension of 9’ is of the form Es
of the construction)
for
a suitably chosen well-ordering < of nmon(Y) of order type o where E + is constructed via the countable forward chaining algorithm. FC-normal
NRS’s also possess the “semi-monotonicity”
Theorem 4.6. Let Y,
= (U, N,)
that N1 E Nz but mon(Y1)
and Y,
property.
= (U, N2) be two FC-normal NRS such
= mon(YP2). Assume, in addition, that both are FC-normal
with respect to the same consistency property. Then for every extension El of Y1, there is an extension E, of Yz such that 1. El z E2 and 2. NG(E,,
YI)
z NG(E,,
Here given and extension rules, i.e., r=
Yz). E, we let NG(E, 9’) denote
the set of all E-applicable
El, ‘.., q.l: PI, .‘.> Bk *
is in NG(E, 9) if and only if a,, . . . , LX~EEand fll, . . . , flk are not in E. FC-normal NRS’s also satisfy the orthogonality of extensions property with respect to their consistency property. That is, we have the following analogue of Theorem 2.3. Theorem 4.7. Let Y = (U, N ) be an FC-normal NRS with respect to a consistency property Con. Then if E, and E2 are two distinct extensions of 9, El v E, 6 Con. Similarly
we also have the following
analogue
Theorem 4.8. Let Y = (U, N) be an FC-normal property Con. Suppose that cl,,,(c(r):rEnmon(Y)} extension.
of Theorem
2.4.
NRS with respect to a consistency is in Con. Then Y has a unique
We now have two more results which are also analogues of the results of Reiter’s [26]. We say that cpeU has a consistent proof scheme with respect to a consistency property Con over Y = (U, N) if and only if there is a proof scheme P = (
ro,
GO),
. . . . (cp,,
r,,,,
such that (P,,, = cp and (cpo, . . . . cp,)ECon.
G,))
(7)
We then have the following.
Theorem 4.9. Let Y = (U, N } be an FC-normal NRS with respect to a consistency property Con. Then cp~U is an element of some extension of Y if and only if cp has a consistent proof scheme with respect to Con.
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283
Theorem 4.10. Suppose Y = (U, N ) is an FC-normal NRS and that D G nmon(Y). Supposefurther that E; and E; are distinct extensions of (U, D u man(Y)). Then Y has distinct extensions El and E, such that E; c E, and E; G EZ.
5. FC-normal nonmonotonic
rule systems and other nonmonotonic
reasoning
formalisms In this section, we shall explicitly translate the definitions and theorems of the previous section into the language of logic programming, default logic, and truth maintenance
systems.
5.1. Logic programming, general case Now because general logic programs are probably the most widely studied type of nonmonotonic reasoning we shall take some time to give the translation of FCnormal nonmonotonic rule systems and the results above into the language of logic programming. A similar translation can be done for all other nonmonotonic formalisms to follow but we shall not carry out the translation in detail in the other cases. A general program clause is an expression of the form C=ptq, where p, ql,
,...,
qn,irl
,..,,
ir,
(8)
, qn, rl, . , r, are atomic formulas
possibly
with variables
in some first-
order language 2. A program is a set of clauses of the form (8). A clause C is called a Horn clause if m = 0. We let H(P) denote the set of all Horn clauses of P. XP is the Herbrand base of P, that is, the set of all ground atomic formulas of the language of P. ground(P) is the set of ground Herbrand substitutions of clauses in P. Given a set MGX’~, the Gelfond-Lifschitz [9] reduct of P, PM is the set of ground Horn clauses p +--ql, . . . . qn such that for some rl, . . . . r,$ M, the clause pegI,..., qn,~r~,...,~rm~ground(P).Misca11edastab1emode1ofPifMcoincides with the least model of PM. Assign to a ground clause p c ql, . . . , q,,, lrl, . . . . lr,Eground(P) the rule r(C)
=
41, . . . . qn:rl,
....
rrn.
(9)
P Let r(P) = (yi”p, {r(C):CEground(P)}). model of P if and only if M is an extension
Then, as shown of r(P).
in [13], M is a stable
This given, it is easy to see that sP plays the role of the universe U and ground(P) plays the role of the set of rules N in Section 4. Moreover, the Horn part of ground(P), i.e. ground(H(P)), plays the role of the monotonic part of the rules N. Then of course the immediate provability operator associated with H(P), THcpj (cf. Apt [3]) is monotonic.
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In this setting
the notion
of a consistency
a family of subsets of yi”p, Con, a consistency conditions:
property
becomes
the following.
Call
property over P if it satisfies the following
1. (bECon. 2. If A c B and BECon, then AECon. 3. Con is closed under directed unions. 4. If AECon Conditions
then A u T,(,,(A)ECon. l-3
nects “consistent” adding prove:
atoms provable
Proposition TWP)
are Scott’s conditions
5.1.
for information
systems.
sets of atoms to the Horn part of the program; from A preserves
If Con is a consistency
“consistency”.
property
Condition
if A is consistent
The following
4 conthen
fact is easy to
with respect to P and A~con,
then
II 4AWon.
Here, for a program Q, T, fi o(A) is the cumulative fixpoint of T, over A. Proposition 5.1 says that our condition 4 in the definition of consistency property implies that the cumulative closure of a “consistent” set of atoms under THcpj is still “consistent”. Here T, fi w(A) is the analogue of the monotonic closure of the set A. Given a consistency property, we define the concept of an FC-normal program with respect to that property in analogy with our definition of FC-normal NRS. Definition 5.2. (a) Let P be a general program, let Con be a consistency property with respect to P. Call P FC-normal with respect to Con if for every clause such that CEground(P) - ground(H(P)), and for C = p + ql, . . . . qn, lrl, . . . . lr, every consistent
fixpoint
A of THcpl, if ql,. .., q,,EA, p, rlr . . . . r,$ A we have:
(1) Au (p)ECon, (2) Au(p,ri}$Con for all 1
Example
4.1 into the language
property
of general
Con such that P is
logic programs.
Example 5.1. Let the Herbrand base consist of atoms a, b, c, d, e,f: Let the consistency property be defined by the following condition: A # Con Now consider (I) a +,
if and only if
either
{c, d) 5 A or {e, f} s A.
this program: t-4 b + c,
(4) c t a, 7 d,
(3) c + b,
(5) e +- c, if:
This program is FC-normal with respect to the consistency property described above and one can easily check that there is a unique stable model M = {a, b, c, e}.
A context for belief revision
If we add to this program the clause f t c, FC-normal but now there are two stable
285
the resulting models,
program
is still
M1 = {a, b, c, e}
and
Mz = {a, b, c,f}. We now have the following
analogues
of Theorem
4.1 and Theorem
4.2.
Theorem 5.3. Zf P is an FC-normal program, the P possesses a stable model. Theorem 5.4. If P is an FC-normal program with respect to the consistency property Con and IECon, then P v {i +- : ie I} possesses a stable model. The analogue of our forward chaining construction of extensions of an FC-normal NRS becomes the following. Let < be a well-ordering of ground(P) - ground@(P)). That is, the well-ordering < determines some listing of the clauses of c,: GIE~} where y is some ordinal. Let 0, be the least ground(P) - ground(H(P)), { cardinal such that y < 0,. Our forward chaining construction will define an increasing sequence of subsets of Ye,, {T>jaeo,. This given, we will then define T+ = UasO,Tb and show that T < is always an stable model of P. Moreover, we shall show that all stable models
of P arise from this construction.
The forward chaining construction of T< Case 0: a = 0. T,jj = Tn(r) fi o(0). Case 1: CI= ye+ 1 is a successor ordinal. Given T,jj, let /(cr) be the least il~y such thatc,=cpca,,..., a,,, 1 pl, . , lb,,, where CI~,. . . , CI,ET,’ and fil, . . . , fl,,, , q 4 T,‘. If there is no such /(cc), let Tb = T,“. Otherwise let
where pfcnJ is the head of c/(,). Case 2: a is a limit ordinal, Then let Td = UBEa T> We then get: Theorem 5.5. If P is an FC-normal program and < is any well-ordering of ground(P) - ground(H(P)), then: (1) T< is a stable model of P. (2) (Completeness of the construction) Every stable model model of P is of the form T< for a suitably chosen ordering i of ground(P) - ground(H(P)). Example 5.2. If we consider the final extended program of Example 5.1, it is easy to check that any ordering <1 in which the clause C1 = e + c, if precedes the clause C2 = f c c, 1 e will have T
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Once again, we note that if we restrict ourselves to countable programs P, then we can restrict ourselves to orderings of order type o. That is, suppose we fix some well-ordering
i
of ground(P)
- ground(H(P))
of order
type o. Thus,
the well-
ordering < determines some listing of the clauses of ground(P) - ground(H(P)), can be presented in {c n : neo}. Again, in this case, our forward chaining construction a more straightforward
manner.
Our construction
will define an increasing
of sets (T,‘i jllEW in stages. This given, we will then define T< = u,,,, The countable forward chaining construction
sequence
T,,, .
of T<
Stage 0. Let T$ = THcpIfl o(8). Stage
n + 1. Let /(n + 1) be the least SEW such that c, = cp c c~i, . . . . ak, where CQ, . . . . QE Tnx and /II, . . . . Pm, cp$ T$. If there is no such /(n + l), let Tn3 1 = T,,, . Otherwise let
1
PI, . . . . ifim
where pLcn+1j is the head of ctcn+ r). Theorem 5.6. Zf P is a countable
FC-normal
program,
and < is any well-ordering
ground(P) - ground(H(P)) of order type w, then: (1) T + is a stable model of P where T< is constructed
of
via the countable forward
chaining algorithm. (2) (Completeness of the construction) Every stable model of P is of the form T< for a suitably chosen ordering < of ground(P) - ground(H(P)) of order type o where T< is constructed via the countable forward chaining algorithm. Theorem 5.7. If P is an FC-normal model M of P is in Con. FC-normal
programs
logic program with respect to Con, then every stable
possess the “semi-monotonicity”
property.
such that P, G P2 but Theorem 5.8. Let PI, Pz be two general programs H(P,) = H(P,). Assume, in addition, that both are FC-normal with respect to the same consistency property. Then for every stable model MI of PI, there is a stable model M2 of P2 such that 1. M1sM2and 2. NG(M1, P,) E NG(M2, Pz). Here given a logic program P and a stable model M, we let NG (M, P) equal the set of all clauses c = cp c txl, . . . . &, i pl, . . . . ip,,, in ground(P) such that ~1~). . . . cxkEM and PI, . . . . P,,,$M. This is a very useful result. That is, if we consider our paradigm of the robot who moves are determined by a hardwired logic program as described in the introduction, this results tells us that if the robot is operating with respect to certain belief set or
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point of view and subsequently new clauses, FC-normal with respect to the consistency property ruling the behavior of the robot, are added, then the beliefs at that point do not need to be recomputed, just new point of view extending the current point of view can be formed. Adding new Horn rules, however, may require backtracking. The reason is that we may have chosen a belief explicitly contradicting these new facts. The analogues of the remaining theorems of Section 4, also hold for FC-normal logic programs. Theorem
5.9. Let P be an FC-normal
property Con. Then ifE,
logic program
with respect
to a consistency
and E2 are two distinct stable models of P, then El v E2 $ Con.
Theorem 5.10. Let P be an FC-normal program with respect to a consistency property Con. Suppose that To o( {h ea d( c ): CEg round(P) - ground(H(P)))) is in Con wherefor any clause, c, head(c) denotes the head of the clause. Then P has a unique stable model. Theorem 5.11. Suppose P is an FC-normal logic program and that D c ground(P) - ground(H(P)). Suppose further that E; and E; are distinct stable models of the program of ground(P) u D. Then P has distinct stable models El and E2 such that E; G El and E; G EZ. We need to define the notion of proof scheme for p with respect to P is a sequence of triples number, such that the following conditions all 1. Each pt is in ZP. Each Cl is in ground(P). 2. pn is p. 3. The S[, C1 satisfy the following conditions. below holds. (a) Crisp,c,andS,isS,_,, (b) Cl is pt+-lsl,..., is, and SI is S1_r (c) CL is pl + pm,, . . . . pmr, SL-1 u (s1, . . . . s,}. (We put SO = 8.)
is1,
for a logic program P. A proof scheme (( pI, CI,SI))l $tGn, with n a natural hold. Each Sr is a finite subset of Ye,. For all 1 d 1 d n, one of (a), (b), (c)
u {sl, . . . . s,}, or m,
and
SI
is
is a proof scheme. Then cln(cp) denotes the Suppose that cp = (
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5.2. Default logic In this subsection, we shall translate our results of Section 4 back into the language of default logic. We shall start with the translation between default logic and nonmonotonic rule systems as described in [13] and [16]. Let U be the collection of all formulas of propositional logic 9. Recall a default theory (D, W) is a pair where D is a collection of default rules, that is, rules of form c(:MP,, . . . . Mp,,, *
(10)
’
(where ~1,PI, . . . . Pm, and II/ are formulas) and W a collection of formulas language 9. Represent such a default theory as a rule system consisting of three lists: (i) Elements YEW are represented as rules:
(ii) Rules of form (10) are represented
of the
as
Y
(That is, the restraints of the rule representing a default rule r have an additional negation in front.) (iii) Processing rules of logic. That is, all the monotonic rules of the system of classical logic. We then have the following proposition from [13]:
Proposition 5.13. A collection S c U is an extension of a system consisting of rules of types (i), (ii), and (iii) if and only if S is a default extension of (D, W). Given a default rule r as in (lo), we let prem(r) = {a}, cons(r) = {l/II, . . . . l/I,,,}, and cln(r) = t,b.If m = 0, then we say that r is a monotonic rule and otherwise we will say that r is a nonmonotonic rule. We let mon( (D, W)) denote the set of monotonic rules of (D, W) and nmon((D, W)) d enote the set of monotonic rules of (D, W). We say that a subset S c 9’ is monotonically closed relative to (D, W), if WCS, Cn(S) = S and for any monotonic rule $ in (D, W), it is the case that $ES if ads. Thus a set S containing W is monotonically closed relative to (D, W), if S is closed under the application of all monotonic rules of (D, W) as well as being closed under logical consequence. It is easy to see that the intersection of any two monotonically closed sets relative to (D, W) is also monotonically closed relative to (D, W) so that for any set T G Y”, there is a smallest set S which contains T and is monotonically closed. We let cl,,,(T) denote the smallest monotonically closed set relative to (D, W) which contains T.
A context
for belief revision
Call a family of subsets of 9, Con a consistency following conditions: 1. 2. 3. 4.
289
property for (D, W) if it satisfies the
Qkcon. If A E B and BECon then AECon. Con is closed under directed unions. If AECO~ then cl,,,(A)~Con.
Given a consistency property Con, we say that a default rule as in (10) is FC-normal with respect to Con if for any theory AE Con such that CX~Aand $, 1 pi, . . . ,l p,,,, are not in A, then Cn(A u {$))~Con
but Cn(A u {tj, 1/3i))$
Con for any i. Then we say
that a default theory (D, W) is FC-normal with respect to Con if each rule rED is FC-normal with respect to Con. Our next result will show that our definition of FC-normal default theories actually extends Reiter’s original definition of normal default theories. Theorem 5.14. Every normal default theory (D, W> is an FC-normal
default theory.
Proof. First observe that since every rule in a normal default theory is of the form 7, there are no monotonic rules in (D, W). Thus a set S is monotonically closed relative to (D, W) if and only if Cn(S u W) = S (that is W c S and Cn(S) = S). There are two cases. First, suppose that W is a logically consistent set of formulas. In this case, it is easy to see that the set of logically consistent subsets S of the language 97 such that WV S is logically consistent is a consistency property for (D, W). We note that if Con is just the set of logically consistent sets S in ~3’ such that S u W is logically consistent, then every rule of the form y is certainly FC-normal. That is, no consistent set can contain both p and l/? and if A is a consistent theory which contains Wsuch that neither /I, 1,!3 are in A, then Th(A u {a}) is also consistent. Thus if W is logically consistent set, then (D, W) is FC-normal relative to the consistency property consisting of all logically consistent sets S such that S u W is logically consistent. Second, if W is not logically consistent, then it is easy to see that for any set SC 9, &,,(S) = Cn(S u W) = 2. Thus in this case, the only possible consistency property with respect to (D, W) is the set of all subsets of 9. Moreover, the only monotonically closed set is 9. But then every rule of the form y is FC-normal since there is no monotonically closed set T such that both /I and l/3 are not in T. Thus if W is logically inconsistent, then (D, W) is FC-normal with respect to the consistency property consisting of all subsets of 9. q Of course, it is easy to see that there are many FC-normal default theories which are not normal default theories since in FC-normal default theories we allow monotonic rules and we do not restrict nonmonotonic rules to be of the form y. We then have the following analogues of the results of Section 4. Theorem 5.15. Let (D, W) be an FC-normal default theory with respect to consistency property Con, then there exists an extension of (D, W).
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Theorem 5.16. Let (D, W) be an FC-normal default theory with respect to consistency property Con and let IECon. Then there exists an extension 1’ of (D, W u I >. The analogue is the following. <
of the forward chaining construction Given an FC-normal default theory
of D. That
is, the well-ordering
<
determines
for FC-normal default theories (D, W), fix some well-ordering some listing
of the rules of D,
{r,: cq} where y is some ordinal. Let 0, be the least cardinal such that y d 0,. In what follows, we shall assume that the ordering among ordinals is given by E. Our forward
chaining
construction
will define an increasing
The forward chaining construction of E < Case 0: M= 0. Let E$ = c&,,~,,(W). Case 1: c( = n + 1 is a successor ordinal. that rA =
at,...,
E,,,
of sets {E>}aoB,.
let e(a) be the least il~y such
up: Ml%, . . . . MA IL
where c1r, . . . , c+EV+ and 1 bl, . . . , lPk, E&i = E,,, = E,<. Otherwise, let E&l
Given
sequence
= Ed = &,,(EV~
$ $ E,,< If there is no such /(cc), then let
u {cWre(,,)>).
Case 2: o! is a limit ordinal. Then let Ed = u psa.EB<. This given, we have the following. Theorem 5.17. If (D, W> is an FC-normal default theory and < is any well-ordering of D, then 1. E < is an extension of (D, W). 2. (Completeness of the construction) Every extension of (D, W) is of the form E X for a suitably chosen well-ordering i of D. Corollary 5.18. Let (D, W) be an FC-normal default theory with respect to consistency property Con. Then every extension of (D, W> is in Con. If we restrict ourselves to countable default theories,,, in stages.
291
A context for belief rtwision
The countable forward chaining construction of E X Stage 0. Let E$ = cl,,,(W). Stage n + 1. Let !(n + 1) be the least SEO such that rs =
czl, . . . . c(~:Mbk, . . . . M/3,‘ *
where c(~,. . . . QEE,+ andip,, E,5 1 = Ens. Otherwise, let
. . . . 1 bk, $4 E,< If there is no such [(PI + l), then let
En3 I = &on(En~ u {cW/c,+ We then define E+ = u,,,
d>).
E2.
This given, we then have the following. Theorem 5.19. If (D, W) is a countable FC-normal default theory then 1. E < is an extension of (D, W) where E + is constructed via the countable forward chaining algorithm with respect to <, where < is any well-ordering of D of order type w. 2. (Completeness of the construction) Every extension of (D, W> is of the form E < for a suitably chosen well-ordering < of D of order type o, where E < is constructed via the countable forward chaining algorithm.
The “Semi-monotonicity”
property holds for FC-normal default theories.
Theorem 5.20. Let AI = (DI, W) and A2 = (D2, W) be two FC-normal default theories with respect to a consistency property Con such that mon(A,) = mon(Az) and nmon(A,) s nmon(Az). Then for every extension E, of (DI, W>, there is an extension E2 of (D2, W) such that 1. E, c Ez and 2. GD(E,, A,) c GD(E2, AZ).
FC-normal default theories also satisfy the orthogonality with respect to their consistency property.
of extension property
Theorem 5.21. Let (D, W) be an FC-normal default theory with respect to the consistency property Con. Then, ifEl and E, are two distinct extensions of (D, W>, then El u E, 4 Con.
Similarly we have the following analogues of Theorems 2.4 and 2.5. Theorem 5.22. Let (D, W) be an FC-normal default theory with respect to a consistency property Con. Suppose that Wu {cln(r) : rED) is in Con. Then (D, W) has a unique extension.
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Theorem 5.23. Let
Then (D, W) has distinct extensions El and E, such that
E; c El and E; E Ez. of Theorem
4.9. First
we assume
underlying logic has some proof systems consisting and finitely many rules ol, . . . . ok. An (annotated) sequence
Finally
there
is also
an analogue
of finitely
many
proof
scheme
axiom
that
the
schema
for q is a finite
P = ((cpO, ro, Go>, . . . . (cp,, r,, G,))
(11)
such (1) (a) (b) (2)
that (P,,, = cp and If m = 0 then: cpo is an instance of an axiom schema for 9 or (POEW, r. = qo, and Go = 8, or cp is a conclusion of a rule r = ‘MP’*;.*MBr, r. = r, and Go = cons(r). If m > 0, ((Cpi, ri, Gi))~Yol is a proof scheme of length m and either of r = ai’MBz;.‘*MBr where i < m, rm = r, and (a) qrn is a conclusion G, = G,_ 1 u cons(r), or (b) there is some i. < ... < i, < m such that (P,,,is the result of applying one of the rules of proof Bj to 4i0, . . . . $iS, rm = Bj, and G, = G,_ 1. The formula (P,,,is called the conclusion of p and is written cln( p). The set G, is called the support of p and is written supp(p). We say that cp~_Y’ has a consistent proof scheme with respect to a consistency property Con over (D, W) if and only if there is a proof scheme P=
<
ro,
GO),
. . . . (cp,,
rm,
such that (P,,, = cp and {cpo, . . . , cp,}ECon.
G,))
We then have the following.
Theorem 5.24. Let (D, W> be an FC-normal default theory with respect to a consistency property Con. Then cpis an element of some extension of (0, W) tfand only ifq has a consistent proof scheme with respect to Con,
5.3. Logic programming with classical negation We now discuss the so-called “logic programming with classical negation” of [lo] as a chapter in the theory of nonmonotonic rule systems. Recall the basic notions introduced in [lo]. The collection of objects appearing in heads or bodies of clauses is the set of all literals, that is, atoms or negated atoms. In particular, a negated atom may appear in the head of a clause. Consider first “general Horn” clauses in which literals may appear in arbitrary places. To each set P of such clauses assign its answer set, the least collection A of literals satisfying the following two conditions: (1) If a+ bI, . . . . b, is in P and bI, . . . . b&A then aEA.
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293
(2) If for some atom p, p and 1 p are both in A, then A is the whole collection Lit of all literals. Introduce a collection Str of structural processing rules over the set U = Lit. These are all monotone
rules of the form:
for all atoms p and literals Translate
a.
the clause: a c b, , . . . , b, as rule:
b 1, . . . . b,: a and let tr(P) be the collection Str. Then we have
of translations
of clauses in P plus the structural
rules
Proposition 5.25. A subset A c Lit is an answer set for P if and only if A is an extension of tr(P). Since tr(P) is a set of monotonic rules, such an answer set is the leastjxpoint of the (monotonic) operator associated with the translation. Gelfond and Lifschitz then introduce general rules. Since the negation used in literals is not the “negation-as-failure” of general logic programming, Gelfond and Lifschitz introduce another negation symbol “not” and a general logic clause with classical negation in the form: a c b,, . . . . b,, not(c,),
. . . . not(c,).
Then the answer set for a set P of clauses of this form is introduced by merging the operational procedure for the construction of stable models for a program (as introduced in [9]) with the procedure above. They define the answer set for a program with classical negation as follows: Let A4 c Lit and P be a general program. Define P/M as a collection of clauses lacking not obtained as follows: (1) If a clause C contains a substring not(a) where aeM, then eliminate C altogether. (2) In remaining clauses eliminate all substrings of the form not(a). The resulting program P/M lacks the symbol not, so the answer set is well defined. Let M’ be the answer set for P/M. We call M an answer set for P precisely when M’ = M. Gelfond and Lifschitz give a computational procedure for finding such answer sets, and subsequently reduce computing them to computing default logic extensions. Here we give a general result showing that the construction of Gelfond and Lifschitz is faithfully represented within nonmonotonic rule systems; here is how. Define U to be
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Lit, and translate
the clause:
a c bl, . . . . b,, not(q), . . . . not(c,) as the rule: b 1,..., b,:cI,
. . . . c,
a The translation of the program P then consists of the translations of individual clauses C of P, incremented by the structural rules Str. We get the following result: Proposition 5.26. Let P be a general logic program with classical negation and Np be the translation described above. Then a collection M is an answer set for P if and only ifM is an extension for the rule system (U, Np). Let Lit, = yi”pu {lp: p~yi”p>. It is easy to see that Lit, plays the role of the universe U and ground(P) u Str plays the role of the set of rules N in Section 4. Moreover, the not-free part of ground(P) together with the structural rules Str, plays the role of the monotonic part of the rules N. Define man(P) to be the the not-free part of P incremented by Str. The immediate provability operator, T,(M), associated with man(P) is monotonic. Moreover, if the input M contains a pair of contradictory literals then T,(M) is the whole set Lit,. In this setting the notion of a consistency property becomes the following. Call a family of subsets of Lit,, Con a consistency property over P if it satisfies the following conditions: 1. @ECon. 2. If A E B and BECon then AECon. 3. Con is closed under directed unions. 4. If AECon then A u T,,,&A)ECon. 5. Lit, # Con. We have, as before, Proposition 5.27. If Con is a consistency property with respect to P and AECon, then Tman(P) II4A k Con. fi o(A) is the cumulative fixed point of Here for a general logic program Q, TmoncpJ Tp over A. Given a consistency property, we define the concept of an FC-normal CN logic program with respect to that property in analogy with our definition of FC-normal NRS. Definition 5.28. (a) Let P be a CN logic program, let Con be a consistency property with respect to P. Call P FC-normal with respect to Con if for every clause C = p+ 41, . . . . qn, not(rl), . . . . not@,) such that CEground(P) - ground(mon(P)),
A context for belief revision
and for every consistent
A of TmoncP,,if ql, . . . . q,,gA, p, rl, . . . . r,$ A we have:
fixpoint
(1) Au {p}~Con, (2) AU{p,ri)~Conforall
1
(b) P is called FC-normal if there exists a consistency FC-normal with respect to Con. Our condition consistency
(5) implies
property
that there is a weakest
based on the the absence
We now have the following
analogues
Theorem 5.29. If P is an FC-normal
The
Con and IECon,
analogue
normal
NRS
of our becomes
property
consistency
Con such that P is
property.
of pair of complementary
of Theorems
This is the literals.
4.1 and 4.2.
CN logic program, then P possesses an answer set.
Theorem 5.30. Zf P is an FC-normal property
295
CN logic program
with respect to the consistency
then P u {i +- : iE I} possesses an answer set I ‘.
forward the
chaining following.
construction Let <
of extensions of an FCbe a well-ordering of
ground(P)
- ground(mon(P)). That is, the well-ordering < determines some listing of the clauses of ground(P) - ground(mon(P)), {c,: ME?} where y is some ordinal. Let 0, be the least cardinal such that y < 0,. Our forward chaining construction will define an increasing sequence of subsets of Lit,, {T,< },, @,. This given, we will then define
T< = U.0. , T;< and show that TS is always an answer set for P. Moreover, show that all answer sets for P arise from this construction. of TX
The forward chaining construction Case 0: IX= 0. T,jj = Tman(P)0
we shall
40).
Case 1: cx= ye+ 1 is a successor
ordinal. Given
T;‘,
let /(a) be the least AEY such that c1 = cp c gl, . . . . tl,, not(fil), . . . . not(b,) where CY~,. . . . CI,ET~< and fil ,..., pm, cp$ T,“. If there is no such /(a), let T,’ = T,‘. Otherwise let
where prcaj is the head of c/(,). Case 2: u is a limit ordinal. Then let Tb = u BEnTz. We then get: Theorem 5.31. If P is an FC-normal ground(P)
- ground(mon(P)),
CN logic program
1. T< is an answer sef for P. of the construction)
2. (Completeness
a suitably chosen ordering
and < is any well-ordering
of
then:
<
of ground(P)
Every answer set for P is of the form - ground(mon(P)).
T< for
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V. W. Marek et nl,
Once again, we note that if we restrict ourselves to countable programs P, then we can restrict to orderings of order type o. That is, suppose we fix some well-ordering < of ground(P) - ground(mon(P)) of order type o. Thus, the well-ordering 4 determines some listing of the clauses of ground(P) - ground(mon(P)), {c,,: n~o}. Again in this case, our forward ward manner.
chaining
Our construction
construction
can be presented
will define an increasing
stages. This given, we will then define TX = U,,, The countable forward chaining construction
in a more straightfor-
sequence
of sets (7’n<}.Bo in
T”<.
of T+
Stage 0. Let T,’ = Tmon(rjfi o(8). Stage n + 1. Let a(n + 1) be the least natural number s such that c, = cp +- al, . . . . uk, not, . . . . not(/?,) where c1r, . . . . akETn+ and /?r, . . . . fi,,,, cp$! T”<. If there is no such /(n + l), let T> 1 = T:. Otherwise let K< I = THVJ 0 dT,z~ u {P/W+ I,>, where P~(,,+~) is the head of c~(~+i). Theorem 5.32. If P is a countable FC-normal CN logic program, and < is any well-ordering of ground(P) - ground(mon(P)) of order type w, then: (1) T+ is a answer set for P where T < is constructed via the countable forward chaining algorithm. (2) (Completeness of the construction). Every answer set for P is of the form T< for a suitably chosen ordering < of ground(P) - ground(mon(P)) of order type o where T< is constructed via the countable forward chaining algorithm. Theorem 5.33. If P is an FC-normal CN logic program with respect to Con, then every answer set model M for P is in Con. FC-normal
programs
possess “semi-monotonicity”
property.
Theorem 5.34. Let P,, P2 be two CN logic programs such that PI E P2 but mon(P,) = mon(P,). Assume, in addition, that both are FC-normal with respect to the same consistency
property.
Then for every answer set MI for PI, there is an answer set
M2 for Pz such that
1. MI c M2 and 2. NG(M1, P,) z NG(Mp, P2). Here given a CN logic program P and an answer set M, we let NG(M, P) equal the set of all clauses c = (PC ml, . . . . &, not(P1),..., not@,) in ground(P) such that CY~ ,..., C(,&M and fll ,..., &$M. The analogues of the remaining theorems of Section 4, also hold for FC-normal logic programs with classical negation.
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Theorem 5.35. Let P be an FC-normal CN logic program with respect to a consistency property Con. Then, if El and E2 are two distinct answer sets for P, then El u E, $ Con. Theorem 5.36. Let P be an FC-normal property
CN logic program
Con. Suppose that Tfi w( (head(c): cEground(P)
where for any clause c, head(c)
denotes
with respect to a consistency
- ground(mon(P))})
the head of the clause.
is in Con
Then P has a unique
answer set.
5.37. Suppose
Theorem
D G ground(P)
P
is
an
- ground(mon(P)).
sets for the program
of ground(P)
FC-normal
CN
Suppose further
logic
program
and
that
that E; and E; are distinct answer
u D. Then P has distinct answer sets E, and E, such
that E; 5 El and E; E E,.
To state the analogue of Theorem 4.9, we must define the notion of proof scheme for a logic program P. A proof scheme for p with respect to P is a sequence of triples <( pt, Cl, SI))16LSnr with n a natural number, such that the following conditions all hold. 1. Each pt is in Lit,.
Each C1 is in ground(P).
Each SL is a finite subset of Litr.
2. pn is p.
3. The SI, C1 satisfy the following below holds.
conditions.
For all 1 d 1 < n, one of (a), (b), (c)
(a) CL is pl +, and SI is Sl_,, (b) C1 is pt c not(sI), . . . . not(s,) and SI is S,- 1 u (sl, . . . . s,}, or (c) CI is pr+ pm,, . . . . pm*, not(s,), . . . . not(&), m, < 1, . . . . mk < 1, and SI is si_r u {Si, . ..) &>. (We put SO = 0.) is a proof scheme. Then cln(cp) denotes atom Swposethatcp = <
a consistent
5.4.
CN logic program with respect to a consistency
Con. Then (PE Lit, is an element of some answer set for P if and only if9
has
proof scheme with respect to Con.
Truth maintenance
systems
In this and the next subsection, we shall discuss two other nonmonotonic formalism which have appeared in the literature. In each of these cases, we shall be content to simply translate these formalisms into nonmonotonic rules systems and leave it to the reader to translate the definition of FC-normal and the statements of its various properties.
V. W. Marek et al.
298
Our description
takes care of both truth maintenance
[7] and De Kleer
[6], with subsequent
Brewka [25]. Let At be a collection
of atoms.
contributions
systems as defined by Doyle of Reinfrank,
sequence (al, . . . . a,) (1) a, = a. (2) For
M G At, an M-derivation
satisfying
every j < n, either
and
By a rule over At we mean a object of the form
Y = (A 1B) -+ c where A, B s At, ceAt. A truth maintenance is a collection of rules. Let S be a TMS. Given
Dressler,
system (TMS for short)
of an atom
aeAt
is a finite
the conditions: a rule
(8 10) + aj belongs
to S or there
is a rule
(A~B)+ajinSsuchthatA~{a,,...,aj-l},BnM=@ We call M a TMS-extension of S if and only if M has the property that M consists of precisely these atoms that possess an M-derivation. It is easy to reconstruct, in this setting, truth maintenance as logic programming. Namely, we can translate a rule (A 1B) + c as a program clause c +- a,, . . . . a,,,, not@,), . ..) not(b,) where A = {ai, . . . . a,} and B = {b,, . . . . b,}. In this fashion, TMS-extensions become stable models of the that one can also construct truth-maintenance To do this properly one considers (as in the negation) rules which involve literals and also
resulting programs. This also implies systems with literals. case of logic programs with classical “built-in” rules ((a, 1 a}, 0} + c. For
more about these, see [21]. The results on FC-normal logic programs as well as on FC-normal logic programs with classical negation allow us to prove analogues results for TMS, and also TMS with literals. We shall not pursue this matter further in this paper. 5.5.
McDermott
and Doyle systems
McDermott and Doyle [23] and McDermott [22] investigated another system of nonmonotonic reasoning. This system is based on modal logic. Here is a short description of that approach and the description how it fits into our framework. Let .4pL be the propositional modal language based on modal operator L (expressing the necessity operator). We consider a strong notion of proof based on the application of the necessitation rule to all formulas, not just all theorems, of the logic under consideration. That is, this notion of proof from a set of formulas I allows us to apply necessitation to all formulas previously proved. Let 9’ be a modal logic. Examples of such a logic include the familiar S4, S5, K or even a logic that does not include the schemes of K. We associate with Y its consequence operation based on the above strong notion of proof. We denote it by Cn& .). We now introduce the notion of Y-expansion. Given a set of formulas I s YL, we say that a theory T E 5?L is a Y-expansion of I if T=
Cny(lu{lLq::cp$T}).
(12)
Notice that the role of the logic 9’ here is slightly different than in the usual applications of modal logic. 9’ serves as means of reconstruction of T from the initial
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for belief revision
assumptions I and the negatioe introspection with respect to T. It should be clear that regardless of what .Y is (it does not even need to be included in S5) that an expansion of any theory is closed under SS-consequence. It is the discipline of reconstruction that makes the difference. Note the weaker the logic, the more difficult it is to reconstruct. We show now how this formalism can be faithfully represented as a nonmonotonic rule system. Let Y be a fixed modal
logic, axiomatized
define a rule system SY as follows. The universe N consists of the following five groups of rules: 1. $,
where cp ranges YL, treating
2. i, cp 3. 5
over all the axioms
every formula
where cp ranges
of propositional
AX. We
is _!YL. The set
logic in the language
of the form LIl/ as an atom.
over all the axioms
for all the formulas
by a set of axioms
U of our system
of the logic 9.
(PE_Y’~.
4 cp,cp~@ . ~ for all the formulas II/ :cp 5. ~ for all 9~9~. 1LV
cp, $EY~.
Notice that the groups 1, 2, 3, and 4 of rules are monotonic, consists of nonmonotonic rules. We have the following result.
only the group
Theorem 5.39. Let 9’ be a modal logic. Let I s $PL. Then T is an Y-expansion and only if T is an extension of I in the nonmonotonic rule system Sy.
5
of I if
This result allows us to apply the theory developed in this paper to Y-expansions. That is, we can introduce the notion of consistency property for a modal logic (notice that we have just one system for any given modal logic). If the system Sy is FC-normal for such property then, in particular, every set of formulas consistent with respect to that property
possesses
6. The complexity
an extension.
of extensions
for a recursive FC-normal NRS
This section will be divided into two subsections. First we shall introduce some preliminaries from recursion theory and various classes of recursive rule systems so that we can make our statements about the complexity of extensions for recursive FC-normal NRS’s precise. We shall also provide a brief review of what is known about the complexity of extensions in recursive nonmonotonic rule systems. In the second subsection, we shall state our new results about the complexity of extensions in recursive FC-normal nonmonotonic rule systems. The proofs of these new results will be deferred until Section 8.
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et al.
6. I. Preliminaries Let o denote the set of natural numbers. The canonical index, can(X), of a finite set x = {x1 < ..’ < x,) E o is defined as 2”’ + ... + 2”n and the canonical index of 0 is defined as 0. Let Dk be the finite set whose canonical index is k, i.e., can(Q) = k. We
shall
identify
a rule
Y with
a triple
(k, 1, cp) where
Dk = prem(r),
D, = cons(r), cp = c(r). In this way, when U E w we can think about
and
N as a subset of
w as well. This given, we then say that a NRS Y = (U, N ) is recursive if U and N are recursive subsets of w. Next ox
we shall define
various
o -+ o be a fixed one-to-one
types
of recursive
and onto
recursive
trees and pairing
IZ(: classes. function
Let [,I:
such that the
projection functions rci and 7r2 defined by rcl([x, y]) = x and rc2([x, y]) = y are also recursive. Extend our pairing function to code n-tuples for n > 2 by the usual inductive definition, that is, let [x1, . . . . x,] = [x1, [x2, . . . . x,]] for n 2 3. Let cecw be the set of all finite sequences from w, let 2’” be the set of all finite sequences of O’s and 1’s. Given CI= (a,, . . . . a,) and p = (fir, . , ljk) in cP”, write c( c b if (x is an initial segment of /3, i.e., if n 6 k and ai = pi for i 6 n. In this paper, we identify each finite sequence a = (a,, . . . . cr,,) with its code c(g) = [n, [a,, . . , a,]] in o. Let 0 be the code of the empty sequence 0. When we say that a set S c o<@ is recursive, recursively enumerable, etc., what we mean is that the set {C(U): UES) is recursive, recursively enumerable, etc. Define a tree T to be a nonempty subset of gco such that T is closed under initial segments. Call a functionf: w + o an infinite path through T provided that for all n, (f(O), . . . , f(n))E T. Let [T] be the set of all infinite paths through T. Call a set A of functions of I7:-class if there exists a recursive predicate R such that A = {f: o + w:Vn(R([f(O), . . ..f(n)])}. Call a Z7,0-class A recursively bounded if there exists a recursive function g: o + o such that V’EA V’,(f(n) 6 g(n)). It is not difficult to see that if A is a UF-class, then A = [T] for some recursive tree T c coCw. Say that a tree T E coCo is highly recursive if T is a recursive, finitely branching tree, and also there is a recursive procedure which, a canonical index of the set of immediate
applied to c( = (c(i) . . . . a,) in T, produces successors of a in T. Then if A is a recursively
nf-class, it is easy to show that A = [T] for some highly recursive tree T G wCw, see [12]. For any set A c o, let A’ = {e: {e}A(e) is defined} be the jump of A, let 0’ denote the jump of the empty set 0. We write A
bounded
to B and A EBB if A dTB and B GOB. Formally, say that there is an effective, one-to-one degree-preserving correspondence between the set of extensions 6(Y) of a recursive nonmonotonic rule system 9 =
= EfEcT(9), (i) VRCTI{eI >g*(J”) (ii) V’EEF(Y){e2}E =_&E[T], and v(f) = E if and only if (ez}E =f) (iii) Vjtz[T]VEd(Y)({e,} where {e}B denotes the function computed by the eth oracle machine Also, write {e}” = A for a set A if {e}B is a characteristic function
with oracle B. of A, and for
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A context for belief revision
a functionf: w -+ 0, gr(f) = { [x, f(x)]: the tree T uniformly produce extensions
XEO}. Condition
(i) says that the branches
via an algorithm
with index e,. Condition
of (ii)
says that extensions of Y uniformly produce branches of the tree T via an algorithm Turing equivalent with index e2. Condition (iii) asserts that if {e,} gr(/) = E,, thenfis to E,. In the sequel we shall not explicitly construct the indices e1 and e2, but it will be clear that such indices
can be constructed
in each case.
There are two important subclasses of recursive NRS’s introduced in [14], namely locally finite and highly recursive nonmonotonic rules systems. Say that the system (U, N ) is locally jinite if for each (PE U, there exist only finitely proof schema
with conclusion
cp. If (U, N ) is locally
many
<-minimal
finite, then for every cp, there
exists a finite set of derivations Dr, such that all the derivations of cp are inessential extensions of derivations in Dr,. That is, if p is a derivation of cp, then there is a derivation p1 EDrq such that pi 4 p. Finally, say that (U, N ) is highly recursive if (U, N) is recursive, locally finite, and the map cp H can(Dr,) is partial recursive. The latter means that there is an effective procedure which, when applied to any VE U, produces a canonical index of the set of all <-minimal proof schema with conclusion cp. This given, we can now state some basic results from [14-161 on the complexity of extensions in recursive nonmonotonic rule systems. Theorem 6.1. For any highly recursive NRS system 9’ = (U, N ), there recursive tree T, such that there is an efective one-to-one degree-preserving ence between [Ty] and k?(9). Vice versa, for any highly recursive tree a highly recursive NRS system Y’r = (U, N ) such that there is an effective degree-preserving correspondence between [T] and b(Y,).
is a highly correspondT, there is one-to-one
Theorem 6.2. For any locallyjnite recursive NRS system 9’ = (U, N ), there is a tree TY which is highly recursive in 0’ such that there is an effective one-to-one degreepreserving correspondence between [ TY] and 8(Y). Vice versa, for any highly recursive tree Tin 0’, there is a locallyjnite recursive NRS system Yr = (U, N > such that there is an efective one-to-one degree-preserving correspondence between [T] and &(Y’r). Theorem 6.3. For any recursive NRS system Y = (U, N ), there is a recursive tree Ty such that there is an eflective one-to-one degree-preserving correspondence between [TY] and &(9). Vice versa, for any recursive tree T, there is a recursive NRS system LfT = (U, N > such that there is an eflective one-to-one degree-preserving correspondence between [T] and &(Yr). Because the set of degrees of paths through trees has been extensively studied in the literature, we immediately can derive a number of corollaries about the degrees of extensions in recursive NRS systems. We shall give a few of these corollaries below. We begin with some consequences of Theorem 6.1. First there are some basic results which guarantee that there are extensions of a highly recursive NRS system which are
V. W. Marek et al
302
not too complex.
Recall 0 denotes
the degree of recursive
sets and 0’ its jump.
Call
A low if A’ =rO’. This means that A is called low provided that the jump of A is as small as possible with respect to Turing degrees. The following corollary is an immediate
consequence
of Theorem
Corollary 6.4. Let Y = (U, N)
4.1 and the work of Jockusch
be a highly recursive nonmonotonic
and Soare [12].
rule system such
that &(Y) # 8. Then (i) There exists an extension (ii) If 9 has onlyfinitely
E of Y such that E is low.
many extensions,
then every extension E of Y is recursive.
In the other directions, there are a number of corollaries of the Theorem 4.1 which allow us to show that there are highly recursive NRS systems 9 such that the set of degrees realized by elements of E(9) are quite complex. Again all these corollaries follow by transferring results of Jockusch and Soare [ll, 121.
Corollary 6.5. 1. There is a highly recursive nonmonotonic rule system (U, N ) such that (U, N) has 2’0 extensions but no recursive extensions. 2. There is a highly recursive nonmonotonic rule system (U, N ) such that (U, N ) has 2”0 extensions and any two extensions El # E2 of (U, N ) are Turing incomparable. 3. Ifa is any Turing degree such that 0
enumerable
degrees b b T a.
Now the situation for locally finite recursive NRS rule systems is very similar to the situation for highly recursive NRS systems except that the degrees of extensions may be more complex. Essentially every result in Corollaries 6.4 and 6.5 holds for locally finite recursive NRS systems where each statement is taken relative to an 0’ oracle. See [16] for further details. However, the situation for general recursive NRS systems is quite different. That is, for general recursive NRS systems the set of elements of g(9) may be extremely complex. For example, all we can say in the positive direction is the following.
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A context for belief reaision
Corollary 6.6. 1. Every recursive NRS system Y = (U, N ) which has an extension has an extension E such that E dT B where B is a complete II: -set. 2. If Y = (U, N)
is a recursive
NRS
system with a unique extension
E, then E is
hyperarithmetic.
In the opposite
direction
we have the following
Corollary
6.7. 1. There
extension
but Y has no extension
2. For each recursive possessing
6.2.
NRS
system Y = (U, N)
such that Y
has an
which is hyperarithmetic.
ordinal u, there exists a recursive
a unique extension
Recursion
In this a recursive will show tonic rule extension
a recursive
results, see [15].
NRS
system 9’ = (U, N >
E such that E = T O@‘.
theory of extensions
of FC-normal
NRS
subsection, we shall state our results on the complexity of extensions in FC-normal nonmonotonic rule systems. Our first result of this subsection that under the assumption of FC-normality that even recursive nonmonosystems are guaranteed to have at least one relatively well behaved which is in great contrast to Corollary 6.7.
Theorem 6.8. Suppose that Y = (U, N)
is a recursive nonmonotonic
Y is FC-normal.
E such that E is r.e. in 0’ and hence E dT 0”.
Then Y has an extension
We note that Theorem 6.8 is in some sense the best possible. [17] show that the following holds. Given sets A, B G o, let
rule system and
That is, results from
A @ B = (2x: XEA} u {2x + 1 : XEB}. Theorem 6.9. Let A be any r.e. set and B any set which is r.e. in A, i.e., B = (x: p:(x) Then there is a recursive extension
FC-normal
NRS
E and E = T A @ B. In particular,
then there is an FC-normal
Y = (U, N > such that Y
1 }.
has a unique
if B is any set which is r.e. in 0’ and B > T 0’,
NRS Y such that 9
has a unique extension
However, we shall show that if either man(Y) or nmon(Y) improve on Theorem 6.8. That is, we have the following.
E and E =T B.
is finite, then we can
Theorem 6.10. Let Y = (U, N) be a recursive rule system such that 9 and nmon(Y) is jnite, then every extension of Y is r.e.
is FC-normal
nonmonotonic rule system 9 = (U, N ) is monotonically closure of any finite set F is recursive and there is a uniform effective procedure to go from a canonical index of a finite set F to a recursive index of the cl,,,(F), i.e., if there is a recursive function f such that for all k, 4f(kJ is the We say that a recursive
decidable if the monotonic
304
V. W. Marek et al.
characteristic function of c1,,,(&). It is easy to see that if man(Y) is finite, then the recursive nonmonotonic rule system Y = (U, N ) is automatically monotonically decidable. Theorem 6.11. Let 9’ = (U, N)
be a recursive nonmonotonic rule system such that
Y is FC-normal and monotonically decidable. Then Y has an extension which is r.e. Next we consider the case where Y = (U, N ) is a finite FC-normal nonmonotonic rule system, i.e. we assume that both U and N are finite. In this case, we shall see that our forward chaining algorithm runs in polynomial time. we shall assume that the elements of U are coded by C. Thus every aE U will have some length which we
For complexity considerations, strings over some finite alphabet denote by 11 a 11.Next, for a rule r=
aI, . . . . a,:bl,
. . . . b, 3
we define lIrII = (ir, Finally,
1H:JJ + (zm
1ibj1i) + 114.
for a set Q of rules, we define
IIQII = c IId. reQ
Theorem 6.12. Suppose Y = (U, N) is a finite FC-normal nonmonotonic rule system and < is some well-ordering of nmon(S). Then E< as constructed via our forward man(Y) (I . IInmon(9) 11+ chaining algorithm can be computed in time 0( 11 IInmoWP)l12). We note that we did not make any explicit assumptions that the underlying consistency property of a recursive FC-normal NRS 9’ = (U, N ) is any way effective. Indeed none of the above results require that the underlying consistency property has any effective properties. We define a consistency property Con for 9’ to befinitely decidable if there is an effective procedure which applied to the canonical index of any finite subset X of U determines whether XECon. Then the following result easily follows from our completeness theorem for FC-normal NRS’s. Theorem 6.13. Let Y = (U, N) be a highly recursive rule system such that Y is FC-normal with respect to a decidable consistency property Con. Then {uEU: u is in some extension of Y”> is recursive. Finally systems.
we have the following
result about recursive
FC-normal
nonmonotonic
rule
A context
305
for belief revision
Theorem 6.14. Let T be a recursive tree in 2’” such that [T] # 8. Then there is an FC-normal recursive NRS 9 = (U, N) such that there is an efSective one-to-one degree-preserving
correspondence
between
[T] and 8(Y).
Reiter [26] proved that there is a recursive
normal
default theory with no recursive
extension. Theorem 6.14 generalizes his result but also gives much finer information even for recursive normal default theories since the set of degrees of paths through highly recursive trees have been extensively studied. Our correspondence will allow us to transfer all results about the possible degrees of paths through highly recursive trees to results about the degrees of extensions of recursive FC-normal NRS systems. That is, all the results of Corollary 6.5 continue to hold if we replace recursive nonmonotonic rule system by FC-normal recursive nonmonotonic rule system in each part of its statement. Moreover, we note that the proof of Theorem 6.14 explicitly constructs recursive FC-normal NRS’s such that the underlying consistency property is not finitely decidable. Corollary 6.15. 1. There is an FC-normal recursive nonmonotonic rule system (U, N ) such that (U, N ) has 2”0 extensions but no recursive extensions. 2. There is an FC-normal recursive nonmonotonic rule system (U, N ) such that (U, N) has 2”o extensions and any two extensions El # E2 of (U, N) are Turing incomparable. 3. If a is any Turing degree such that 0 < a ~~01, then there is a highly recursive nonmonotonic rule system (U, N ) such that (U, N ) has 2 ‘0 extensions but no recursive extensions and (U, N) has an extension of degree a. 4. If a is any Turing degree such that 0 cTa ~~01, then there is an Fe-normal recursive nonmonotonic rule system (U, N) such that (U, N) has K,, extensions, (U, N ) has an extension E of degree a and if E’ # E is an extension of (U, N ), then E’ is recursive. 5. There is an FC-normal recursive nonmonotonic rule system (U, N) such that (U, N ) has 2’0 extensions and if a is the degree of any extension E of (U, N ) and b is any recursively enumerable degree such that a cT b, then b ~~0’. 6. Ifa is any recursively enumerable Turing degree, then there is an FC-normal recursive nonmonotonic rule system (U, N ) such that (U, N ) has 2”o extensions and the set of recursively enumerable degrees b which contain an extension of (U, N ) is precisely the set of all recursively enumerable degrees b b Ta. Theorem 6.8 allows us to compare the FC-normal nonmonotonic rule systems (and thus FC-normal logic programs) with another class of rule systems that always have extensions. To this end, motivated by Apt, Blair and Walker [2] and Przymusinski [24] consider the class of locally stratified rule systems. These are rule systems 9’ = (U, N ) with the following property: There exists a function rank: U -+ Ord such
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V. W. Marek
et al.
that for every rule a,,
. ..>
CL:813
. . . . Pn
Y
in N, for all 1 < i < m, rank(cci) < rank(y) and for all 1 < j < n, rank(pj)
< rank(y).
Apt, Blair and Walker [2] (see also [21]) show that a stratified NRS possesses a unique extension. Hence, stratified NRS systems share with FC-normal NRS systems the property
that an extension
always exist. This similarity
will show that the classes of stratified
and FC-normal
is, however, superficial.
systems
We
are very different.
Example 6.1. 1. There exists a locally stratified NRS system which is not FC-normal, 2. There exist FC-normal NRS system which is not stratified. (1) It follows from the results of [S] that for every recursive ordinal number a there exists a recursive locally stratified nonmonotonic rule system Ya such that its unique for a > 3, as extension has degree 2 T O(‘). By Theorem 6.8,5$ cannot be FC-normal every recursive FC-normal NRS system possesses an extension of degree < 0”. (2) Since there are FC-normal NRS systems with more than one extension, such systems cannot be locally stratified. Next, we notice that the locally stratified systems have another property that makes their use for belief revision awkward. Namely, that the locally stratified NRS systems do not possess the semimonotonicity property. Example 6.2. Let U = {a, . . . , z} and let N be composed 7 -2 W
q:s r ’
Thus (U, N ) is stratified
of these three rules:
p:
ii’ and {w > is its unique
extension.
Now add the rule
:t -.
P
The new system semimonotonicity
is again stratified, fails for stratified
but its unique programs.
extension
7. Proofs of general results on FC-normal non-monotonic In this section,
is now {p, q, r}. Thus
rule systems
we shall give the proofs of the results stated in Section
From now on we assume all FC-normal consistency property given by Con.
nonmonotonic
Theorem 4.1. Every FC-normal
rule system has an extension.
nonmonotonic
rule systems
4. have the
A context
Proof. We shall show that our forward extension. Thus fix some well-ordering
307
for belief reuision
chaining construction will always produce an < of nmon(Y). Our well-ordering < deter-
mines some listing of the rules of nmon(Y), {r,: Amy}, where y is some ordinal. Let 0, be the least cardinal such that y d 0,. In what follows, we shall assume that the ordering increasing
among
ordinals
sequence
is given by E. Recall our forward
chaining
construction
an
of sets {Eb }neo, as follows. of E x
The forward chaining construction Case 0: a = 0. Let E$ = d,,,(0). Case 1: c1= q + 1 is a successor that El,
..., q7:P1,
..v
ordinal. Given
E,<, let /(a) be the least AEY such
P/i
l-J,= *
where c~i, . . , ape E,,< and /II, . . . , fik, II/ # E,+. E,5 1 = Ed = E,,, . Otherwise, let E,? 1 = Ed
= 4,,,(E7~
Case 2: CIis a limit ordinal. We then let Es
If there
is no such
G(M), then
let
u {cln(r&>). Then let Em<= U OEaE,jj.
= UaEO,,~>.
It is straightforward to prove by (transfinite) induction that Con(Eb) holds for all ~(~63~and hence Con(E <) holds. Next we want to prove by (transfinite) induction that Em
Z EV5 I, there exists one rule
wherea,,..., tx,~E,,,,/3~,..., /$$E,,and Eq<+l = cI,,,(E,,u{$}). Butsincerl(,+,,is FC-normal, we know that E2 u {$, pi} is not consistent for all i < k. Since Es is consistent, it must be the case that E,< u {II/, pi} $ E < for all i < k since subsets of consistent sets are consistent. Thus for all i d k, /3i$ E <. Hence rfCq+lj shows that $EC~<(~). But then E; u {$} c C,<(O) from which it easily follows that that &,,(E~~ u {Icl}) = E ,
308
V. W. Marek et al.
some aGO,.
Suppose
where i0 < . . . < i, < m and PI, . . . . Pk 4 E <. Now it is easy to see that our construction ensures that if r((,,+r) is defined, then cln(rfcs+ 1j) q+E,<. Hence if A # q and rlclj and r/(V) are defined, then r/(A) # r/(tlI. Thus the function L( ) is one-to-one on its domain. Now suppose that a, 4 E <. Then for all A + 1 greater than a, ram is a candidate to be recA+ 1J at stage A + 1. Hence it must be that case that rfo, + t) is defined and /(A + ~)EQ,,. But this is impossible. That is, if 0, is infinite, then the cardinality of {rfcA+ 1j : v,E~E@~} is equal to the cardinality of 0, which is strictly greater than the cardinality of (6: REV,}. Similarly if 0, is finite, then the fact that r/(A) is defined for all A d 0, and 8( ) is one-to-one would mean that the cardinality of {rfcij : 11~0,) is equal to the cardinality of nmon(9) so that every rule rEnmon(Y) must be equal to r((ij for some 1. Thus in either case we have shown that if CI,$ E <, then for some ,uLE@~,r,,,, is the least rule r=
such
that
61, . ..> &:?I,
. ..> Yr
*
6r, . . . . &eEW<
and
yr, . . . , yr,
u,EE,<+ 1 G E <. Thus CI, must be in E <. Hence
as claimed.
Ic/$ E,<. But then by construction CE< (0) c E < and E < is an extension
0
Note that the proof of Theorem 4.1, remains unchanged if instead of starting with &,,(0) at stage 0, we start with &,,(I) where ZECon. Thus we also have the following. Theorem
4.2. Let Y = (U, N)
respect to consistency exists an extension
property
be an FC-normal
nonmonotonic
rule system
Con. Let I be a subset of U such that IsCon.
with
Then there
I’ of I in 9.
Next we want to show that every extension of an FC-normal NRS Y = (U, N ) can be constructed by our forward chaining construction relative to an appropriate ordering of nmon(Y). Theorem 4.3. If 9’ = (U, N) nmon(Y),
is an FC-normal
NRS,
and <
is any well-ordering
of
then:
(1) E + is an extension of 9. (2) (Completeness of the construction) a suitably chosen ordering
<
Every extension
of 9’ is of the form E + for
of nmon(Y).
Proof. (1) follows from our proof of Theorem 4.1. (2) We prove the following fact: Let F be an extension of an FC-normal NRS Y = (U, N ). Let p = card(NG(F, 9’)) and let < be some well-ordering of nmon(Y)
A context
such that the listing of nmon(Y) NG(F, Y) = {r,: CIE~}. Then
for belief revision
determined
(i) F = cl,,,({cln(r): VENG(F, Y)}), (ii) F = E x where E < is constructed
309
by <, {r,: cr~y}, is such that p < y and
and by our forward
chaining
construction.
For (i), note that for each El,
an:P1,..., Pk ENW’, S),
‘..,
I=
*
cln(r)~F. Moreover, for any set W L C,(8), d,,,(W) c1,,,( {&r(r): ~ENG(F, 9))) c C,(0). Then a straightforward length then
of a minimal proof scheme p cln(p)ecl,,,( {c/n(r) : rENG(F, 9’))).
will It
show then
L C,(0) induction
that if follows
so that on the
supp(p) n F = 8, that C,(0) =
cI,,,( (h(r): rENG(F, 9’))). be constructed by the forward chaining construction For (ii), let {Ed : wOy} relative to the well-ordering of rules {r, : Amy }. Then we claim E,< = F and E,< = E,< for CI> ,u. First it is easy to show by induction that cl,,,(Ed) = Ed for all CLNext we claim that if asp, then E,< c F and moreover if E,< # E,< 1, then [(a + l)~/*. That is, E> = cl,,.@) s F. Next suppose by induction that ED< s F for all BECKThen if tl is a limit ordinal, E> = up,, EPs E F. If a is a successor ordinal, we can assume by induction that Eqx c F where r] + 1 = ~1.Now consider E,<. If Eqx = F, then for any rule r=
al>
. ..>
‘%:P1,. . . . Pk
EnmOn(
*
it must be the case that either { B1, . . . , &} n F # 0, {aI, . . . . LX,,> c/F, or I,!JEFsince F is an extension. That is, if E, < = F then /(q + 1) must be undefined and hence Esrl = E,,, 1 = F. If E,j # F, then consider some UEF - E,+. Since UEF, there is some minimal proof scheme p = ((a,, fO, G,), . . . . (a,, r,, G,)) where CI, = a and G, n F = 0 which witnesses that aEF. Since a$E,<, there must be some k < m such that czi, . . . . uk_lEEv’ and akqkEqX. Then consider Fk =
ai,,
...>
clij:81,
...>
Bt
where i. < . . < ij < k.
@k
., fit} = 8 since otherwise ukEcl,,,(E~) = E,,$. Thus 1,...,/31}~G,andG,nF=0,itmustbethecasethat iii, . . . . A} # 0. But since (P ence F,,,ENG(F, 9) and F,,,is a can1,..., /$}nE,,5=0and{/3, ,..., P,}nF=@H didate to be rf(,,+ 1l. But this means that if f,,, = rs in our ordering of rules in nmonh(Y), then a(~ + 1) d p < p. But for any REP, cln(rb)eF by our choice of our well-ordering. Thus cln(r~~,+,,)~F so that E,< u {cln(r~t,+,,)} L F and hence
Now
it cannot
be that
cL,,(E7~
{fil,.
u {cln(rf(,+ i))}) = E,-$ I = -%< c f’.
It follows that E> c F since E, = lJaslr Eb and Eb E F for all IXE,U.We claim that it must be the case that E,< = F for otherwise EPx c F and hence for all asp, E,+ c F.
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V. W. Marek et al
But our argument fact, in turn, scheme
above shows that if Eb c F, then Eb c E,< 1 and d(cr + l)~p. This
will allow us to prove by induction
that for all ~ENG(F,
Y), cln(r)~E,‘.
on the length
That
of a minimal
is, suppose
proof
I,$ = cln(r) for some
reNG(F,
9). Now Ic/ = cln(p) for some minimal proof scheme p = where G, n F = 8 and E, = II/. Now assume by <
conclusion of some minimal proof scheme q such that supp( q) n F = 8 and length of q < m is in E,,<. Note that each rk for k < m is either in man(Y) or in NG(F, 9’) since p shows c(i, . . . . M,EF and cons(rk) s G, where G, n F = 0. It follows that each Clifor i < m such that rieNG(F, 9’) is in E ,’ by our induction hypothesis. But then = E:. So consider r,,,. Now {a 1, . . . . c~,_i} c cl,,,(E>) c~,_i} G E,<. If r,ENG(F, Y), then r, = * = %14nLJ,{~1, . . . . rules where 5 < p. Moreover, there is some AE~ such that {c(i, then for any 2 < 6 d ,u, if $6 Ed< then r5 is a possible candidate must be that case that rfc6+ 1j is defined and L’(8 + l)~& But this if ,u is infinite, then the cardinality of {rp(6+1j: AE~E~} is equal
if r,Emon(Y),
then
ry in our orderings
of . . . . CI,_ i} c En<. But to be rLcd+1j. Hence it is impossible. That is, to the cardinality of .Dwhich is strictly greater than the cardinality of (cr: CIE[}. Similarly if ,Uis finite, then the fact that r!(1) is defined for all 2 Q p and 8( ) is one-to-one would mean that the cardinality of {r((lj: 1 d u} is equal to the cardinality of ,u so that every rule rg with 6 < ,Dmust be equal to rLcAjfor some /z Q p. This in either case we have shown that if $ $ E,$, then for some 6 6 p, ry = rfcaj. But then by construction tj~E> s E> . Thus $ must be in E,<. Thus we have shown that {cln(r): rENG(F, 9)) E E,< if E,
{cln(r): rENG(F,
Y)})
c cl,,,(E2)
= E>.
Thus it must be the case that E,< = F. Note we have already shown that if Eb = F, then Ed = E,< 1. Thus since EPX = F it easily follows that E A< = F for all p < II < 0,. Hence E< = uorsB E, = F as Y claimed. 0 Since every E < is in Con, we immediately Corollary respect
get the following
4.4. Let 9’ = (U, N > be an FC-normal
to consistency
property
corollary.
nonmonotonic
Con. Then every extension
of Y
rule system
with
is in Con.
Next we show that if our FC-normal nonmonotonic rule system (U, N ) is countable, i.e., if U is countable which automatically implies that N is countable, then every extension of (U, N ) can be constructed via the countable forward chaining construction relative to some well-ordering < of nmon( (U, N )) of the order type of w. Theorem 4.5. If 9 = (U, N) is a countable FC-normal nonmonotonic rule system, then 1. E X constructed via the countable forward chaining construction with respect to <, where < is any well-ordering of nmon(Y) of order type w, is an extension of Y.
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311
2, (Completeness of the construction) Every extension of Y is of the form E-C for a suitably chosen ordering < of nmon(Y) of order type o where E-X is constructed via the countable forward chaining construction. Proof. We note that if 4 is a well-ordering of nmon(Y) of order type w, the countable forward chaining algorithm is just the first o steps of the forward chaining algorithm. Thus
to prove
1, we must
show that if we construct
E + with respect
the forward
chaining algorithm, then E, -X = En< for all A 2 w. In fact, we need only show that E’=E+ o+ Now suppose that w I
.
El,
r/(,+
1) =
“‘,
%l:B1,
‘..,
Bk
$
is defined. Thus tli, . . . . CC,,CZE>and fir, . . . . &, $$ E>. Moreover, rgC,+ll = rq for of rules determined by <. But since some q where {rnjnEO is the ordering E$, there must be some s such that aI,. ., a,~Eb. Hence for all t >, s, EZ = un,, PI, . ..> hr ICI$E,< so that r4 is candidate to be rfC1)for all t > s. Since the function e(t) is one-to-one, it easily follows that there would have to be some finite t such that r4 = r/(,). Thus rfcw+ 1) must not be defined and hence E: = E <. Next we consider the proof of 2. Note that if we apply the proof of Theorem 4.3 to F in this case the most natural thing to do is to order the rules of NG(F, 9) first, say NG(F, 9) = {so, si, . ..}. and then follow this ordering by a listing all the rules of nmon(Y) - NG(F, Y) = (to, tl, . ..}. Now, if NG(F, 9) is finite, then our listing of rules determines a well-ordering < of order type o in which the proof of Theorem 4.3 shows that F = E <. If NG(F, 9) is infinite, then our listing of rules determines a well-ordering < of order type < o + cc).It then follows from the proof of Theorem 4.3 that
and that E;
= F. The key point
to note is that for any
is not in NG(F, 9’), it must be the case that {/zI~,...,~~}~F#QI or a,} $ F. But since F = Uiew Ei<, it also follows that either {@l,. . . . (a) for some i, {P1,...,Pk)r\Ei~ #@or (b) for all j, (~(1, . . . . a,,} $ EJ. In case (a) if we insert r between sk and ski 1 where k = t(i), then this change will have no effect on the construction of the Ei+‘s. That is, the construction of E-C up to stage i can depend only on s,,, , S/(i) and hence we will get the same sets, Ej< forj < i, for any ordering which starts out s O, . . . . S/(i). Thus if we take the ordering s,,, . . . . Sl(i), , fik > n EiS # 8, r is not a candidate to be rfCkJfor r, s 1+/(i), . . . , then because {/Ii, any k > i and hence the insertion of r does not effect the rest of the construction of E <. which
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In case (b), we can insert r anywhere in the initial o part of the list and it will have no effect on the construction of the Ei<‘s for i~o because the premises of r are never contained
in such Ei<‘S. In this way, we can see that it is possible
to interleave
all the
r’s in nmon(Y)
ordering
- NG(F, 9’) into the basic ordering sO, sr, . . . so as to create an of order type o but with out changing the sequence E,<, El<, . . . Thus it will
still be the case that F = E$ Theorem Theorem
= Uiiw
4.6 follows immediately
0
from the following
Suppose
(Semi-monotonicity).
D c nmon(Y).
EC<.
9’ = (U, N)
result. is an FC-normal
NRS.
Let
Then
1. 9” = (U, man(Y)
u D) is FC-normal
2. if E’ is an extension
(a) E’cEand (b) NG(E’, 9’)
of 9’,
c NG(E,
NRS, and
then there is an extension
E of Y such that
9).
Proof. The fact that Y’ is an FC-normal definitions. For part 2, let p = cardinality
NRS is an immediate consequence of our of NG(E’, 9”) and choose a well-ordering
of NG(E’, Y’), {r,: UE~}. Th en extend this well-ordering to a well-ordering {r,: C(EY} of nmon(S). It follows that if E < 1s constructed via our forward chaining algorithm with respect to the well ordering < determined by (r, : Amy}, then proof of Theorem 4.3 shows E’ = E,+ so that E’ z E+. It remains to be proved that NG(E’, 9’) G NG(E<, 9’). Now suppose r=
Then since hence since
al,
..‘,
%:Pl, . . . . Pk
9’).
{c1i, . . . . cl,,} c E’ E E+ and {pi,..., jjk} n E’ = 8. But 4.3, E’ = c1,,,( {cln(r): reNG(E’, by the FC-normality of r, E’ u {$, pi} IS not consistent E + is consistent, E’ u {$, pi} $ E X for any i = 1,. . . ,
E’ = E,<. By Theorem
3..., k and reNG(E<,
i=l
ENG(E’,
*
9’).
q
We prove now the result on the orthogonality Theorem 4.7 (Orthogonality with respect to a consistency ofY,
note that Con(E’) holds 9”))). Thus $EE’ and for any i = 1, . . . . k. But k. Hence Bile s for all
of Extensions). property
of extensions.
Let Y = (U, N)
be an FC-normal
NRS
Con. Then, if E and F are two distinct extensions
EvF$Con.
Proof. By Theorem 4.3, E = uneO E$ where {Ees}a.O, by the forward chaining construction relative to some well t( be the least ordinal such that Ed cF but E,2, $F. an a since otherwise E s F and then by the minimality
is the sequence constructed ordering < of nmon(S). Let Note there must be such of extensions, E = F. Thus
A contextfor belief relrision
313
the rule Xl, r/c,+
..‘>
G:/jl,...,
Pk
1) = +
8 # {PI, . . . . fik}, {PI, . . . . ljk} n E,< = 8 and {cxl, . . . . c(,} c Ed, u {I/J}). Since E,3 1 $ F, it must be that $$F. But this means that /3iEF for some i since otherwise Y/(,+ Ij ENG(F, Y) which would imply that $EF
is
such
that
E,5 1 = cl,,,,(E,<
because F is an extension. By the FC-normality of r/(,+ 1j, E,< u {t,b, pi} is not 0 consistent. But since Eb u {lc/, Bi} c E u F, E u F is also not consistent. Theorem 4.8. Suppose Y = (U, N ) is an FC-normal property
Con such that cl,,,({cln(r):r~nmon(Y)})
NRS with respect to consistency is in Con. Then Y
has a unique
extension.
Proof. For a contradiction, assume P’ has two distinct extentions, El and E2. Then by our proof of Theorem 4.3, Ei = cl,,,( {cln(r): rENG(Ei, Y)}) for i = 1,2. But then Thus E, u E2 is contained in a consistent for i = 1, 2, Ei s cl,,, {cln(r) : rEnmon(Y)}. set so that E, u E2 is consistent, contradicting Theorem 4.7. 0 Theorem 4.9. Let 9 = (U, N) property
be an FC-normal
Con. Then VE U is an element
a consistent
proof scheme with respect
NRS
with respect to a consistency
of some extension
of Y
if and only if cp has
to Con.
Proof. Clearly if PEE where E is an extension, then Con(E) by Corollary 4.4. Thus since fl~C,(fl), there is a consistent minimal proof scheme for cp. Conversely suppose p = ( (cpO, r,,, G,), , (cp,, r,, G,)) is a consistent minimal < i, < m be set of all i < m such that riEnmon(Y). proof scheme for p. Let 0 < iI < ... Now well-order nmon(S) so that ri, . . . , rik are the first k elements in the list. Then if we construct an extension via our forward chaining construction, it is easy to show by induction on k that BEEP. Hence BEE< which is an extension. 0 Theorem 4.10. Suppose Y = (U, N) Suppose further
is an FC-normal
that E; and E; are distinct extensions
distinct extensions
NRS and that D c nmon(Y). of (U, D u man(9)).
Then Y has
El and E2 such that E; c El and E; c E2.
Proof. By Theorem 4.9, we know that there are extensions of 9, El and E2, such that E; s E, and E; E E2. But then the orthogonality of extensions for (U, D u mon(9’)) ensures E; u E; is not consistent. Hence E, u E, is not consistent so that E,#&.
0
8. Recursive FC-normal
In this section,
NRS systems and the complexity of their extensions
we shall give all the proofs of theorems
stated in Section
6.
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Our first result will show that under the assumption of FC-normality that once again even recursive NRS systems are guaranteed to have at least one relatively well behaved extension. Theorem 6.8. Suppose that Y = (V, N) is a recursive nonmonotonic rule system and 9 is FC-normal. Then Y has an extension E such that E is r.e. in 0’ and hence E ~~011. Proof. It is easy to see that since N is recursive, man(9) is also recursive. It then easily follows that if X E U and X is r.e., then clmon(X) is also r.e. In fact, there is a recursive function f such that for all e, WfceJ= clmon(W,) where W, is the eth r.e. set = {x : q,(x) 1 }. We shall show that for any recursive well-ordering < of nmon(Y) of order type o, E < is r.e. in 0’ where E X is constructed via the countable forward chaining algorithm with respect to <. That is, i determines an effective listing of nmon(S), rO, rlt . . . . Then the countable forward chaining construction relative to i constructs a sequence of sets, E> -c E> c .... It is straightforward to prove by induction that E,< is r.e. for all e. That is, suppose E,< = W,,, for some e,. Es=E’ Then either that OlI E,-:zE,& SO E,?1 = &A-@ u {cln(ri,.+lJ!)j) = ~~~~~(W,,u {cln(r(,,+ 1j))) which is clearly r.e. Now there are two cases. Either there is an n such that E,% = E,$1 in which case E,+ = E < so that E < is r.e. Otherwise, E,< # E,-$ 1 for all n and hence rfln+ r) is defined for all n. In this case, given an 0’-oracle, we can test for membership in E,,+ and hence it is easy to see that given aI, rk
=
. ..>
%:A, ,.., Pt
E nmon(P),
*
we can test whether cL1,. . ., QzE,,‘, PI, . . ., /ilk, I)$ E,,<. Thus given an O’-oracle, we can effectively find r((“+ Ij. However, gtven ry(,,+ 1l and an r.e. index e, such that Wen= E,<, then we can effectively produce an index r.e. index e,,, such that Thus there is an 0’-effective sequence We,+ 1 = 4,,,05~ u (cWrt,+lJ)) = En?+,. is r.e. in 0’ in this case. e0, el, . . . such that E,> = Wen for all n. Hence E < = un Eb Thus in either case, E X will be r.e. in 0’. 0 Note that Theorem 6.8 is in great contrast to the case of arbitrary recursive NRS Y = (U, N ) where, for example as in Theorem 6.7(a), there exists a recursive NRS 9’ such that Y’ has an extension but 9” has no hyperarithmetic extension. Theorem 6.10. Let Y = (V, N) b e a recursive rule system such that 9’ is Fe-normal and nmon(Y) is finite, then every extension of Y is r.e. Proof. This theorem follows immediately from our argument in Theorem 6.8 once we observe that if nmon(Y) is finite, then there is some finite stage such that E,< equals E < for any well-ordering < of nmon(Y). 0
A context for belief reuision
Another situation where a recursive FC-normal have an r.e. extension is when Y is monotonically finite, then 9 = (U, N)
is guaranteed
315
NRS Y = (U, N) is guaranteed to decidable. In particular if man(Y) is
to have an r.e. extension.
Theorem 6.11. Let Y = (U, N) be a recursive rule system such that Y is FC-normal and monotonically decidable. Then Y has an extension which is r.e.
Proof. We shall show exactly as in Theorem 6.8 that for any recursive well-ordering via the countable < of nmon(Y) of order type o, E + is r.e. where Es is constructed forward chaining algorithm with respect to 4. Again, there are two cases. If there is some n such that E,.? = EL 1, then E,,, = E < and E < is r.e. Otherwise, E,< # E,? 1 for all n and hence r/(,+1I is defined for all n. In this case, it is easy to prove by induction that E2 = clmon((c(rf(Ij), . . . , c(rf(,))}) for all n 3 1. Because 9’ is monotonically decidable, it follows that E, -X is recursive for all n. Indeed there is a recursive function of cl,,,(DJ. Thus given function f such that qfckj is characteristic rfcnI}, we can effectively find k, such that Dkn = (c(rf(lI), . . . . c(rg(,))} and (r41j, . . . . then (arty,,) will be the characteristic function of E,,<. But then we can use (P/(~,) to effectively find rpcn+ 1J. Thus we can effectively find the sequence of rules rytlj, rfc2), . . . and since E < = cl,,,( {c(rf(lj), c(r/(2j), . ..}). Es is r.e. q
Next we consider the case where Y = (U, N) is a finite FC-normal nonmonotonic rule system, i.e. we assume that both U and N are finite. In this case, we shall see that our forward chaining algorithm runs in polynomial time. For complexity considerations, we shall assume that the elements of U are coded by strings over some finite alphabet C. Thus every aeU will have some length which we denote by 11 a I/. Next, for a rule
r=
aI, . . . . a,:bI,
. . . . b,
C
we define
Finally,
for a set Q of rules, we define
IIQII = c IId. rsQ
Theorem 6.12. Suppose Y = (V, N) is a Jinite FC-normal nonmonotonic rule system and < is some well-ordering of nmon(S). Then E-X as constructed via our forward chaining algorithm can be computed in time 0( IIman(Y) II - II nmon(Y) (I +
IInmoW)l12).
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Proof. First observe that our forward
chaining
algorithm
will stop after stage n where
n = )nmon(Y)I since 8(k) is a one-to-one function, i.e., at then end of stage n, there will be no possrble candidate for r((,+ r) sothatE,$=Ez,=E<. Next consider a stage k + 1 in the forward chaining construction. Given Ek<, we must make a pass through whether maintaining
the rules in order to check for each rule r = ‘I”‘.’ a”;b”‘.” bm
(aI, . . . . a,) s E,,< and
for some constant
{b,, . . . . b,, c} n Ek< = 8. Notice
that
at a cost of
this check in C
11steps
C and we call such a check a rule check. Now assuming
that we
an appropriate
data structure
we can perform
the rules in order, if r is the first rule such that {al, . . . , a,} G E,,? and b,, c} n Ek< = 8, then r = rk+lr E& 1 = cl,,,({c} u E,$). Moreover, if 1, . . . . {b (b,, . . . . b,, c> n Ek. # 8, then we know that r can never be a candidate to rj for any j > k so that we can just mark rule r and never consider it again. Of course, we also mark rk+ 1 at stage k + 1 if it is defined so that at each stage will mark at least one rEnmon(Y). Moreover, if rk+r is not defined, then we can stop since then we know Ek+ = E<. process
It follows that at stage k + 1, we need to look at most Inmon(Y) I - k rules and hence perform at most Inmon(Y)I - k rule checks. Since the construction must stop at requires at most stage Inmon(Y)I, it follows that the entire construction (a) of computing (b) )nmon(Y)I operations of c&,,,,(0). (4 the computation
cI,,,,(Ek< u {c}), and
Now if 11nmon(Y)/I = klnmon(Y)( for some k, then the rule checks could require 0( I/nmon(Y) II”) steps. Next consider the computations cl,,,(A) which are required for (b) and (c) above. We claim all this can be done 0( I(man(Y) II - Ij nmon(Y) 11)steps. Since in our construction all the elements of EkX must appear in one of the rules, we can assume IIA I) < IIman(Y) /I + IInmon(Y) 11.Now we can first make a pass through all the rules of nmon(Y) to get a list of all the elements of U which occur in one of the rules, Call this set V. Another pass through the rules will allow us to set up a system of pointers from each CE V to the set of rules rEmon such that c occurs in the set of premises of r. We can also mark which c are in A. All this will require CI (II monW4P)II + IInmofl(Y)II1d CI (II man(Y) ((* ((nmon(Y) II) steps. Now for each CEA, use the pointers from c to the rules remon to update each r by marking each premise of r in A. Now if a rule rGmon(.Y) has all of its premises marked, we mark the conclusion of r, i.e., we add the c/n(r) to cl,,,(A) and use the pointers from cln(r) to rules in man(Y) to further update the premises of each rule by marking h(r). We continue in this fashion until there are no more rules to update in which case A together with the marked conclusions will form the &,,(A). Now assuming that updates can be performed in constant time, each rule remon can require at most I\r )I updates in this process since once all the premises of a rule have been marked we no longer have to consider it. Thus we require at most IIman(Y) 11updates so the entire process takes at most 0( (Iman(Y) 11) steps. Thus to compute the monotonic
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A context for belief revision
closures
required
0( ((man(Y)
in (b) and
I(- ((nmon(Y)
(c) above
(() steps.
We pause to make one further
takes
0((1
+ Inmon(Y)l)
observation
about
any highly recursive
NRS Y = (U, N) for which the underlying consistency property decidable. That is, if ,Y is highly recursive, then we can effectively minimal
proof
schemes
for any
XEU. If Con is finitely
decidable,
effectively tell if any of the proof schemes for x are consistent. a consistent proof scheme if and only if x is in some extension we can effectively decide whether following
* 11man(Y)
11)d
q
x in some extension
FC-normal
Con is finitely find the set of then
we can
By Theorem 4.9, x has of Y’. Thus in this case,
of Y. Thus we have proved the
result.
Theorem 6.13. Let 9’ = (U, N) be a highly recursive rule system such that 9’ is FC-normal with respect to a decidable consistency property Con. Then {uEU: u is in some extension qf Y is recursive.
We end this section with a result that shows that recursive least as expressive
as highly recursive
FC-normal
NRS’s are at
NRS.
Theorem 6.14. Let T be a recursive tree in 2’” such that [T] # 8. Then there is an FC-normal recursive NRS Y =
(2) i for all cx~T which are terminal G! (3) L& -cc-o, a-1: (4) ~ E tl :
(5) a^l
a-0
=&
nodes.
for all CXE T such that M is not a terminal
for all C(ET such that CIis not a terminal a:a-1 cc^o
node.
node.
for all cr~T such that cz is not a terminal
node.
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V. W. Marek et al.
(6) s
for all EE T such that M is not a terminal
s
(7) F
for all CZT - (8) and all @LJ.
(8) ;
for all CET and /I I= CL
CX, lx-o: (9) a^l
c1,a-1: cr^o
node.
for all C(ET such that c1is not a terminal
node.
It is easy to see that P’ = (U, N ) is a recursive nonmonotonic rule system. Given a path rc=(rc(O),rr(l),...) through T, let E,= {a:ct is a node on rr}u{E:a is not a node on rc>. We claim that E is an extension of Y if and only if E = E, for some m[T]. It is easy to see that E, is an extension for every XE[T]. That is, (1) shows that QkCE,(@) and repeated use of the rules in (5) will allow us to show that rc(n))~C~$) for all n. Then the rules in (6) will allow us to show that for every MO), ’‘. 2 4 (40), . . ., n(n - l), &n))~C~,(ti) 6(n) =
Now
suppose
0
if n(n) = 1,
i 1
if n(n) = 0.
MET and
where
LY is not
on
rr. Then
there
is an
n such
that
MO), . . . > n(n - l), 6(n)) L cc Hence repeated use of the rules in (3) will allow us to show cl~C,~(@). Thus E, E C,(@). A straightforward proof by induction on the length of a derivation will show that C&J) c E,. Hence E, = C,r(8) and E, is an extension of Y. Next assume that E is an extension . First we claim that E # U. For if E = U, then rule cannot be used to derive C”(0) = &o.(0) since every strictly nonmonotonic elements of C,(0). Since [T] # 8, there is some path ICE[ T]. By our previous remarks, E, is an extension. But clearly E, # U and &,,,($!I) s E,. Thus C,(0) c E, # U so that U is not an extension. Once we know that U is not an extension, the rules in (7) show that for any CIET - (8}, E can contain at most one of a and E. Now assume by induction on n that there is a node b,, = (p(O), . . . , /?(n - 1)) of length n such that fi, is on some infinite path through T and for all MET with 1CI1< n, C(EE if and only if a c fl,, and iie:E if and only if c( & p,,. Now the rules in (3) ensure that if YET is such that IyI > n and y does not extend b,,, then TEE. Thus the only possible nodes, y of length n + 1 such that yeE are y = Pn^O and y = j$,^l. Because we are assuming that fi,, is on an infinite path through T, we know that at least one of /&^O and /I,,^1 must be on an infinite path through T. Now suppose Pn^O is not on an infinite path through T. Then A, = {ye:q J fin-O& VET} is finite by Konig’s lemma. Note that if qcAO and q is a terminal node, then ij~E by (2). But then it is easy to see that since T is a (0,2)-tree,
A context for belief revision
repeated
319
use of the rules of (4) will allow us to show GEE for all SEA,,. In particular,
P,,*OEE and hence by rule (9)
Thus we have shown that if j?,,^Ois not on an infinite path through T, then P,,*l EE, /&*l is on an infinite path through T, and for all UE T such that 1~) d n + 1, VEE if and only if g & /3,,*1 and FREEif and only if ye& /3,-l. Thus our inductive
hypothesis
n + 1 with /3,,+i = /$,^l. A similar argument will show that our inductive holdsatn+ 1 with/?,+, = /?,,^Oif pn^l is not on an infinite path through can assume that both @n-0 and /?,,^l lie on infinite rules
paths through
holds at hypothesis T. Thus we
T. Now consider
the
from (5). Note that R. shows Pn^l EE if /?/O$ E and RI shows /$^OEE if /$^l #E. Thus at least one of Bn*Oand /$ll must be in E. We claim that it cannot be the case that both PnAOand Bn^l are in E. For if /$*O, /?,,ll EE, then consider the minimal proof scheme P = ((Q,
TO, G,),
. . . .
of shortest possible length such that G, n E = 8 and c(, = P,,^O. Note that any rule Y with conclusion ,Bn^Ois either RI or comes from (7) or (8). Now I, # RI since RI is blocked for E if { P,,*O,/3,*1) E E. Since U # E, we cannot use any rule from (7). Thus
where yz$,^O. It follows that y = C(ifor some i. It cannot otherwise the proof scheme (ff02 ro> Cm>,...> (ai, ri, Gi),
P’ =
fin-o, A,
be that i < m - 1 since
Gi
”
>>
would violate our choice of p since it could be refined to a minimal proof scheme p” of length < the length of p such that cln(p”) = /I.^0 and supp(p”) n E = 0. Thus y = tl,_ i. But then rm- 1 = f where y c 6 from (8) or r,_1
=
cc:cC(l - i) ff*i
where a^i = y from (5). If rm_ 1 = $, then 6 = Cljfor some j < m - 1 and as before, the proof scheme <~o,ro,Go),.~.,
,
>
320
would
V. W. Marek et al.
violate
our choice of p. If
rm-l =
cc:cc^(l - i)
then either (i) az/3,^0
’ and c( = Cljfor some j < m - 1 or (ii) a = /InA and PnAO= C(jfor
some j < m - 1. In either case, we would violate our choice of p. Thus there can be no such p and hence /In*0 4 C,(0) if {/In-O, p,*l} z E. Hence if (p”*O, /?,,^l} G E, E # C,(0) which contradicts the fact that E is an extension. Thus we must conclude one of B,,lO and /I,,^1 is in E. Now if P,,^OEE, then the rule
from (9) shows BnAlEE and our induction If /?,^l E E, then the rule
hypothesis
that exactly
holds at n + 1 with fi,+ i = fi,^O.
p,*1:
shows that Pn^OeE and our induction hypothesis holds at n + 1 with /I, + I = 0,-l. This completes our induction and shows that 0 = /I,, t pi c Bz c . . . determines a path l7 through T and that E = E,. It now follows that the corresponding 7cH E, is our desired effective one-to-one degree-preserving correspondence between Y(Y) and J?(Y). Thus to complete our proof of the theorem, we need only check that Y is FC-normal with respect to some consistency property. We define X _c U to be consistent, Con(X), if and only if X 5 E, for some ne[T]. It is easy to check that properties (l), (2), (4) hold for Con. For property (3), note that if X G E,, then c&,,,,(X) c C,$) = E, since E, is an extension. Thus Con defines a consistency property. Finally we must check that all rules r~nmon(Y) are FC-normal with respect to Con. Note the only rules renmon(Y) are of the form 9 and g from (5). Now clearly (a-0, a-1) is not consistent so that all we need to show is that if Xc U is such that Con(X), cl,,,(X) = X, aeX, a-(1 - i)$X, then X u (u-i} is consistent. Now consider the possibilities for a monotonically closed consistent set X. For some rc = (rc(O), rc(l), . ..)e[T]. X c E, = {a:ct is on rr} u {cC:a is not on rc}. First suppose for infinitely many n, (n(O), . . . . z(n))eX. Then rules of the form (8) show (rc(O), . . . , n(m))EX = cl,,,(X) for every m. Next rules of the form (6) allow us to show (K(O), . . , x(n - l), 1 - x(n))EX = cl,,,(X) for all n and then rules of the form (3) will allow us to show VEX = c&,,,,(X) for all CLnot on rr. That is, X = E, if for infinitely many n, (n(O), . . . . x(n))EX. But in such a case, MEE, = X, a*(1 - i) $ E, = X implies a*iEE, = X so that Con(X u {Eli>) holds. Thus we must assume that there is a largest m, say n, such that (n(O), . . . , n(m))eX. Then rules of the form of (8) will allow us to show that (n(O), . . . . n(m))EX = clmO,,(X) for all m G n. Next rules of the form (6) will allow us to show that for all m < n, (x(O), . . . . x(m - l), 1 - x(m))EX
= cl,,,(X)
and then rules of the form (3) will allow
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A context .for belief revision
us to show that ZEX = c1,,,(X) for all MET such that CIand q are incomparable where n(n)). Next we claim that it must be the case that both q-0 and y-1 lie on V = MO), . *. 2 infinite through
paths
through
T. For if not, suppose
T. Then by K&rig’s Lemma
q^i does not lie on an infinite
path
{/I : PE T & fi J q-i} is finite and we can argue as
above that repeated use of the rules in (2) and (4) will show ~Ai~clmon(8) E X. But then -=7 from (9) shows ~“(1 - i)~cl~~~(X) = X violating our choice of q. Next the rule e suppose
that
q-i does not lie on z and ylA(l - i) lies on 7~. Now
consider
the set
A = (a: c17qAi and E$X}. Note that rules of the form (3) ensure that if EEA and ~“i c /I F ~1, then BEA. That is, if P$A, then VEX and hence EE&,,(X) = X by repeated use of the rules in (3). Thus the set of nodes in A determine a subtree of Trooted a q. We claim that there are no nodes PEA which are terminal with respect to A and hence A is infinite. For suppose BEA is a node which is terminal with respect to A. Thus fl# X and hence fl is not a terminal node of T because otherwise the rules of (2) would ensure P;clmon(8) E X. But then it must be the case that p^O, p1~T and -p-0, pll~X. But then the rule piEm from (4) would show that /?EcZ,,,(X) = X violating our choice of 8. Thus A has no terminal nodes and hence A is infinite since Neil A. But this means that there is a least one path rc* = (x*(O), *(l), . . .) through T such that q-i is on z* and (n*(O), . . . , rc*(k)) # X for all k. Thus X c E, and X c E#. Finally, it is easy to see that the only rules of the form & such that crux, M~$X are such that (a)
Mc n
and
a*(1 -j)
(b)
a = ‘I,
j = i,
(c)
a = I?,
j=l-i,
and
r= ye, ~‘(1 - j) = ~~(1 - i), and
or
a*(1 - j)= q^i.
In cases (a) or (b), a^( 1 - j)is on the path 7cand hence X u {cl-(1 - j)} G E, so that Con(X u (a-(1 -j)}) holds. In case (c), a-(1 -j) is on the path rr* and X u (a^(1 - j)} c E,* so that Con(X u {cz^(l -j)}) also holds in this case. Thus we have shown that all rules of nmon(S) are FC-normal with respect to Con. 0 Note that in the case where [T] has no recursive elements, (9~ T: ‘1 is an infinite path through Tj is not recursive and hence we will not be able to effectively decide whether a finite set S G U is consistent. Thus it is crucial that we make no assumptions about the effectiveness of the consistency property.
9. Conclusions In this paper we exhibited and investigated a natural condition which ensures that a logic program always has a stable model or a default theory or truth maintenance system always has an extension. This condition, called in our paper “FC-normality” is an abstraction of Reiter’s notion of normality in default logic coupled with Scott’s notion of Information System. This condition can be formulated in the general setting
V. W. Marek et al.
322
of nonmonotonic rules systems and hence can be immediately translated into other nonmonotonic reasoning formalisms including logic programming with negation as failure, default logic, and truth maintenance systems. We have shown that FC-normal nonmonotonic rule systems FC-normal logic programs, FC-normal logic programs normal truth maintenance systems) have all the desirable
(and, consequently.
with classical negation, FCproperties of Reiter’s normal
default theories such as the existence of extensions, the semimontonicity extensions, and the orthogonality of extensions. Indeed, when FC-normal tonic rule systems default
theories
are translated
into the language
which we call FC-normal
property of nonmono-
of default logic, we get a class of
default theories,
which strictly contain
the
class of normal default theories of Reiter and yet continue to have all the desirable properties of Reiter’s normal default theories. We gave a general construction, which we called the forward chaining construction, which constructed all possible extensions of an FC-normal nonmonotonic rule system based on well orderings of the set of nonomontonic rules of the system. By analyzing the forward chaining construction, we were able to establish bounds on the complexity of reasoning with FC-normal systems. For example, while there are recursive non-monotonic rule systems which have extensions but which have no hyperarithmetic extensions, we showed that every recursive normal nonmonotonic rule systems has an extension which is r.e. in 0’. Moreover, for any finite nonmonotonic rule system, we showed that one can always construct an extension in polynomial time. We hope that the technique presented in this paper can serve as an indication to designers of intelligent systems, in particular to monitoring systems, as well as to logic programmers why some programs behave better than other programs. FC-normal programs offer a “smooth transition” from one belief state to another. In particular modification of belief rules (but not of “hard facts”) can be handled smoothly by such systems. This is one way to maintain programs in a way which permits one to expand the present belief set to a larger one in the presence of new rules. It is expected that a further research into the nature of consistency properties for non-monotonic rule systems and their complexity will produce general techniques for writing normal
and maintaining FC-normal nonmonotonic rule system (and hence FClogic programs, extended normal default logics, and FC-normal truth main-
tenance
systems) for specific applications.
References [l]
K. Apt and H.A. Blair, Arithmetic classification of perfect models of stratified programs, Fund. Inform. 12 (1990) 1-17. [2] K. Apt, H.A. Blair, and A. Walker, Towards a theory of declarative knowledge, in: J Minker, ed., Foundations of Deductive Databases and Logic Programming (Morgan Kaufmann, Los Altos, CA, 1987) 89-142.
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[3] [4] [S] [6] [7] [S] [9]
[IO]
[11] [12] 1131 [14]
[IS] [16] [17] Cl83 [19] [20] [21] 1221 [23] [24]
[25] [26] [27] [28]
323
K. Apt, Logic programming, in: J. van Leeuven, ed., Handbook of Theoretical Computer Science, (MIT Press/Elsevier, Cambridge, MA/Amsterdam) 4933574. N. Bidoit and Ch. Froidevaux, Negation by default and unstratifiable logic programs, Theoret. Comput. Sci. 78 (1991) 85-112. H. Blair and W. Marek and J. Schlipf, Expressiveness of locally stratified programs, Ann. Math. Artificial Intelligence, to appear. J. de Kleer, An assumption-based TMS, Artificial Intelligence 28 (1986) 127-162. J. Doyle, A truth maintenance system, Artificial Intelligence 12 (1979) 231-272. D.W. Etherington, Reasoning with Incomplete Information (Pitman, London, 1988). M. Gelfond and V. Lifschitz, The stable semantics for logic programs, in: R. Kowalski and K. Bowen, eds., Proceedings of the 5th International Symposium on Logic Programming (MIT Press, Cambridge, MA, 1988) 1070-1080. M. Gelfond and V. Lifschitz, Logic programs with classical negation, in: D. Warren and P. Szeredi, eds., Logic Programming: Proceedings of the 7th International Conference (MIT Press, Cambridge, MA, 1990) 579-597. C.G. Jockusch and R.I. Soare, Degrees of members of Z: classes, Pacific J. Math. 40 (1972) 605-616. C.G. Jockusch and RI. Soare, rcy classes and degrees of theories, Trans. Amer. Math. Sot. 173 (1972) 33-56. W. Marek, A. Nerode, and J.B. Remmel, Nonmonotonic rule systems I, Ann. Math. Artificial Intelligence 1 (1990) 241-273. W. Marek, A. Nerode, and J.B. Remmel, The stable models of predicate logic programs, in: K.R. Apt, ed., Proceedings of International Joint Conference and Symposium on Logic Programming (MIT Press, Boston, MA, 1992) 446460. W. Marek, A. Nerode, and J.B. Remmel, How complicated is the set of stable models of a recursive logic program?, Ann. Pure Appl. Logic 33 (1992) 229-263. W. Marek, A. Nerode, and J.B. Remmel, Nonmonotonic rule systems II, Ann. Math. Artificial Intelligence 5 (1992) 229-263. W. Marek, A. Nerode, and J.B. Remmel, Computing jumps with an FC-normal nonmonotonic rule system, in preparation, 1993. W. Marek, A. Nerode, and J.B. Remmel, Rule systems and well-orderings, forward chaining construction, in preparation, 1993. W. Marek and M. Truszczynski, Relating autoepistemic and default logics, in: Principles of Knowledge Representation and Reasoning (Morgan Kaufmann, San Mateo, CA, 1989) 276288. W. Marek and M. Truszczynski, Autoepistemic logic, J. ACM 38 (1991) 588-619. W. Marek and M. Truszczyriski, Nonmonotonic Logic-Context-dependent Reasonings (Springer, Heidelberg, 1993). D. McDermott, Nonmonotonic logic II: nonmonotonic modal theories, J. ACM 29 (1982) 33-57. D. McDermott and J. Doyle, Nonmonotonic logic I, Artificial Intelligence 13 (1980) 41-72. T. Przymusinski, On the declarative semantics of stratified deductive databases and logic programs. in: J. Minker, ed., Foundations of Deductive Databases and Logic Programming, (Morgan Kaufmann, Lost Altos, CA, 1987) 193-216. M. Reinfrank, 0. Dressler, and G. Brewka, On the relation between truth maintenance and nonmonotonic logics, in: Proceedings of IJCAI-89 (Morgan Kaufmann, San Mateo, CA, 1989) 12061212. R. Reiter, A logic for default reasoning, Artificial Intelligence 13 (1980) 81-132. D. Scott, Domains for denotational semantics, in: Proceedings of ICALP-82 (Springer, Heidelberg, 1982) 577-613. A. Tarski, Logic, Semantics, Metamathematics (Oxford University Press, Oxford, 1956).
269
Logic 67 (1994) 269-323
A context for belief revision: forward chaining-normal nonmonotonic rule systems V.W. Marek” Department
qf Computer Science, University qf Kentucky,
Lexington.
KY 40.506-0027. USA
A. Nerode** Mathematical
Sciences Institute,
Cornell University,
Ithaca, NY 14853, USA
J.B. Remmel*** Department
of Mathematics.
University
of California
at San Diego, La Jolla, CA 92903, USA
Communicated by D. van Dalen Received 1 December 1992 Revised 15 June 1993
Abstract Marek, V.W., A. Nerode and J.B. Remmel, A context for belief revision: forward chaining-normal nonmonotonic rule systems, Annals of Pure and Applied Logic 67 (1994) 269-323. A number of nonmonotonic reasoning formalisms have been introduced to model the set of beliefs of an agent. These include the extensions of a default logic, the stable models of a general logic program, and the extensions of a truth maintenance system among others. In [13] and [16], the authors introduced nonmonotonic rule systems as a nonlogical generalization of all essential features of such formalisms so that theorems applying to all could be proven once and for all. In this paper, we extend Rieter’s normal default theories, which have a number of the nice properties which make them a desirable context for belief revision, to the setting of nonmonotonic rule systems. Reiter defined a default theory to be normal if all the rules of the default theory satisfied a simple syntatic condition. However, this simple syntatic condition has no obvious analogue in the setting of nonmonotonic rule systems. Nevertheless, an analysis of the proofs of the main results on normal default theories reveals that the proofs do not rely on the particular syntactic form of the rules but rather on the fact that all rules have a certain consistency property. This led us to extend the notion of normal default theories with respect to a general consistency property. This extended notion of normal default theories, which we call Forward Chaining normal (FC-normal), is easily lifted to nonmonotonic rule systems and hence applies to general logic programs and truth maintenance
Correspondence to: V.W. Marek, Department of Computer Science, University of Kentucky, Lexington, KY 40506-0027, USA. Email:[email protected] * Work partially supported by NSF grant IRI-9012902. ** Work partially supported by NSF grant DMS-8902797 and AR0 contract DAAG629-85-C-0018. *** Work partially supported by NSF grant DMS-9006413. 016%0072/94/$07.00 0 1994- Elsevier Science B.V. All rights reserved SSDI 0168-0072(93)E0053-Q
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et al.
systems as well. We prove that FC-normal nonmonotonic rule systems have all the desirable properties possessed by normal default theories and a number of other important properties as well. We also give an analysis of the complexity of the set of extensions of FC-normal nonmonotonic rule systems. For example, we prove that every recursive FC-normal nonomontonic rule system has an extension which is r.e. in the halting set. Similarly any finite FC-normal nonomonotonic rule system S has an extension which can be computed in polynomial time in the size of S. Again these results automatically apply to general logic programs, truth maintenance systems, and default theories. Moreover, even in the case of normal default theories, our complexity results are new.
1. Introduction and motivation A number of nonmonotonic reasoning formalisms have been introduced to model the set of beliefs of an agent. For example, Reiter [26] introduced default logic where the set of beliefs of an agent reasoning with incomplete information corresponded to an extension of a default theory (D, W). In the realm of logic programming, the stable models of a general logic program, as introduced by Gelfond and Lifschitz [9], can be used to model the set of beliefs of an agent. Similarly, extensions of truth maintenance systems as defined by Doyle [‘i’] and De Kleer [6], with subsequent contributions of Reinfrank, Dressler, and Brewka [25] can be used to model the set of beliefs of an agent. We introduced nonmonotonic rule systems as a non-logical generalization of all essential features of default theories, truth maintenance systems, and logic programs so that theorems applying to all could be proven once and for all, see [13] and [16]. We put forth the hypothesis that any of the above systems can be used to effectively model the beliefs of an agent as long as we restrict ourselves to a rather wide class of default theories, general logic programs, or truth maintenance systems which correspond to the forward chaining-normal nonmonotonic rule systems, or FC-normal systems, introduced in this paper. For example, suppose that belief sets are identified with stable models of an FC-normal logic program. To explain the role of FCnormality, we employ the paradigm of a blocks world for a robot with a hardwired motion planner. We think of the robot as making moves based on facts about where blocks are and based on rules of thumb such as “as long as such and such configuration is not observed, move as follows” based on logic programming. We choose as our current belief set a stable model, if any, incorporating known facts and rules. First difficulty: a general logic program may have no stable model. A second difficulty can arise even if our logic program has a stable model. That is, we want to deduce robot moves in this stable model. But if the robot observes a new fact, a block position not in the stable model, we have to abandon the previous stable model of the old program and find a new stable model of the logic program extended by the new facts. Again, for arbitrary logic programs, there may be no such stable model. So if stable models are to model beliefs and the robot is to have a belief set no matter what facts arise, with a hardwired underlying logic program, we have to find a useful and general condition on the logic program which guarantees that the program, extended by new facts,
A context
for
belief
always has a stable model. Such a condition
271
revision
is the irreducible
model belief sets as stable models and to specify belief revision
minimum operations
in order to as program-
ming operations on stable models. In default logic, Reiter [26] (see also Etherington [S]) defined a simple class of default rules, called normal, for which one never gets stuck in finding extensions. His normality
condition is syntactic and unduly restrictive, i.e., in a normal default theory It turns out that this syntactic condition does not all rules must be of the form y. generalize directly to nonmonotonic rule systems. However, an analysis of the proofs of the main results of Reiter on normal default theories reveals that his proofs do not rely on the particular the form 7 generalization
syntactic
form of his rules but rather on the fact that all rules of
have a certain consistency property. This led us to define a far reaching of normal default theories with respect to a general consistency prop-
erty. Moreover, this generalization is easily extendible to nonmonotonic rule systems and hence applies to general logic programs and truth maintenance systems as well. Thus we introduce in this paper the definition of what we call FC-normal nonmonotonic rule systems. We shall see that when we translate FC-normal nonmonotonic rule systems back into default theories, we will define a large class of default theories which we call FC-normal default theories. We shall see that the class of FC-normal default theories strictly contains the class of normal default theories and that FC-normal default theories have all the desirable properties of normal default theories. Finally, there are natural analogues of FC-normal default theories or FC-normal nonmonotonic rule systems in the formalisms of logic programming and truth maintenance systems as well. To repeat, we claim that extensions of nonmonotonic rule systems can be used to model beliefs of an agent, provided that we restrict ourselves to the class of FC-normal nonmonotonic rule systems introduced in this paper. Here an extension of a nonmonotonic rule system is the common generalization of extensions of default theories, stable models of logic programs, and extensions of truth maintenance systems. Restricting ourselves to FC-normal nonmonotonic rule systems avoids our ever getting stuck in finding new extensions in the face of new facts and there is no syntactic limitation. Rather, we employ an axiomatic “consistency property”, different for each application, which covers all areas of intended application we have examined. Our notion of consistency property can be thought of as a version of Scott’s “information systems” [27] tailored to extensions of nonmonotonic rule systems. In future papers this will allow us to use, for example, metaprogramming as a belief revision programming language operating on “FC-normal” logic programs and representations of stable models. Now the problem of whether there exists a stable model of a finite propositional program is known to be NP-complete [20]. The same result applies to the problem of whether there exists an extension of a nonmonotonic rule system. So the problem of finding a consistency property under which a logic program P or a nonmonotonic rule system (U, N ) is FC-normal is, at least, NP-hard. But in all applications, we can see directly what consistency notion to introduce, dictated by natural considerations of consistency for the semantics of the application.
272
V. W. Marek et al.
We shall not only prove that FC-normal nonmonotonic rule system have the desirable properties possessed by normal default theories but we shall also prove that FC-normal nonmonotonic rule systems have a number of other important properties as well. For example, given an FC-normal nonmonotonic rule system 9’ = (U, N), we shall show that every extension of Y can be constructed via a simple forward chaining
algorithm
which is based on a well-ordering
< of the strictly nonmonotonic
rules of Y which we denote by nmon(Y). That is, each ordering mines an extension of Y and every extension of 9’ is determined However,
many orderings
ings themselves
may determine
the same extension.
may be taken as computational
surrogates
of nmon(Y) deterby an ordering <.
This means
for extensions.
the orderIn a com-
panion paper [lS], we introduce a more general forward chaining process based on a well-ordering of the nonmonotonic rules which can be applied to arbitrary nonmonotonic rule systems 9’. For general nonmonotonic rule systems, our forward chaining process also yields an extension, but for a possibly smaller rule system than our original rule system. It turns out that one can view our definition of FC-normal nonmonotonic rule systems as a sufficient condition which guarantees that the general forward chaining algorithm always produces an extension of the original rule system. This is important because we will show that the forward chaining algorithm, when applied to finite FC-normal nonmonotonic rule systems, produces an extension in polynomial time in the sum of the lengths of the rules of the system. More precisely one can construct an extension of an FC-normal nonmonotonic rule system in time which is of order the square of the sum of the lengths of the rules of system. As usual, this same result also applies to constructing extensions of FC-normal default theories, stable models of “FC-normal” logic programs, and extensions of “FC-normal” truth maintenance systems. Thus FC-normal nonmonotonic rules system have the property that one can construct an extension or go from one extension to another in a highly efficient manner. Finally we explore the complexity of the sets of extensions of arbitrary recursive FC-normal nonmonotonic rule systems. Briefly, a nonmonotonic rule system 9 = (U, N) is called recursive if the universe U is a recursive set of integers and the set of rules N is a recursive set. We showed in [15] that for every countably branching recursive tree, there is a recursive nonmonotonic rule system 9’ = (U, N) such that there is a recursive one-to-one correspondence between maximal branches of the tree and the extensions of the 9’. Conversely, the set of all extensions of a recursive nonmonotonic rule system always so arises. Since there are recursive trees without hyperarithmetic maximal branches, but with a continuum of maximal branches, it follows there are recursive NRS with a continuum of extensions but no hyperarithmetic extensions. This does not happen with FC-normal recursive NRS. They always have extensions recursive in 0”. However, recursive FC-normal nonmonotonic rule systems are very expressive. We show that given any highly recursive tree, i.e. a recursive tree which is finitely branching and which has the property that we can effectively find all the successors of any node in the tree, there is an FC-normal nonmonotonic rule system Y = (U,N) such that there is a recursive one-to-one
A context for belief reoision
correspondence
between
the maximal
branches
273
of the tree and the extensions
9’. This is shown via coding trees into suitably constructed a simultaneous construction of a suitable consistency property. for example,
that there are recursive
FC-normal
nonmonotonic
of the
recursive NRS with This will imply that, rules systems which
have no recursive extensions. The outline of this paper is as follows. In Section 2, we shall briefly review Reiter’s [26] definition of normal default theories and state the main properties of such theories. Then in Section 3 we introduce the basic definitions of nonmonotonic rule systems.
In Section
define FC-normal
4, we shall introduce nonmonotonic
our abstract
consistency
properties
rules systems. We shall also introduce
and
our forward
chaining construction and state our major results about FC-normal nonmonotonic rules systems. The proofs of these theorems will be postponed until Section 7. In Section 5, we shall show how to translate the definitions and results of Section 4 back into logic programming, default logic, and truth maintenance systems. In Section 6, we shall provide a background on the complexity of general recursive nonmonotonic rule systems and state our basic complexity results for recursive FC-normal nonmonotonic rule systems. The proofs of the complexity results in Section 6 will be postponed until Section 8.
2. Normal default
default rule is a rule of proof of the form c~:M$i,...,
M$,,,
(1)
Y
where q, $ 1, . . . , Grn, y are formulas of a propositional language 9. A default theory is a pair (0, W) where D is a set of default rules and W G 9. For any subset of formulas S c 9, we let Cn(S) denote the set of all logical consequences of S. Also if D is a set of default rules, let
v:M$,,..., Y
Given a subset S c 9, define T(S) as the least set T(under conditions: 1. WLT; 2. Cn(T) = T; 3. Whenever rED is a default lIc/j$S, then YET.
inclusion)
satisfying
these
rule of the form (1) and VET and for all j d m,
214
V.W. Marek
It is easy to see that T(S) always
et al.
exists. We say that S E _Y’ is an extension of
(D, W) if T(S) = S. A default rule of the form (1) is called generating for S if cp~S, i$,#S. Let S c 9’. Then we define GD(D, S) as the set of all generating 1$1,..., rules for S in D and c(GD(D, S)) is the set of their conclusions. A rule r is normal if it is of the form
q:Mti II/ A default proved
theory
(2)
. (D, W) is normal if every rED is a normal
the following
theorems
about
normal
default
default
rule. Reiter [26]
theories.
Theorem 2.1. Every normal default theory possesses an extension. Theorem 2.2 with D’ E D. A = (D, W). 1. E’ G E
(Semi-monotonicity). Suppose that D and D’ are sets of normal defaults Let E’ be an extension of the normal default theory A’ =
2. GD(E’, A’) G GD(E, A). Theorem 2.3 (Orthogonality of Extensions). Zf a normal default theory (D, W) has distinct extensions E and F, then E u F is inconsistent. Theorem 2.4. Suppose that A = (D, W) is a normal default theory and WV c(D) is consistent. Then A has a unique extension. Theorem 2.5. Suppose that A = (D, W> is a normal default theory and that D’ c D. Suppose further that E; and E; are distinct extensions of (D’, W). Then A has distinct extensions E, and Ez such that El c E, and E; c E,.
3. Monotonic and nonmonotonic rule systems In this section we recall the basic definitions of monotonic and nonmonotonic rule systems from [13, 163. Tarski [28] characterized monotonic formal systems by means of monotonic rules of inference. Such systems include intuitionistic logics, classical logics, modal logics, and many others. Suppose that a nonempty set U is given. In a particular application U may be the collection of all formulas of a propositional or first-order logic, of all legal strings of a formal system, or of all atomic statements as in logic programming. A monotonic rule of inference is a tuple r = (P, cp) where P = (aI, . . . , a,) is a finite (possibly empty) list of objects from U and cpis an element of U. Such a rule r is usually
215
A context for belie1 revision
written
in the suggestive al,
form
a”
.‘.,
(3)
Y=
.
cp
We call fxl , . . . , a, the premises
of r and cp the conclusion
of r.
Definition 3.1. (a) A monotonic formal system is a pair (U, M), where U is a nonempty set and M is a collection of monotonic rules. (b) A subset S c U is called deductively closed over (U, M) if for all rules r=
Ml,
. ..>
%
EM,
al,
....
(x,ES implies
cpeS.
cp Inspired
by Reiter [26], and Apt [3], we introduced
the notion
of a non-monotonic
system (U, N) in [13, 161. A nonmonotonic rule of inference is a triple (p, R, cp>, where P = {ul, . . . . an}, R = {PI, . . . . Pm} are finite lists of objects from U and cp~U. Each such rule is written in a more suggestive form as
formal
r=
al> . . . . u,: 81, .“> Pm
(4)
cp
Here c1i, . . . . CI, are called the premises of rule r, PI, . . . , /3,,,are called the constraints of rule r, and cp is called the conclusion of rule r. For any rule r as in (4), we shall write prem(r) = (a,, . . . . a,}, cons(r) = {j3,, . . . . pm}, and c(r) = cp. Either prem(r), or cons(r), or both may be empty. If prem(r) = cons(r) = 0, then the rule r is called an axiom. A nonmonotonic rule system is a pair (U, N ), where U is a nonempty set and N is a set of nonmonotonic rules such that prem(r), cons(r), and {c(r)} are subsets of U for all rEN. Each monotonic formal system can be identified with the nonmonotonic which every monotonic rule has an empty set of constraints. A subset S s U is called deductively closed if for all r=
El, .‘., a,:
Bl,
cp
...,
Pm
system in
EN,
whenever all the premises cx1, , a, of r are in S and all the constraints r are not in S, then the conclusion cp of r belongs to S. In nonmonotonic systems, deductively closed sets are not generally
/11, . . . . /?,,,of closed under
arbitrary intersections as in the monotone case. But deductively closed sets are closed under intersections of descending chains. Since U is deductively closed, by the Kuratowski-Zorn Lemma, for any I E U, there is at least one minimal deductively closed superset of I. Given sets S E U and I s U, an S-deduction of cp from I in (U, N) is a finite sequence (cp 1, . . . , qk) such that (Pk = cp and, for all i < k, each Cpiis in I, or is an axiom, or is the conclusion of a rule rEN such that all the premises of r are included
276
V. W. Marek et al.
in (cpl,.... vi-
1 } and
all constraints
S-consequence of I is an element
of r are in U - S (see [19],
of U occurring
in some S-deduction
also [25]).
An
from I. Let C,(Z)
be the set of all S-consequences of I in (U, N ). Clearly I is a subset of Cs( I). However, note that S enters solely as a restraint on the use of the rules imposed by the constraints in the rules. A single constraint in a rule in N may be in S and thus prevent the rule from ever being applied in an S-deduction from I, even though all the premises
of that rule occur earlier
in a deduction.
Thus S contributes
no members
directly to C,(Z), although members of S may turn up in C,(Z) by an application of a rule which happens to have its conclusion in S. For a fixed S, the operator Cs( .) is monotonic.
That is, if I c J, then C,(Z) c C,(J).
Also, C,(C,(I))
= C,(I).
However,
for fixed I, the operator C,(Z) is anti-monotonic in the argument S. That is if S’ E S, then C,(Z) s C,*(Z). Generally, C,(I) is not deductively closed in (U, N ). It is perfectly possible that all the premises of a rule be in C,(I), the constraints of that rule are outside C,(I), but a constraint of that rule be in S, preventing the conclusion from being put into C,(Z). Example
3.1. U = {CI,8, y}, N = ($, y>,
S = {/I}. Then
C,(O) = (cz> is not deduct-
ively closed. However, Proposition
the following 3.2.
holds:
If S G C,(I),
then C,(I)
is deductively closed.
We say that S c U is grounded in I if S E C,(I). We say that S s U is an extension of I if C,(Z) = S. The notion of groundedness is related to the phenomenon of “reconstruction”. S is grounded in I if all elements of S are S-deducible from I (remember that S influences only the negative sides of rules). S is an extension of I if two things happen. First, every element of S is deducible from I, that is, S is grounded in I (this is an analogue of the adequacy property in logical calculi). Second, the converse holds: all the S-consequences of I belong to S (this is the analogue of completeness). Thus extensions are analogues for a nonmonotonic systems of the set of all consequences for monotonic systems. Both properties (adequacy and completeness) need to be satisfied-if we want S to be an extension. The notion of an extension is related to that of a minimal deductively closed set. Indeed, the following propositions are proved in [13]. Proposition 3.3. Zf S is an extension of I, then: (1) S is a minimal deductively closed superset of I. (2) For every Z‘ such that Z E Z‘ G S, Cs( I ‘) = S. Proposition 3.4. The set of extensions of I forms an antichain. That is, if S,, Sz are extensions of I and S1 E S2, then S1 = Sz.
211
A context for belief rel;ision
With each rule r of form (4), we associate
obtained
from r by dropping
a monotonic
all the constraints.
rule of form (3)
Rule r’ is called the projection
of rule
r. Let NG (S, Y) be the of all S-applicable rules. That is, a rule r belongs to NG (S, 9) if all the premises of r belong to S and all constraints of r are outside of S. We write M(S) for the collection of all projections of all rules from NG(S, 9’). The projection (U, N ) Is is the monotone system (U, M(S)). Thus (U, N ) Is is obtained as follows: First, non-S-applicable rules are eliminated. Then, the constraints are dropped altogether. We have the following characterization theorem. Theorem 3.5. A subset S c U is an extension deductive closure of I in (U, N )is.
of I in (U, N)
if and only if S is the
For the rest of this paper, we shall only consider extensions of 8 unless explicitly stated otherwise. We say that T is an extension of Y if T is an extension of 0 in 9. We shall end this section by giving yet another characterization of extensions. For this we need the concept of a proof scheme. A proof scheme for cp is a finite sequence P = (
ro,
GO),
. . . . (cp,,
rm,
G,>>
(6)
such that (P,,, = cp and (1) If m = 0 then: (a) cpo is an axiom Go = 0, or (b) cp is a conclusion
(that is, there exists a rule reN of a rule r =
(2) If m > 0, ((Cpi, ri, Gi))yCol
‘b:..“br,
such that r = $), r. = r, and
r0 = r, and Go = cons(r).
is a proof scheme of length m and (P,,,is a conclusion
of
where io, . . . . i, < m, r, = r, and G, = G,_ 1 u cons(r). The formula qrn is called the conclusion of p and is written cln(p). The set G, is called the support of p and is written supp(p). The idea behind this concept is as follows. An S-derivation in the system (U, N ), say p, uses some negative information about S to ensure that the constraints of rules that were used are outside of S. But this negative information is finite, that is, it involves a finite subset of the complement of S. Thus, there exists a finite subset G of the complement of S such that as long as G n Si = 0, p is an S1 -derivation as well. Our notion of proof scheme captures this finitary character of S-derivation. We can then characterize extensions of (U, N) as follows.
V. W. Marek et al.
278
Theorem 3.6. Let Y = (U, N > be a nonmonotonic rule system and let S c U. Then S is an extension of 9 if and only if (i) for each cp~S, there is a proof scheme p such that cln(p) = cpand supp(p) A S = $79, and
(ii) for
each
cp$ S, there
is no proof
scheme
p such
that cln(p) = cp and
SUPP(P) c-l s = 0.
There is a natural utilize.
Given
preordering)
preordering
a proof
scheme
proof scheme
is always
a minimal
with the same conclusion.
recursion-theoretic considerations. more detail in [13, 161
4. FC-normal nonmonotonic
of proof schemes according p, there
The concept
to the set of rules they (with respect
to that
This fact will be used in our
of minimal
proof scheme is treated
in
rule systems
In this section we shall define forward chaining-normal nonmonotonic rule systems (FC-normal nonmonotonic rule systems) and state the analogues of all the theorems of the previous section. We shall postpone the proof of all theorems in this section until Section 7. Let 9 = (U, N ) be a nonmonotonic rule system. Let man(Y) be the set of all rules rEN such that r has no constraints. Thus man(Y) = {rEN :cons(r) = 0}. We let nmon(Y) = N - mon(9’). We shall refer to man(Y) as the monotonic part of Y and nmon(Y) as the nonmonotonic part of 9’. We say a set W c U is monotonically closed if whenever r = -Emon and CI~,. . . . CI,,EW, then YEW. Given any set A G U, the monotonic-closure of A, written cl,,,(A), is defined to be the intersection of all monotonically closed sets containing A. It is easy to see that such a set is itself monotonically closed. So far our investigations of nonmonotonic rule systems did not get beyond the information established already in [13, 161. Next we introduce the notion of consistency property over (U, N ), which leads us to the main subject of this paper. We say that a subset Con E 9(U) (where Y(U) is the power set of U) is a consistency property over 9’ = (U, N ) if 1. @Con; 2. VA, B E U(A E B A Con(B) * Con(A)); 3. VA z U (Con(A) * Con(cl,,,(A))); 4. whenever A! c Con has the property
that
then Con( U JZ?). Condition 1 says that the empty set is consistent. Condition 2 requires that a subset of a consistent set is also consistent. Condition 3 postulates that the closure of a consistent set under monotonic rules is consistent. Finally, the last condition says
A context for b&f
of a directed family of consistent
that the union
219
revision
sets is also consistent.
We note that
conditions 1, 2, and 4 are Scott’s conditions for information systems. Condition 3 connects “consistent” sets to the monotonic part of the rule system; if A is consistent then adding
elements
derivable
from A via monotonic
ency property r=
is FC-normal
rules preserves
“consistency”.
rule system and let Con be a consist-
Now suppose ,Y = (U, N ) is a nonmonotonic over (U, N ), Then we say a rule @l>
‘..>
/31,. .. . Bk
&I:
E nmon(Y)
Y
(with respect
to Con) if Con(Vu
{r})
and not Con(Vu
{y, pi}) for all
i d k whenever V G U is such that Con(V), clmO,,(V) = V, c(i, . . . . C(,EV, and 1,. . . , pk 4 V. We say that Y = (U, N ) is a FC-normal (with respect to Con) if all Y>a rEnmon(.Y) are FC-normal with respect to Con. Finally, we say that (U, N) is FC-normal nonmonotonic rule system if for some consistency property Con s P(U), (U, N ) is FC-normal with respect to Con. Example 4.1. Let U = {a, b, c, d, e, j>. Let the consistency following condition: A $ Con
if and only if
either
(2) ;,
(3) ;,
be defined by the
{c, d } c A or {e, f } c A.
Thus {u,b,c,e}, {u,b,c,f), {u,b,4e},and {u,b,Lfj which are in Con. Now consider the following set of rules, N: (1) ;,
property
(4) q,
are the maximal
subsets of Y(U)
(5) 5
Then for the nonmonotonic rule system Y = (U, N ), rules (l), (2), and (3) form the monotonic part of Y and rules (4) and (5) form the nonmonotonic part of 9. First it is easy to check that Con is a consistency property over 9. The monotonically closed subsets of P(U) which are in Con are {a}, (a, d}, {a, e}, {a,f}, {a, b, c}, {a, d, e}, {a, d,f}, {a, b, c, e}, and {a, b, c,f}. It is then easy to check that Y is FC-normal with respect to Con. Moreover, one can easily check that Y has a unique extension A4 = {a,b,c,e}. If we add to N the rule : to get a set of rules N’, then Con is no longer a consistency property over 9” = (U, N’) because {c}ECon but the monotonic closure of {c} relative to Y’ = (U, N ‘) which equals {a, b, c, d} is not in Con. If we add the rule y to N to form a new NRS 9” = (U, N”), Con will still be a consistency property over 9” = (U, N”) because the property of being a consistency property depends only on the monotonic part of the rule system. However 9” = (U, N”) is not FC-normal with respect to Con because r = 7 is not FC-normal with respect to Con. That is, for the monotonically closed set {a, b, c, e}, we have prem(r) E {a, b, c, e}, cons(r) n {a, b, c, e} = @, but cI,,,( {c(r)} u {a, b, c, e}) = {a, b, c, d, e} is not in Con.
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Finally if we add to N the rule 7 9”’ = (U, N”‘) is still FC-normal extensions,
M1 = {a, 6, c,e) and M2 = {a, b, c,f).
We have the following Theorem
to get a set of rules N”‘, then the resulting NRS with respect to Con but now there are two
analogue
4.1. Let 9 = (U, N)
of Theorem
0
2.1.
be an FC-normal
nonmonotonic rule system with
respect to consistency property Con. Then there exists an extension of 9. It is easy to adopt
the proof of Theorem
4.1 to get the following
result.
Theorem 4.2. Let Y = (U, N ) be a normal nonmonotonic rule system with respect to consistency property Con and let I be a subset of U such that IECon. Then there exists an extension I’ of I in 9. In fact, we will show that there is a uniform construction of extensions of FCnormal NRS Y = (U, N > which depends on a well-ordering < of the nonmonotonic rules of 9, i.e. the rules in nmon(Y). We shall call this construction the forward chaining construction with respect to < (and this is the reason why we call our systems, for which this construction always succeeds in producing an extension, forward chaining-normal). To this end, fix some well-ordering < of nmon(Y). That is, the well-ordering < determines some listing of the rules of nmon(Y), {rn : LXE~}where y is some ordinal. Let 0, be the least cardinal such that y < 0,. In what follows, we shall assume that the ordering among ordinals is given by E. Our forward chaining construction will define an increasing sequence of sets {Eg<}aE~,. We will then define E< = uorso. Ed and show that ES is always an extension of Y. Moreover we shall show that all extensions
of Y arise from this construction.
The forward chaining construction of E < Case 0: CI= 0. Let E$ = cl,,,(O). Case 1: ~11 = y + 1 is a successor ordinal. Given that r1 = where E&I
a,,
.'., a,:B1,
E,<, let {(cc) be the least %EY such
.'.> Pk
*
CI~, ., CY,EE,~ and pl, . . . . Pk, Ic/$ E,+. = Ed = E,,,. Otherwise, let E6 1 = Eh = &,,(E~
If there
u (c(rc(,))>).
Case 2: CYis a limit ordinal. Then let Ed = U pEaED<. This given, we have the following.
is no
such
C(a), then
let
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Theorem 4.3. If Y = (U, N ) is an FC-normal well-ordering of nmon(Y), then 1. E X is an extension of Y. 2. (Completeness of the construction) a suitably
chosen ordering
It is quite straightforward with respect to consistency Thus the following
nonmonotonic
Every extension
rule system and < is any
of 9
is of the form E X for
< of nmon(9’). to prove by induction property
is an immediate
that if 9’ = (U, N ) is FC-normal
Con, then E,< icon consequence
for all CIand hence E -Xicon.
of Theorem
4.3(2).
Corollary 4.4. Let Y = (U, N) be an FC-normal nonmonotonic rule system respect to consistency property Con, then every extension of Y is in Con.
with
Example 4.2. If we consider the final extended program of Example 4.1, it is easy to check that any ordering <1 in which the rule r1 = $ precedes the rule r2 = “4 will have E ),,a
in stages. This given, we will then define E-X = u,,,
E,<.
The countable forward chaining construction of E < Stage 0. Let E$ = cl,,,(@). Stage n + 1. Let L(n + 1) be the least SEO such that
where c(i) . ., a&E,< and pi, . . . . Pk, $4 E,+. If there is no such /(n + l), then let E,? 1 = E,<. Otherwise, let
This given, we then have the following. Theorem 4.5. If 9 = (U, N ) is a countable FC-normal nonmonotonic rule system, then 1. Es is an extension of Y if E-X is constructed via the countable forward chaining algorithm with respect to < where < is any well-ordering of nmon(Y) of order type w.
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2. (Completeness
Every extension of 9’ is of the form Es
of the construction)
for
a suitably chosen well-ordering < of nmon(Y) of order type o where E + is constructed via the countable forward chaining algorithm. FC-normal
NRS’s also possess the “semi-monotonicity”
Theorem 4.6. Let Y,
= (U, N,)
that N1 E Nz but mon(Y1)
and Y,
property.
= (U, N2) be two FC-normal NRS such
= mon(YP2). Assume, in addition, that both are FC-normal
with respect to the same consistency property. Then for every extension El of Y1, there is an extension E, of Yz such that 1. El z E2 and 2. NG(E,,
YI)
z NG(E,,
Here given and extension rules, i.e., r=
Yz). E, we let NG(E, 9’) denote
the set of all E-applicable
El, ‘.., q.l: PI, .‘.> Bk *
is in NG(E, 9) if and only if a,, . . . , LX~EEand fll, . . . , flk are not in E. FC-normal NRS’s also satisfy the orthogonality of extensions property with respect to their consistency property. That is, we have the following analogue of Theorem 2.3. Theorem 4.7. Let Y = (U, N ) be an FC-normal NRS with respect to a consistency property Con. Then if E, and E2 are two distinct extensions of 9, El v E, 6 Con. Similarly
we also have the following
analogue
Theorem 4.8. Let Y = (U, N) be an FC-normal property Con. Suppose that cl,,,(c(r):rEnmon(Y)} extension.
of Theorem
2.4.
NRS with respect to a consistency is in Con. Then Y has a unique
We now have two more results which are also analogues of the results of Reiter’s [26]. We say that cpeU has a consistent proof scheme with respect to a consistency property Con over Y = (U, N) if and only if there is a proof scheme P = (
ro,
GO),
. . . . (cp,,
r,,,,
such that (P,,, = cp and (cpo, . . . . cp,)ECon.
G,))
(7)
We then have the following.
Theorem 4.9. Let Y = (U, N } be an FC-normal NRS with respect to a consistency property Con. Then cp~U is an element of some extension of Y if and only if cp has a consistent proof scheme with respect to Con.
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283
Theorem 4.10. Suppose Y = (U, N ) is an FC-normal NRS and that D G nmon(Y). Supposefurther that E; and E; are distinct extensions of (U, D u man(Y)). Then Y has distinct extensions El and E, such that E; c E, and E; G EZ.
5. FC-normal nonmonotonic
rule systems and other nonmonotonic
reasoning
formalisms In this section, we shall explicitly translate the definitions and theorems of the previous section into the language of logic programming, default logic, and truth maintenance
systems.
5.1. Logic programming, general case Now because general logic programs are probably the most widely studied type of nonmonotonic reasoning we shall take some time to give the translation of FCnormal nonmonotonic rule systems and the results above into the language of logic programming. A similar translation can be done for all other nonmonotonic formalisms to follow but we shall not carry out the translation in detail in the other cases. A general program clause is an expression of the form C=ptq, where p, ql,
,...,
qn,irl
,..,,
ir,
(8)
, qn, rl, . , r, are atomic formulas
possibly
with variables
in some first-
order language 2. A program is a set of clauses of the form (8). A clause C is called a Horn clause if m = 0. We let H(P) denote the set of all Horn clauses of P. XP is the Herbrand base of P, that is, the set of all ground atomic formulas of the language of P. ground(P) is the set of ground Herbrand substitutions of clauses in P. Given a set MGX’~, the Gelfond-Lifschitz [9] reduct of P, PM is the set of ground Horn clauses p +--ql, . . . . qn such that for some rl, . . . . r,$ M, the clause pegI,..., qn,~r~,...,~rm~ground(P).Misca11edastab1emode1ofPifMcoincides with the least model of PM. Assign to a ground clause p c ql, . . . , q,,, lrl, . . . . lr,Eground(P) the rule r(C)
=
41, . . . . qn:rl,
....
rrn.
(9)
P Let r(P) = (yi”p, {r(C):CEground(P)}). model of P if and only if M is an extension
Then, as shown of r(P).
in [13], M is a stable
This given, it is easy to see that sP plays the role of the universe U and ground(P) plays the role of the set of rules N in Section 4. Moreover, the Horn part of ground(P), i.e. ground(H(P)), plays the role of the monotonic part of the rules N. Then of course the immediate provability operator associated with H(P), THcpj (cf. Apt [3]) is monotonic.
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In this setting
the notion
of a consistency
a family of subsets of yi”p, Con, a consistency conditions:
property
becomes
the following.
Call
property over P if it satisfies the following
1. (bECon. 2. If A c B and BECon, then AECon. 3. Con is closed under directed unions. 4. If AECon Conditions
then A u T,(,,(A)ECon. l-3
nects “consistent” adding prove:
atoms provable
Proposition TWP)
are Scott’s conditions
5.1.
for information
systems.
sets of atoms to the Horn part of the program; from A preserves
If Con is a consistency
“consistency”.
property
Condition
if A is consistent
The following
4 conthen
fact is easy to
with respect to P and A~con,
then
II 4AWon.
Here, for a program Q, T, fi o(A) is the cumulative fixpoint of T, over A. Proposition 5.1 says that our condition 4 in the definition of consistency property implies that the cumulative closure of a “consistent” set of atoms under THcpj is still “consistent”. Here T, fi w(A) is the analogue of the monotonic closure of the set A. Given a consistency property, we define the concept of an FC-normal program with respect to that property in analogy with our definition of FC-normal NRS. Definition 5.2. (a) Let P be a general program, let Con be a consistency property with respect to P. Call P FC-normal with respect to Con if for every clause such that CEground(P) - ground(H(P)), and for C = p + ql, . . . . qn, lrl, . . . . lr, every consistent
fixpoint
A of THcpl, if ql,. .., q,,EA, p, rlr . . . . r,$ A we have:
(1) Au (p)ECon, (2) Au(p,ri}$Con for all 1
Example
4.1 into the language
property
of general
Con such that P is
logic programs.
Example 5.1. Let the Herbrand base consist of atoms a, b, c, d, e,f: Let the consistency property be defined by the following condition: A # Con Now consider (I) a +,
if and only if
either
{c, d) 5 A or {e, f} s A.
this program: t-4 b + c,
(4) c t a, 7 d,
(3) c + b,
(5) e +- c, if:
This program is FC-normal with respect to the consistency property described above and one can easily check that there is a unique stable model M = {a, b, c, e}.
A context for belief revision
If we add to this program the clause f t c, FC-normal but now there are two stable
285
the resulting models,
program
is still
M1 = {a, b, c, e}
and
Mz = {a, b, c,f}. We now have the following
analogues
of Theorem
4.1 and Theorem
4.2.
Theorem 5.3. Zf P is an FC-normal program, the P possesses a stable model. Theorem 5.4. If P is an FC-normal program with respect to the consistency property Con and IECon, then P v {i +- : ie I} possesses a stable model. The analogue of our forward chaining construction of extensions of an FC-normal NRS becomes the following. Let < be a well-ordering of ground(P) - ground@(P)). That is, the well-ordering < determines some listing of the clauses of c,: GIE~} where y is some ordinal. Let 0, be the least ground(P) - ground(H(P)), { cardinal such that y < 0,. Our forward chaining construction will define an increasing sequence of subsets of Ye,, {T>jaeo,. This given, we will then define T+ = UasO,Tb and show that T < is always an stable model of P. Moreover, we shall show that all stable models
of P arise from this construction.
The forward chaining construction of T< Case 0: a = 0. T,jj = Tn(r) fi o(0). Case 1: CI= ye+ 1 is a successor ordinal. Given T,jj, let /(cr) be the least il~y such thatc,=cpca,,..., a,,, 1 pl, . , lb,,, where CI~,. . . , CI,ET,’ and fil, . . . , fl,,, , q 4 T,‘. If there is no such /(cc), let Tb = T,“. Otherwise let
where pfcnJ is the head of c/(,). Case 2: a is a limit ordinal, Then let Td = UBEa T> We then get: Theorem 5.5. If P is an FC-normal program and < is any well-ordering of ground(P) - ground(H(P)), then: (1) T< is a stable model of P. (2) (Completeness of the construction) Every stable model model of P is of the form T< for a suitably chosen ordering i of ground(P) - ground(H(P)). Example 5.2. If we consider the final extended program of Example 5.1, it is easy to check that any ordering <1 in which the clause C1 = e + c, if precedes the clause C2 = f c c, 1 e will have T
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Once again, we note that if we restrict ourselves to countable programs P, then we can restrict ourselves to orderings of order type o. That is, suppose we fix some well-ordering
i
of ground(P)
- ground(H(P))
of order
type o. Thus,
the well-
ordering < determines some listing of the clauses of ground(P) - ground(H(P)), can be presented in {c n : neo}. Again, in this case, our forward chaining construction a more straightforward
manner.
Our construction
will define an increasing
of sets (T,‘i jllEW in stages. This given, we will then define T< = u,,,, The countable forward chaining construction
sequence
T,,, .
of T<
Stage 0. Let T$ = THcpIfl o(8). Stage
n + 1. Let /(n + 1) be the least SEW such that c, = cp c c~i, . . . . ak, where CQ, . . . . QE Tnx and /II, . . . . Pm, cp$ T$. If there is no such /(n + l), let Tn3 1 = T,,, . Otherwise let
1
PI, . . . . ifim
where pLcn+1j is the head of ctcn+ r). Theorem 5.6. Zf P is a countable
FC-normal
program,
and < is any well-ordering
ground(P) - ground(H(P)) of order type w, then: (1) T + is a stable model of P where T< is constructed
of
via the countable forward
chaining algorithm. (2) (Completeness of the construction) Every stable model of P is of the form T< for a suitably chosen ordering < of ground(P) - ground(H(P)) of order type o where T< is constructed via the countable forward chaining algorithm. Theorem 5.7. If P is an FC-normal model M of P is in Con. FC-normal
programs
logic program with respect to Con, then every stable
possess the “semi-monotonicity”
property.
such that P, G P2 but Theorem 5.8. Let PI, Pz be two general programs H(P,) = H(P,). Assume, in addition, that both are FC-normal with respect to the same consistency property. Then for every stable model MI of PI, there is a stable model M2 of P2 such that 1. M1sM2and 2. NG(M1, P,) E NG(M2, Pz). Here given a logic program P and a stable model M, we let NG (M, P) equal the set of all clauses c = cp c txl, . . . . &, i pl, . . . . ip,,, in ground(P) such that ~1~). . . . cxkEM and PI, . . . . P,,,$M. This is a very useful result. That is, if we consider our paradigm of the robot who moves are determined by a hardwired logic program as described in the introduction, this results tells us that if the robot is operating with respect to certain belief set or
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A context for belief revision
point of view and subsequently new clauses, FC-normal with respect to the consistency property ruling the behavior of the robot, are added, then the beliefs at that point do not need to be recomputed, just new point of view extending the current point of view can be formed. Adding new Horn rules, however, may require backtracking. The reason is that we may have chosen a belief explicitly contradicting these new facts. The analogues of the remaining theorems of Section 4, also hold for FC-normal logic programs. Theorem
5.9. Let P be an FC-normal
property Con. Then ifE,
logic program
with respect
to a consistency
and E2 are two distinct stable models of P, then El v E2 $ Con.
Theorem 5.10. Let P be an FC-normal program with respect to a consistency property Con. Suppose that To o( {h ea d( c ): CEg round(P) - ground(H(P)))) is in Con wherefor any clause, c, head(c) denotes the head of the clause. Then P has a unique stable model. Theorem 5.11. Suppose P is an FC-normal logic program and that D c ground(P) - ground(H(P)). Suppose further that E; and E; are distinct stable models of the program of ground(P) u D. Then P has distinct stable models El and E2 such that E; G El and E; G EZ. We need to define the notion of proof scheme for p with respect to P is a sequence of triples number, such that the following conditions all 1. Each pt is in ZP. Each Cl is in ground(P). 2. pn is p. 3. The S[, C1 satisfy the following conditions. below holds. (a) Crisp,c,andS,isS,_,, (b) Cl is pt+-lsl,..., is, and SI is S1_r (c) CL is pl + pm,, . . . . pmr, SL-1 u (s1, . . . . s,}. (We put SO = 8.)
is1,
for a logic program P. A proof scheme (( pI, CI,SI))l $tGn, with n a natural hold. Each Sr is a finite subset of Ye,. For all 1 d 1 d n, one of (a), (b), (c)
u {sl, . . . . s,}, or m,
and
SI
is
is a proof scheme. Then cln(cp) denotes the Suppose that cp = (
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5.2. Default logic In this subsection, we shall translate our results of Section 4 back into the language of default logic. We shall start with the translation between default logic and nonmonotonic rule systems as described in [13] and [16]. Let U be the collection of all formulas of propositional logic 9. Recall a default theory (D, W) is a pair where D is a collection of default rules, that is, rules of form c(:MP,, . . . . Mp,,, *
(10)
’
(where ~1,PI, . . . . Pm, and II/ are formulas) and W a collection of formulas language 9. Represent such a default theory as a rule system consisting of three lists: (i) Elements YEW are represented as rules:
(ii) Rules of form (10) are represented
of the
as
Y
(That is, the restraints of the rule representing a default rule r have an additional negation in front.) (iii) Processing rules of logic. That is, all the monotonic rules of the system of classical logic. We then have the following proposition from [13]:
Proposition 5.13. A collection S c U is an extension of a system consisting of rules of types (i), (ii), and (iii) if and only if S is a default extension of (D, W). Given a default rule r as in (lo), we let prem(r) = {a}, cons(r) = {l/II, . . . . l/I,,,}, and cln(r) = t,b.If m = 0, then we say that r is a monotonic rule and otherwise we will say that r is a nonmonotonic rule. We let mon( (D, W)) denote the set of monotonic rules of (D, W) and nmon((D, W)) d enote the set of monotonic rules of (D, W). We say that a subset S c 9’ is monotonically closed relative to (D, W), if WCS, Cn(S) = S and for any monotonic rule $ in (D, W), it is the case that $ES if ads. Thus a set S containing W is monotonically closed relative to (D, W), if S is closed under the application of all monotonic rules of (D, W) as well as being closed under logical consequence. It is easy to see that the intersection of any two monotonically closed sets relative to (D, W) is also monotonically closed relative to (D, W) so that for any set T G Y”, there is a smallest set S which contains T and is monotonically closed. We let cl,,,(T) denote the smallest monotonically closed set relative to (D, W) which contains T.
A context
for belief revision
Call a family of subsets of 9, Con a consistency following conditions: 1. 2. 3. 4.
289
property for (D, W) if it satisfies the
Qkcon. If A E B and BECon then AECon. Con is closed under directed unions. If AECO~ then cl,,,(A)~Con.
Given a consistency property Con, we say that a default rule as in (10) is FC-normal with respect to Con if for any theory AE Con such that CX~Aand $, 1 pi, . . . ,l p,,,, are not in A, then Cn(A u {$))~Con
but Cn(A u {tj, 1/3i))$
Con for any i. Then we say
that a default theory (D, W) is FC-normal with respect to Con if each rule rED is FC-normal with respect to Con. Our next result will show that our definition of FC-normal default theories actually extends Reiter’s original definition of normal default theories. Theorem 5.14. Every normal default theory (D, W> is an FC-normal
default theory.
Proof. First observe that since every rule in a normal default theory is of the form 7, there are no monotonic rules in (D, W). Thus a set S is monotonically closed relative to (D, W) if and only if Cn(S u W) = S (that is W c S and Cn(S) = S). There are two cases. First, suppose that W is a logically consistent set of formulas. In this case, it is easy to see that the set of logically consistent subsets S of the language 97 such that WV S is logically consistent is a consistency property for (D, W). We note that if Con is just the set of logically consistent sets S in ~3’ such that S u W is logically consistent, then every rule of the form y is certainly FC-normal. That is, no consistent set can contain both p and l/? and if A is a consistent theory which contains Wsuch that neither /I, 1,!3 are in A, then Th(A u {a}) is also consistent. Thus if W is logically consistent set, then (D, W) is FC-normal relative to the consistency property consisting of all logically consistent sets S such that S u W is logically consistent. Second, if W is not logically consistent, then it is easy to see that for any set SC 9, &,,(S) = Cn(S u W) = 2. Thus in this case, the only possible consistency property with respect to (D, W) is the set of all subsets of 9. Moreover, the only monotonically closed set is 9. But then every rule of the form y is FC-normal since there is no monotonically closed set T such that both /I and l/3 are not in T. Thus if W is logically inconsistent, then (D, W) is FC-normal with respect to the consistency property consisting of all subsets of 9. q Of course, it is easy to see that there are many FC-normal default theories which are not normal default theories since in FC-normal default theories we allow monotonic rules and we do not restrict nonmonotonic rules to be of the form y. We then have the following analogues of the results of Section 4. Theorem 5.15. Let (D, W) be an FC-normal default theory with respect to consistency property Con, then there exists an extension of (D, W).
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Theorem 5.16. Let (D, W) be an FC-normal default theory with respect to consistency property Con and let IECon. Then there exists an extension 1’ of (D, W u I >. The analogue is the following. <
of the forward chaining construction Given an FC-normal default theory
of D. That
is, the well-ordering
<
determines
for FC-normal default theories (D, W), fix some well-ordering some listing
of the rules of D,
{r,: cq} where y is some ordinal. Let 0, be the least cardinal such that y d 0,. In what follows, we shall assume that the ordering among ordinals is given by E. Our forward
chaining
construction
will define an increasing
The forward chaining construction of E < Case 0: M= 0. Let E$ = c&,,~,,(W). Case 1: c( = n + 1 is a successor ordinal. that rA =
at,...,
E,,,
of sets {E>}aoB,.
let e(a) be the least il~y such
up: Ml%, . . . . MA IL
where c1r, . . . , c+EV+ and 1 bl, . . . , lPk, E&i = E,,, = E,<. Otherwise, let E&l
Given
sequence
= Ed = &,,(EV~
$ $ E,,< If there is no such /(cc), then let
u {cWre(,,)>).
Case 2: o! is a limit ordinal. Then let Ed = u psa.EB<. This given, we have the following. Theorem 5.17. If (D, W> is an FC-normal default theory and < is any well-ordering of D, then 1. E < is an extension of (D, W). 2. (Completeness of the construction) Every extension of (D, W) is of the form E X for a suitably chosen well-ordering i of D. Corollary 5.18. Let (D, W) be an FC-normal default theory with respect to consistency property Con. Then every extension of (D, W> is in Con. If we restrict ourselves to countable default theories
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A context for belief rtwision
The countable forward chaining construction of E X Stage 0. Let E$ = cl,,,(W). Stage n + 1. Let !(n + 1) be the least SEO such that rs =
czl, . . . . c(~:Mbk, . . . . M/3,‘ *
where c(~,. . . . QEE,+ andip,, E,5 1 = Ens. Otherwise, let
. . . . 1 bk, $4 E,< If there is no such [(PI + l), then let
En3 I = &on(En~ u {cW/c,+ We then define E+ = u,,,
d>).
E2.
This given, we then have the following. Theorem 5.19. If (D, W) is a countable FC-normal default theory then 1. E < is an extension of (D, W) where E + is constructed via the countable forward chaining algorithm with respect to <, where < is any well-ordering of D of order type w. 2. (Completeness of the construction) Every extension of (D, W> is of the form E < for a suitably chosen well-ordering < of D of order type o, where E < is constructed via the countable forward chaining algorithm.
The “Semi-monotonicity”
property holds for FC-normal default theories.
Theorem 5.20. Let AI = (DI, W) and A2 = (D2, W) be two FC-normal default theories with respect to a consistency property Con such that mon(A,) = mon(Az) and nmon(A,) s nmon(Az). Then for every extension E, of (DI, W>, there is an extension E2 of (D2, W) such that 1. E, c Ez and 2. GD(E,, A,) c GD(E2, AZ).
FC-normal default theories also satisfy the orthogonality with respect to their consistency property.
of extension property
Theorem 5.21. Let (D, W) be an FC-normal default theory with respect to the consistency property Con. Then, ifEl and E, are two distinct extensions of (D, W>, then El u E, 4 Con.
Similarly we have the following analogues of Theorems 2.4 and 2.5. Theorem 5.22. Let (D, W) be an FC-normal default theory with respect to a consistency property Con. Suppose that Wu {cln(r) : rED) is in Con. Then (D, W) has a unique extension.
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Theorem 5.23. Let
Then (D, W) has distinct extensions El and E, such that
E; c El and E; E Ez. of Theorem
4.9. First
we assume
underlying logic has some proof systems consisting and finitely many rules ol, . . . . ok. An (annotated) sequence
Finally
there
is also
an analogue
of finitely
many
proof
scheme
axiom
that
the
schema
for q is a finite
P = ((cpO, ro, Go>, . . . . (cp,, r,, G,))
(11)
such (1) (a) (b) (2)
that (P,,, = cp and If m = 0 then: cpo is an instance of an axiom schema for 9 or (POEW, r. = qo, and Go = 8, or cp is a conclusion of a rule r = ‘MP’*;.*MBr, r. = r, and Go = cons(r). If m > 0, ((Cpi, ri, Gi))~Yol is a proof scheme of length m and either of r = ai’MBz;.‘*MBr where i < m, rm = r, and (a) qrn is a conclusion G, = G,_ 1 u cons(r), or (b) there is some i. < ... < i, < m such that (P,,,is the result of applying one of the rules of proof Bj to 4i0, . . . . $iS, rm = Bj, and G, = G,_ 1. The formula (P,,,is called the conclusion of p and is written cln( p). The set G, is called the support of p and is written supp(p). We say that cp~_Y’ has a consistent proof scheme with respect to a consistency property Con over (D, W) if and only if there is a proof scheme P=
<
ro,
GO),
. . . . (cp,,
rm,
such that (P,,, = cp and {cpo, . . . , cp,}ECon.
G,))
We then have the following.
Theorem 5.24. Let (D, W> be an FC-normal default theory with respect to a consistency property Con. Then cpis an element of some extension of (0, W) tfand only ifq has a consistent proof scheme with respect to Con,
5.3. Logic programming with classical negation We now discuss the so-called “logic programming with classical negation” of [lo] as a chapter in the theory of nonmonotonic rule systems. Recall the basic notions introduced in [lo]. The collection of objects appearing in heads or bodies of clauses is the set of all literals, that is, atoms or negated atoms. In particular, a negated atom may appear in the head of a clause. Consider first “general Horn” clauses in which literals may appear in arbitrary places. To each set P of such clauses assign its answer set, the least collection A of literals satisfying the following two conditions: (1) If a+ bI, . . . . b, is in P and bI, . . . . b&A then aEA.
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293
(2) If for some atom p, p and 1 p are both in A, then A is the whole collection Lit of all literals. Introduce a collection Str of structural processing rules over the set U = Lit. These are all monotone
rules of the form:
for all atoms p and literals Translate
a.
the clause: a c b, , . . . , b, as rule:
b 1, . . . . b,: a and let tr(P) be the collection Str. Then we have
of translations
of clauses in P plus the structural
rules
Proposition 5.25. A subset A c Lit is an answer set for P if and only if A is an extension of tr(P). Since tr(P) is a set of monotonic rules, such an answer set is the leastjxpoint of the (monotonic) operator associated with the translation. Gelfond and Lifschitz then introduce general rules. Since the negation used in literals is not the “negation-as-failure” of general logic programming, Gelfond and Lifschitz introduce another negation symbol “not” and a general logic clause with classical negation in the form: a c b,, . . . . b,, not(c,),
. . . . not(c,).
Then the answer set for a set P of clauses of this form is introduced by merging the operational procedure for the construction of stable models for a program (as introduced in [9]) with the procedure above. They define the answer set for a program with classical negation as follows: Let A4 c Lit and P be a general program. Define P/M as a collection of clauses lacking not obtained as follows: (1) If a clause C contains a substring not(a) where aeM, then eliminate C altogether. (2) In remaining clauses eliminate all substrings of the form not(a). The resulting program P/M lacks the symbol not, so the answer set is well defined. Let M’ be the answer set for P/M. We call M an answer set for P precisely when M’ = M. Gelfond and Lifschitz give a computational procedure for finding such answer sets, and subsequently reduce computing them to computing default logic extensions. Here we give a general result showing that the construction of Gelfond and Lifschitz is faithfully represented within nonmonotonic rule systems; here is how. Define U to be
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Lit, and translate
the clause:
a c bl, . . . . b,, not(q), . . . . not(c,) as the rule: b 1,..., b,:cI,
. . . . c,
a The translation of the program P then consists of the translations of individual clauses C of P, incremented by the structural rules Str. We get the following result: Proposition 5.26. Let P be a general logic program with classical negation and Np be the translation described above. Then a collection M is an answer set for P if and only ifM is an extension for the rule system (U, Np). Let Lit, = yi”pu {lp: p~yi”p>. It is easy to see that Lit, plays the role of the universe U and ground(P) u Str plays the role of the set of rules N in Section 4. Moreover, the not-free part of ground(P) together with the structural rules Str, plays the role of the monotonic part of the rules N. Define man(P) to be the the not-free part of P incremented by Str. The immediate provability operator, T,(M), associated with man(P) is monotonic. Moreover, if the input M contains a pair of contradictory literals then T,(M) is the whole set Lit,. In this setting the notion of a consistency property becomes the following. Call a family of subsets of Lit,, Con a consistency property over P if it satisfies the following conditions: 1. @ECon. 2. If A E B and BECon then AECon. 3. Con is closed under directed unions. 4. If AECon then A u T,,,&A)ECon. 5. Lit, # Con. We have, as before, Proposition 5.27. If Con is a consistency property with respect to P and AECon, then Tman(P) II4A k Con. fi o(A) is the cumulative fixed point of Here for a general logic program Q, TmoncpJ Tp over A. Given a consistency property, we define the concept of an FC-normal CN logic program with respect to that property in analogy with our definition of FC-normal NRS. Definition 5.28. (a) Let P be a CN logic program, let Con be a consistency property with respect to P. Call P FC-normal with respect to Con if for every clause C = p+ 41, . . . . qn, not(rl), . . . . not@,) such that CEground(P) - ground(mon(P)),
A context for belief revision
and for every consistent
A of TmoncP,,if ql, . . . . q,,gA, p, rl, . . . . r,$ A we have:
fixpoint
(1) Au {p}~Con, (2) AU{p,ri)~Conforall
1
(b) P is called FC-normal if there exists a consistency FC-normal with respect to Con. Our condition consistency
(5) implies
property
that there is a weakest
based on the the absence
We now have the following
analogues
Theorem 5.29. If P is an FC-normal
The
Con and IECon,
analogue
normal
NRS
of our becomes
property
consistency
Con such that P is
property.
of pair of complementary
of Theorems
This is the literals.
4.1 and 4.2.
CN logic program, then P possesses an answer set.
Theorem 5.30. Zf P is an FC-normal property
295
CN logic program
with respect to the consistency
then P u {i +- : iE I} possesses an answer set I ‘.
forward the
chaining following.
construction Let <
of extensions of an FCbe a well-ordering of
ground(P)
- ground(mon(P)). That is, the well-ordering < determines some listing of the clauses of ground(P) - ground(mon(P)), {c,: ME?} where y is some ordinal. Let 0, be the least cardinal such that y < 0,. Our forward chaining construction will define an increasing sequence of subsets of Lit,, {T,< },, @,. This given, we will then define
T< = U.0. , T;< and show that TS is always an answer set for P. Moreover, show that all answer sets for P arise from this construction. of TX
The forward chaining construction Case 0: IX= 0. T,jj = Tman(P)0
we shall
40).
Case 1: cx= ye+ 1 is a successor
ordinal. Given
T;‘,
let /(a) be the least AEY such that c1 = cp c gl, . . . . tl,, not(fil), . . . . not(b,) where CY~,. . . . CI,ET~< and fil ,..., pm, cp$ T,“. If there is no such /(a), let T,’ = T,‘. Otherwise let
where prcaj is the head of c/(,). Case 2: u is a limit ordinal. Then let Tb = u BEnTz. We then get: Theorem 5.31. If P is an FC-normal ground(P)
- ground(mon(P)),
CN logic program
1. T< is an answer sef for P. of the construction)
2. (Completeness
a suitably chosen ordering
and < is any well-ordering
of
then:
<
of ground(P)
Every answer set for P is of the form - ground(mon(P)).
T< for
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V. W. Marek et nl,
Once again, we note that if we restrict ourselves to countable programs P, then we can restrict to orderings of order type o. That is, suppose we fix some well-ordering < of ground(P) - ground(mon(P)) of order type o. Thus, the well-ordering 4 determines some listing of the clauses of ground(P) - ground(mon(P)), {c,,: n~o}. Again in this case, our forward ward manner.
chaining
Our construction
construction
can be presented
will define an increasing
stages. This given, we will then define TX = U,,, The countable forward chaining construction
in a more straightfor-
sequence
of sets (7’n<}.Bo in
T”<.
of T+
Stage 0. Let T,’ = Tmon(rjfi o(8). Stage n + 1. Let a(n + 1) be the least natural number s such that c, = cp +- al, . . . . uk, not, . . . . not(/?,) where c1r, . . . . akETn+ and /?r, . . . . fi,,,, cp$! T”<. If there is no such /(n + l), let T> 1 = T:. Otherwise let K< I = THVJ 0 dT,z~ u {P/W+ I,>, where P~(,,+~) is the head of c~(~+i). Theorem 5.32. If P is a countable FC-normal CN logic program, and < is any well-ordering of ground(P) - ground(mon(P)) of order type w, then: (1) T+ is a answer set for P where T < is constructed via the countable forward chaining algorithm. (2) (Completeness of the construction). Every answer set for P is of the form T< for a suitably chosen ordering < of ground(P) - ground(mon(P)) of order type o where T< is constructed via the countable forward chaining algorithm. Theorem 5.33. If P is an FC-normal CN logic program with respect to Con, then every answer set model M for P is in Con. FC-normal
programs
possess “semi-monotonicity”
property.
Theorem 5.34. Let P,, P2 be two CN logic programs such that PI E P2 but mon(P,) = mon(P,). Assume, in addition, that both are FC-normal with respect to the same consistency
property.
Then for every answer set MI for PI, there is an answer set
M2 for Pz such that
1. MI c M2 and 2. NG(M1, P,) z NG(Mp, P2). Here given a CN logic program P and an answer set M, we let NG(M, P) equal the set of all clauses c = (PC ml, . . . . &, not(P1),..., not@,) in ground(P) such that CY~ ,..., C(,&M and fll ,..., &$M. The analogues of the remaining theorems of Section 4, also hold for FC-normal logic programs with classical negation.
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Theorem 5.35. Let P be an FC-normal CN logic program with respect to a consistency property Con. Then, if El and E2 are two distinct answer sets for P, then El u E, $ Con. Theorem 5.36. Let P be an FC-normal property
CN logic program
Con. Suppose that Tfi w( (head(c): cEground(P)
where for any clause c, head(c)
denotes
with respect to a consistency
- ground(mon(P))})
the head of the clause.
is in Con
Then P has a unique
answer set.
5.37. Suppose
Theorem
D G ground(P)
P
is
an
- ground(mon(P)).
sets for the program
of ground(P)
FC-normal
CN
Suppose further
logic
program
and
that
that E; and E; are distinct answer
u D. Then P has distinct answer sets E, and E, such
that E; 5 El and E; E E,.
To state the analogue of Theorem 4.9, we must define the notion of proof scheme for a logic program P. A proof scheme for p with respect to P is a sequence of triples <( pt, Cl, SI))16LSnr with n a natural number, such that the following conditions all hold. 1. Each pt is in Lit,.
Each C1 is in ground(P).
Each SL is a finite subset of Litr.
2. pn is p.
3. The SI, C1 satisfy the following below holds.
conditions.
For all 1 d 1 < n, one of (a), (b), (c)
(a) CL is pl +, and SI is Sl_,, (b) C1 is pt c not(sI), . . . . not(s,) and SI is S,- 1 u (sl, . . . . s,}, or (c) CI is pr+ pm,, . . . . pm*, not(s,), . . . . not(&), m, < 1, . . . . mk < 1, and SI is si_r u {Si, . ..) &>. (We put SO = 0.) is a proof scheme. Then cln(cp) denotes atom Swposethatcp = <
a consistent
5.4.
CN logic program with respect to a consistency
Con. Then (PE Lit, is an element of some answer set for P if and only if9
has
proof scheme with respect to Con.
Truth maintenance
systems
In this and the next subsection, we shall discuss two other nonmonotonic formalism which have appeared in the literature. In each of these cases, we shall be content to simply translate these formalisms into nonmonotonic rules systems and leave it to the reader to translate the definition of FC-normal and the statements of its various properties.
V. W. Marek et al.
298
Our description
takes care of both truth maintenance
[7] and De Kleer
[6], with subsequent
Brewka [25]. Let At be a collection
of atoms.
contributions
systems as defined by Doyle of Reinfrank,
sequence (al, . . . . a,) (1) a, = a. (2) For
M G At, an M-derivation
satisfying
every j < n, either
and
By a rule over At we mean a object of the form
Y = (A 1B) -+ c where A, B s At, ceAt. A truth maintenance is a collection of rules. Let S be a TMS. Given
Dressler,
system (TMS for short)
of an atom
aeAt
is a finite
the conditions: a rule
(8 10) + aj belongs
to S or there
is a rule
(A~B)+ajinSsuchthatA~{a,,...,aj-l},BnM=@ We call M a TMS-extension of S if and only if M has the property that M consists of precisely these atoms that possess an M-derivation. It is easy to reconstruct, in this setting, truth maintenance as logic programming. Namely, we can translate a rule (A 1B) + c as a program clause c +- a,, . . . . a,,,, not@,), . ..) not(b,) where A = {ai, . . . . a,} and B = {b,, . . . . b,}. In this fashion, TMS-extensions become stable models of the that one can also construct truth-maintenance To do this properly one considers (as in the negation) rules which involve literals and also
resulting programs. This also implies systems with literals. case of logic programs with classical “built-in” rules ((a, 1 a}, 0} + c. For
more about these, see [21]. The results on FC-normal logic programs as well as on FC-normal logic programs with classical negation allow us to prove analogues results for TMS, and also TMS with literals. We shall not pursue this matter further in this paper. 5.5.
McDermott
and Doyle systems
McDermott and Doyle [23] and McDermott [22] investigated another system of nonmonotonic reasoning. This system is based on modal logic. Here is a short description of that approach and the description how it fits into our framework. Let .4pL be the propositional modal language based on modal operator L (expressing the necessity operator). We consider a strong notion of proof based on the application of the necessitation rule to all formulas, not just all theorems, of the logic under consideration. That is, this notion of proof from a set of formulas I allows us to apply necessitation to all formulas previously proved. Let 9’ be a modal logic. Examples of such a logic include the familiar S4, S5, K or even a logic that does not include the schemes of K. We associate with Y its consequence operation based on the above strong notion of proof. We denote it by Cn& .). We now introduce the notion of Y-expansion. Given a set of formulas I s YL, we say that a theory T E 5?L is a Y-expansion of I if T=
Cny(lu{lLq::cp$T}).
(12)
Notice that the role of the logic 9’ here is slightly different than in the usual applications of modal logic. 9’ serves as means of reconstruction of T from the initial
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299
for belief revision
assumptions I and the negatioe introspection with respect to T. It should be clear that regardless of what .Y is (it does not even need to be included in S5) that an expansion of any theory is closed under SS-consequence. It is the discipline of reconstruction that makes the difference. Note the weaker the logic, the more difficult it is to reconstruct. We show now how this formalism can be faithfully represented as a nonmonotonic rule system. Let Y be a fixed modal
logic, axiomatized
define a rule system SY as follows. The universe N consists of the following five groups of rules: 1. $,
where cp ranges YL, treating
2. i, cp 3. 5
over all the axioms
every formula
where cp ranges
of propositional
AX. We
is _!YL. The set
logic in the language
of the form LIl/ as an atom.
over all the axioms
for all the formulas
by a set of axioms
U of our system
of the logic 9.
(PE_Y’~.
4 cp,cp~@ . ~ for all the formulas II/ :cp 5. ~ for all 9~9~. 1LV
cp, $EY~.
Notice that the groups 1, 2, 3, and 4 of rules are monotonic, consists of nonmonotonic rules. We have the following result.
only the group
Theorem 5.39. Let 9’ be a modal logic. Let I s $PL. Then T is an Y-expansion and only if T is an extension of I in the nonmonotonic rule system Sy.
5
of I if
This result allows us to apply the theory developed in this paper to Y-expansions. That is, we can introduce the notion of consistency property for a modal logic (notice that we have just one system for any given modal logic). If the system Sy is FC-normal for such property then, in particular, every set of formulas consistent with respect to that property
possesses
6. The complexity
an extension.
of extensions
for a recursive FC-normal NRS
This section will be divided into two subsections. First we shall introduce some preliminaries from recursion theory and various classes of recursive rule systems so that we can make our statements about the complexity of extensions for recursive FC-normal NRS’s precise. We shall also provide a brief review of what is known about the complexity of extensions in recursive nonmonotonic rule systems. In the second subsection, we shall state our new results about the complexity of extensions in recursive FC-normal nonmonotonic rule systems. The proofs of these new results will be deferred until Section 8.
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et al.
6. I. Preliminaries Let o denote the set of natural numbers. The canonical index, can(X), of a finite set x = {x1 < ..’ < x,) E o is defined as 2”’ + ... + 2”n and the canonical index of 0 is defined as 0. Let Dk be the finite set whose canonical index is k, i.e., can(Q) = k. We
shall
identify
a rule
Y with
a triple
(k, 1, cp) where
Dk = prem(r),
D, = cons(r), cp = c(r). In this way, when U E w we can think about
and
N as a subset of
w as well. This given, we then say that a NRS Y = (U, N ) is recursive if U and N are recursive subsets of w. Next ox
we shall define
various
o -+ o be a fixed one-to-one
types
of recursive
and onto
recursive
trees and pairing
IZ(: classes. function
Let [,I:
such that the
projection functions rci and 7r2 defined by rcl([x, y]) = x and rc2([x, y]) = y are also recursive. Extend our pairing function to code n-tuples for n > 2 by the usual inductive definition, that is, let [x1, . . . . x,] = [x1, [x2, . . . . x,]] for n 2 3. Let cecw be the set of all finite sequences from w, let 2’” be the set of all finite sequences of O’s and 1’s. Given CI= (a,, . . . . a,) and p = (fir, . , ljk) in cP”, write c( c b if (x is an initial segment of /3, i.e., if n 6 k and ai = pi for i 6 n. In this paper, we identify each finite sequence a = (a,, . . . . cr,,) with its code c(g) = [n, [a,, . . , a,]] in o. Let 0 be the code of the empty sequence 0. When we say that a set S c o<@ is recursive, recursively enumerable, etc., what we mean is that the set {C(U): UES) is recursive, recursively enumerable, etc. Define a tree T to be a nonempty subset of gco such that T is closed under initial segments. Call a functionf: w + o an infinite path through T provided that for all n, (f(O), . . . , f(n))E T. Let [T] be the set of all infinite paths through T. Call a set A of functions of I7:-class if there exists a recursive predicate R such that A = {f: o + w:Vn(R([f(O), . . ..f(n)])}. Call a Z7,0-class A recursively bounded if there exists a recursive function g: o + o such that V’EA V’,(f(n) 6 g(n)). It is not difficult to see that if A is a UF-class, then A = [T] for some recursive tree T c coCw. Say that a tree T E coCo is highly recursive if T is a recursive, finitely branching tree, and also there is a recursive procedure which, a canonical index of the set of immediate
applied to c( = (c(i) . . . . a,) in T, produces successors of a in T. Then if A is a recursively
nf-class, it is easy to show that A = [T] for some highly recursive tree T G wCw, see [12]. For any set A c o, let A’ = {e: {e}A(e) is defined} be the jump of A, let 0’ denote the jump of the empty set 0. We write A
bounded
to B and A EBB if A dTB and B GOB. Formally, say that there is an effective, one-to-one degree-preserving correspondence between the set of extensions 6(Y) of a recursive nonmonotonic rule system 9 =
= EfEcT(9), (i) VRCTI{eI >g*(J”) (ii) V’EEF(Y){e2}E =_&E[T], and v(f) = E if and only if (ez}E =f) (iii) Vjtz[T]VEd(Y)({e,} where {e}B denotes the function computed by the eth oracle machine Also, write {e}” = A for a set A if {e}B is a characteristic function
with oracle B. of A, and for
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a functionf: w -+ 0, gr(f) = { [x, f(x)]: the tree T uniformly produce extensions
XEO}. Condition
(i) says that the branches
via an algorithm
with index e,. Condition
of (ii)
says that extensions of Y uniformly produce branches of the tree T via an algorithm Turing equivalent with index e2. Condition (iii) asserts that if {e,} gr(/) = E,, thenfis to E,. In the sequel we shall not explicitly construct the indices e1 and e2, but it will be clear that such indices
can be constructed
in each case.
There are two important subclasses of recursive NRS’s introduced in [14], namely locally finite and highly recursive nonmonotonic rules systems. Say that the system (U, N ) is locally jinite if for each (PE U, there exist only finitely proof schema
with conclusion
cp. If (U, N ) is locally
many
<-minimal
finite, then for every cp, there
exists a finite set of derivations Dr, such that all the derivations of cp are inessential extensions of derivations in Dr,. That is, if p is a derivation of cp, then there is a derivation p1 EDrq such that pi 4 p. Finally, say that (U, N ) is highly recursive if (U, N) is recursive, locally finite, and the map cp H can(Dr,) is partial recursive. The latter means that there is an effective procedure which, when applied to any VE U, produces a canonical index of the set of all <-minimal proof schema with conclusion cp. This given, we can now state some basic results from [14-161 on the complexity of extensions in recursive nonmonotonic rule systems. Theorem 6.1. For any highly recursive NRS system 9’ = (U, N ), there recursive tree T, such that there is an efective one-to-one degree-preserving ence between [Ty] and k?(9). Vice versa, for any highly recursive tree a highly recursive NRS system Y’r = (U, N ) such that there is an effective degree-preserving correspondence between [T] and b(Y,).
is a highly correspondT, there is one-to-one
Theorem 6.2. For any locallyjnite recursive NRS system 9’ = (U, N ), there is a tree TY which is highly recursive in 0’ such that there is an effective one-to-one degreepreserving correspondence between [ TY] and 8(Y). Vice versa, for any highly recursive tree Tin 0’, there is a locallyjnite recursive NRS system Yr = (U, N > such that there is an efective one-to-one degree-preserving correspondence between [T] and &(Y’r). Theorem 6.3. For any recursive NRS system Y = (U, N ), there is a recursive tree Ty such that there is an eflective one-to-one degree-preserving correspondence between [TY] and &(9). Vice versa, for any recursive tree T, there is a recursive NRS system LfT = (U, N > such that there is an eflective one-to-one degree-preserving correspondence between [T] and &(Yr). Because the set of degrees of paths through trees has been extensively studied in the literature, we immediately can derive a number of corollaries about the degrees of extensions in recursive NRS systems. We shall give a few of these corollaries below. We begin with some consequences of Theorem 6.1. First there are some basic results which guarantee that there are extensions of a highly recursive NRS system which are
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302
not too complex.
Recall 0 denotes
the degree of recursive
sets and 0’ its jump.
Call
A low if A’ =rO’. This means that A is called low provided that the jump of A is as small as possible with respect to Turing degrees. The following corollary is an immediate
consequence
of Theorem
Corollary 6.4. Let Y = (U, N)
4.1 and the work of Jockusch
be a highly recursive nonmonotonic
and Soare [12].
rule system such
that &(Y) # 8. Then (i) There exists an extension (ii) If 9 has onlyfinitely
E of Y such that E is low.
many extensions,
then every extension E of Y is recursive.
In the other directions, there are a number of corollaries of the Theorem 4.1 which allow us to show that there are highly recursive NRS systems 9 such that the set of degrees realized by elements of E(9) are quite complex. Again all these corollaries follow by transferring results of Jockusch and Soare [ll, 121.
Corollary 6.5. 1. There is a highly recursive nonmonotonic rule system (U, N ) such that (U, N) has 2’0 extensions but no recursive extensions. 2. There is a highly recursive nonmonotonic rule system (U, N ) such that (U, N ) has 2”0 extensions and any two extensions El # E2 of (U, N ) are Turing incomparable. 3. Ifa is any Turing degree such that 0
enumerable
degrees b b T a.
Now the situation for locally finite recursive NRS rule systems is very similar to the situation for highly recursive NRS systems except that the degrees of extensions may be more complex. Essentially every result in Corollaries 6.4 and 6.5 holds for locally finite recursive NRS systems where each statement is taken relative to an 0’ oracle. See [16] for further details. However, the situation for general recursive NRS systems is quite different. That is, for general recursive NRS systems the set of elements of g(9) may be extremely complex. For example, all we can say in the positive direction is the following.
303
A context for belief reaision
Corollary 6.6. 1. Every recursive NRS system Y = (U, N ) which has an extension has an extension E such that E dT B where B is a complete II: -set. 2. If Y = (U, N)
is a recursive
NRS
system with a unique extension
E, then E is
hyperarithmetic.
In the opposite
direction
we have the following
Corollary
6.7. 1. There
extension
but Y has no extension
2. For each recursive possessing
6.2.
NRS
system Y = (U, N)
such that Y
has an
which is hyperarithmetic.
ordinal u, there exists a recursive
a unique extension
Recursion
In this a recursive will show tonic rule extension
a recursive
results, see [15].
NRS
system 9’ = (U, N >
E such that E = T O@‘.
theory of extensions
of FC-normal
NRS
subsection, we shall state our results on the complexity of extensions in FC-normal nonmonotonic rule systems. Our first result of this subsection that under the assumption of FC-normality that even recursive nonmonosystems are guaranteed to have at least one relatively well behaved which is in great contrast to Corollary 6.7.
Theorem 6.8. Suppose that Y = (U, N)
is a recursive nonmonotonic
Y is FC-normal.
E such that E is r.e. in 0’ and hence E dT 0”.
Then Y has an extension
We note that Theorem 6.8 is in some sense the best possible. [17] show that the following holds. Given sets A, B G o, let
rule system and
That is, results from
A @ B = (2x: XEA} u {2x + 1 : XEB}. Theorem 6.9. Let A be any r.e. set and B any set which is r.e. in A, i.e., B = (x: p:(x) Then there is a recursive extension
FC-normal
NRS
E and E = T A @ B. In particular,
then there is an FC-normal
Y = (U, N > such that Y
1 }.
has a unique
if B is any set which is r.e. in 0’ and B > T 0’,
NRS Y such that 9
has a unique extension
However, we shall show that if either man(Y) or nmon(Y) improve on Theorem 6.8. That is, we have the following.
E and E =T B.
is finite, then we can
Theorem 6.10. Let Y = (U, N) be a recursive rule system such that 9 and nmon(Y) is jnite, then every extension of Y is r.e.
is FC-normal
nonmonotonic rule system 9 = (U, N ) is monotonically closure of any finite set F is recursive and there is a uniform effective procedure to go from a canonical index of a finite set F to a recursive index of the cl,,,(F), i.e., if there is a recursive function f such that for all k, 4f(kJ is the We say that a recursive
decidable if the monotonic
304
V. W. Marek et al.
characteristic function of c1,,,(&). It is easy to see that if man(Y) is finite, then the recursive nonmonotonic rule system Y = (U, N ) is automatically monotonically decidable. Theorem 6.11. Let 9’ = (U, N)
be a recursive nonmonotonic rule system such that
Y is FC-normal and monotonically decidable. Then Y has an extension which is r.e. Next we consider the case where Y = (U, N ) is a finite FC-normal nonmonotonic rule system, i.e. we assume that both U and N are finite. In this case, we shall see that our forward chaining algorithm runs in polynomial time. we shall assume that the elements of U are coded by C. Thus every aE U will have some length which we
For complexity considerations, strings over some finite alphabet denote by 11 a 11.Next, for a rule r=
aI, . . . . a,:bl,
. . . . b, 3
we define lIrII = (ir, Finally,
1H:JJ + (zm
1ibj1i) + 114.
for a set Q of rules, we define
IIQII = c IId. reQ
Theorem 6.12. Suppose Y = (U, N) is a finite FC-normal nonmonotonic rule system and < is some well-ordering of nmon(S). Then E< as constructed via our forward man(Y) (I . IInmon(9) 11+ chaining algorithm can be computed in time 0( 11 IInmoWP)l12). We note that we did not make any explicit assumptions that the underlying consistency property of a recursive FC-normal NRS 9’ = (U, N ) is any way effective. Indeed none of the above results require that the underlying consistency property has any effective properties. We define a consistency property Con for 9’ to befinitely decidable if there is an effective procedure which applied to the canonical index of any finite subset X of U determines whether XECon. Then the following result easily follows from our completeness theorem for FC-normal NRS’s. Theorem 6.13. Let Y = (U, N) be a highly recursive rule system such that Y is FC-normal with respect to a decidable consistency property Con. Then {uEU: u is in some extension of Y”> is recursive. Finally systems.
we have the following
result about recursive
FC-normal
nonmonotonic
rule
A context
305
for belief revision
Theorem 6.14. Let T be a recursive tree in 2’” such that [T] # 8. Then there is an FC-normal recursive NRS 9 = (U, N) such that there is an efSective one-to-one degree-preserving
correspondence
between
[T] and 8(Y).
Reiter [26] proved that there is a recursive
normal
default theory with no recursive
extension. Theorem 6.14 generalizes his result but also gives much finer information even for recursive normal default theories since the set of degrees of paths through highly recursive trees have been extensively studied. Our correspondence will allow us to transfer all results about the possible degrees of paths through highly recursive trees to results about the degrees of extensions of recursive FC-normal NRS systems. That is, all the results of Corollary 6.5 continue to hold if we replace recursive nonmonotonic rule system by FC-normal recursive nonmonotonic rule system in each part of its statement. Moreover, we note that the proof of Theorem 6.14 explicitly constructs recursive FC-normal NRS’s such that the underlying consistency property is not finitely decidable. Corollary 6.15. 1. There is an FC-normal recursive nonmonotonic rule system (U, N ) such that (U, N ) has 2”0 extensions but no recursive extensions. 2. There is an FC-normal recursive nonmonotonic rule system (U, N ) such that (U, N) has 2”o extensions and any two extensions El # E2 of (U, N) are Turing incomparable. 3. If a is any Turing degree such that 0 < a ~~01, then there is a highly recursive nonmonotonic rule system (U, N ) such that (U, N ) has 2 ‘0 extensions but no recursive extensions and (U, N) has an extension of degree a. 4. If a is any Turing degree such that 0 cTa ~~01, then there is an Fe-normal recursive nonmonotonic rule system (U, N) such that (U, N) has K,, extensions, (U, N ) has an extension E of degree a and if E’ # E is an extension of (U, N ), then E’ is recursive. 5. There is an FC-normal recursive nonmonotonic rule system (U, N) such that (U, N ) has 2’0 extensions and if a is the degree of any extension E of (U, N ) and b is any recursively enumerable degree such that a cT b, then b ~~0’. 6. Ifa is any recursively enumerable Turing degree, then there is an FC-normal recursive nonmonotonic rule system (U, N ) such that (U, N ) has 2”o extensions and the set of recursively enumerable degrees b which contain an extension of (U, N ) is precisely the set of all recursively enumerable degrees b b Ta. Theorem 6.8 allows us to compare the FC-normal nonmonotonic rule systems (and thus FC-normal logic programs) with another class of rule systems that always have extensions. To this end, motivated by Apt, Blair and Walker [2] and Przymusinski [24] consider the class of locally stratified rule systems. These are rule systems 9’ = (U, N ) with the following property: There exists a function rank: U -+ Ord such
306
V. W. Marek
et al.
that for every rule a,,
. ..>
CL:813
. . . . Pn
Y
in N, for all 1 < i < m, rank(cci) < rank(y) and for all 1 < j < n, rank(pj)
< rank(y).
Apt, Blair and Walker [2] (see also [21]) show that a stratified NRS possesses a unique extension. Hence, stratified NRS systems share with FC-normal NRS systems the property
that an extension
always exist. This similarity
will show that the classes of stratified
and FC-normal
is, however, superficial.
systems
We
are very different.
Example 6.1. 1. There exists a locally stratified NRS system which is not FC-normal, 2. There exist FC-normal NRS system which is not stratified. (1) It follows from the results of [S] that for every recursive ordinal number a there exists a recursive locally stratified nonmonotonic rule system Ya such that its unique for a > 3, as extension has degree 2 T O(‘). By Theorem 6.8,5$ cannot be FC-normal every recursive FC-normal NRS system possesses an extension of degree < 0”. (2) Since there are FC-normal NRS systems with more than one extension, such systems cannot be locally stratified. Next, we notice that the locally stratified systems have another property that makes their use for belief revision awkward. Namely, that the locally stratified NRS systems do not possess the semimonotonicity property. Example 6.2. Let U = {a, . . . , z} and let N be composed 7 -2 W
q:s r ’
Thus (U, N ) is stratified
of these three rules:
p:
ii’ and {w > is its unique
extension.
Now add the rule
:t -.
P
The new system semimonotonicity
is again stratified, fails for stratified
but its unique programs.
extension
7. Proofs of general results on FC-normal non-monotonic In this section,
is now {p, q, r}. Thus
rule systems
we shall give the proofs of the results stated in Section
From now on we assume all FC-normal consistency property given by Con.
nonmonotonic
Theorem 4.1. Every FC-normal
rule system has an extension.
nonmonotonic
rule systems
4. have the
A context
Proof. We shall show that our forward extension. Thus fix some well-ordering
307
for belief reuision
chaining construction will always produce an < of nmon(Y). Our well-ordering < deter-
mines some listing of the rules of nmon(Y), {r,: Amy}, where y is some ordinal. Let 0, be the least cardinal such that y d 0,. In what follows, we shall assume that the ordering increasing
among
ordinals
sequence
is given by E. Recall our forward
chaining
construction
an
of sets {Eb }neo, as follows. of E x
The forward chaining construction Case 0: a = 0. Let E$ = d,,,(0). Case 1: c1= q + 1 is a successor that El,
..., q7:P1,
..v
ordinal. Given
E,<, let /(a) be the least AEY such
P/i
l-J,= *
where c~i, . . , ape E,,< and /II, . . . , fik, II/ # E,+. E,5 1 = Ed = E,,, . Otherwise, let E,? 1 = Ed
= 4,,,(E7~
Case 2: CIis a limit ordinal. We then let Es
If there
is no such
G(M), then
let
u {cln(r&>). Then let Em<= U OEaE,jj.
= UaEO,,~>.
It is straightforward to prove by (transfinite) induction that Con(Eb) holds for all ~(~63~and hence Con(E <) holds. Next we want to prove by (transfinite) induction that Em
Z EV5 I, there exists one rule
wherea,,..., tx,~E,,,,/3~,..., /$$E,,and Eq<+l = cI,,,(E,,u{$}). Butsincerl(,+,,is FC-normal, we know that E2 u {$, pi} is not consistent for all i < k. Since Es is consistent, it must be the case that E,< u {II/, pi} $ E < for all i < k since subsets of consistent sets are consistent. Thus for all i d k, /3i$ E <. Hence rfCq+lj shows that $EC~<(~). But then E; u {$} c C,<(O) from which it easily follows that that &,,(E~~ u {Icl}) = E ,
308
V. W. Marek et al.
some aGO,.
Suppose
where i0 < . . . < i, < m and PI, . . . . Pk 4 E <. Now it is easy to see that our construction ensures that if r((,,+r) is defined, then cln(rfcs+ 1j) q+E,<. Hence if A # q and rlclj and r/(V) are defined, then r/(A) # r/(tlI. Thus the function L( ) is one-to-one on its domain. Now suppose that a, 4 E <. Then for all A + 1 greater than a, ram is a candidate to be recA+ 1J at stage A + 1. Hence it must be that case that rfo, + t) is defined and /(A + ~)EQ,,. But this is impossible. That is, if 0, is infinite, then the cardinality of {rfcA+ 1j : v,E~E@~} is equal to the cardinality of 0, which is strictly greater than the cardinality of (6: REV,}. Similarly if 0, is finite, then the fact that r/(A) is defined for all A d 0, and 8( ) is one-to-one would mean that the cardinality of {rfcij : 11~0,) is equal to the cardinality of nmon(9) so that every rule rEnmon(Y) must be equal to r((ij for some 1. Thus in either case we have shown that if CI,$ E <, then for some ,uLE@~,r,,,, is the least rule r=
such
that
61, . ..> &:?I,
. ..> Yr
*
6r, . . . . &eEW<
and
yr, . . . , yr,
u,EE,<+ 1 G E <. Thus CI, must be in E <. Hence
as claimed.
Ic/$ E,<. But then by construction CE< (0) c E < and E < is an extension
0
Note that the proof of Theorem 4.1, remains unchanged if instead of starting with &,,(0) at stage 0, we start with &,,(I) where ZECon. Thus we also have the following. Theorem
4.2. Let Y = (U, N)
respect to consistency exists an extension
property
be an FC-normal
nonmonotonic
rule system
Con. Let I be a subset of U such that IsCon.
with
Then there
I’ of I in 9.
Next we want to show that every extension of an FC-normal NRS Y = (U, N ) can be constructed by our forward chaining construction relative to an appropriate ordering of nmon(Y). Theorem 4.3. If 9’ = (U, N) nmon(Y),
is an FC-normal
NRS,
and <
is any well-ordering
of
then:
(1) E + is an extension of 9. (2) (Completeness of the construction) a suitably chosen ordering
<
Every extension
of 9’ is of the form E + for
of nmon(Y).
Proof. (1) follows from our proof of Theorem 4.1. (2) We prove the following fact: Let F be an extension of an FC-normal NRS Y = (U, N ). Let p = card(NG(F, 9’)) and let < be some well-ordering of nmon(Y)
A context
such that the listing of nmon(Y) NG(F, Y) = {r,: CIE~}. Then
for belief revision
determined
(i) F = cl,,,({cln(r): VENG(F, Y)}), (ii) F = E x where E < is constructed
309
by <, {r,: cr~y}, is such that p < y and
and by our forward
chaining
construction.
For (i), note that for each El,
an:P1,..., Pk ENW’, S),
‘..,
I=
*
cln(r)~F. Moreover, for any set W L C,(8), d,,,(W) c1,,,( {&r(r): ~ENG(F, 9))) c C,(0). Then a straightforward length then
of a minimal proof scheme p cln(p)ecl,,,( {c/n(r) : rENG(F, 9’))).
will It
show then
L C,(0) induction
that if follows
so that on the
supp(p) n F = 8, that C,(0) =
cI,,,( (h(r): rENG(F, 9’))). be constructed by the forward chaining construction For (ii), let {Ed : wOy} relative to the well-ordering of rules {r, : Amy }. Then we claim E,< = F and E,< = E,< for CI> ,u. First it is easy to show by induction that cl,,,(Ed) = Ed for all CLNext we claim that if asp, then E,< c F and moreover if E,< # E,< 1, then [(a + l)~/*. That is, E> = cl,,.@) s F. Next suppose by induction that ED< s F for all BECKThen if tl is a limit ordinal, E> = up,, EPs E F. If a is a successor ordinal, we can assume by induction that Eqx c F where r] + 1 = ~1.Now consider E,<. If Eqx = F, then for any rule r=
al>
. ..>
‘%:P1,. . . . Pk
EnmOn(
*
it must be the case that either { B1, . . . , &} n F # 0, {aI, . . . . LX,,> c/F, or I,!JEFsince F is an extension. That is, if E, < = F then /(q + 1) must be undefined and hence Esrl = E,,, 1 = F. If E,j # F, then consider some UEF - E,+. Since UEF, there is some minimal proof scheme p = ((a,, fO, G,), . . . . (a,, r,, G,)) where CI, = a and G, n F = 0 which witnesses that aEF. Since a$E,<, there must be some k < m such that czi, . . . . uk_lEEv’ and akqkEqX. Then consider Fk =
ai,,
...>
clij:81,
...>
Bt
where i. < . . < ij < k.
@k
., fit} = 8 since otherwise ukEcl,,,(E~) = E,,$. Thus 1,...,/31}~G,andG,nF=0,itmustbethecasethat iii, . . . . A} # 0. But since (P ence F,,,ENG(F, 9) and F,,,is a can1,..., /$}nE,,5=0and{/3, ,..., P,}nF=@H didate to be rf(,,+ 1l. But this means that if f,,, = rs in our ordering of rules in nmonh(Y), then a(~ + 1) d p < p. But for any REP, cln(rb)eF by our choice of our well-ordering. Thus cln(r~~,+,,)~F so that E,< u {cln(r~t,+,,)} L F and hence
Now
it cannot
be that
cL,,(E7~
{fil,.
u {cln(rf(,+ i))}) = E,-$ I = -%< c f’.
It follows that E> c F since E, = lJaslr Eb and Eb E F for all IXE,U.We claim that it must be the case that E,< = F for otherwise EPx c F and hence for all asp, E,+ c F.
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V. W. Marek et al
But our argument fact, in turn, scheme
above shows that if Eb c F, then Eb c E,< 1 and d(cr + l)~p. This
will allow us to prove by induction
that for all ~ENG(F,
Y), cln(r)~E,‘.
on the length
That
of a minimal
is, suppose
proof
I,$ = cln(r) for some
reNG(F,
9). Now Ic/ = cln(p) for some minimal proof scheme p = where G, n F = 8 and E, = II/. Now assume by <
conclusion of some minimal proof scheme q such that supp( q) n F = 8 and length of q < m is in E,,<. Note that each rk for k < m is either in man(Y) or in NG(F, 9’) since p shows c(i, . . . . M,EF and cons(rk) s G, where G, n F = 0. It follows that each Clifor i < m such that rieNG(F, 9’) is in E ,’ by our induction hypothesis. But then = E:. So consider r,,,. Now {a 1, . . . . c~,_i} c cl,,,(E>) c~,_i} G E,<. If r,ENG(F, Y), then r, = * = %14nLJ,{~1, . . . . rules where 5 < p. Moreover, there is some AE~ such that {c(i, then for any 2 < 6 d ,u, if $6 Ed< then r5 is a possible candidate must be that case that rfc6+ 1j is defined and L’(8 + l)~& But this if ,u is infinite, then the cardinality of {rp(6+1j: AE~E~} is equal
if r,Emon(Y),
then
ry in our orderings
of . . . . CI,_ i} c En<. But to be rLcd+1j. Hence it is impossible. That is, to the cardinality of .Dwhich is strictly greater than the cardinality of (cr: CIE[}. Similarly if ,Uis finite, then the fact that r!(1) is defined for all 2 Q p and 8( ) is one-to-one would mean that the cardinality of {r((lj: 1 d u} is equal to the cardinality of ,u so that every rule rg with 6 < ,Dmust be equal to rLcAjfor some /z Q p. This in either case we have shown that if $ $ E,$, then for some 6 6 p, ry = rfcaj. But then by construction tj~E> s E> . Thus $ must be in E,<. Thus we have shown that {cln(r): rENG(F, 9)) E E,< if E,
{cln(r): rENG(F,
Y)})
c cl,,,(E2)
= E>.
Thus it must be the case that E,< = F. Note we have already shown that if Eb = F, then Ed = E,< 1. Thus since EPX = F it easily follows that E A< = F for all p < II < 0,. Hence E< = uorsB E, = F as Y claimed. 0 Since every E < is in Con, we immediately Corollary respect
get the following
4.4. Let 9’ = (U, N > be an FC-normal
to consistency
property
corollary.
nonmonotonic
Con. Then every extension
of Y
rule system
with
is in Con.
Next we show that if our FC-normal nonmonotonic rule system (U, N ) is countable, i.e., if U is countable which automatically implies that N is countable, then every extension of (U, N ) can be constructed via the countable forward chaining construction relative to some well-ordering < of nmon( (U, N )) of the order type of w. Theorem 4.5. If 9 = (U, N) is a countable FC-normal nonmonotonic rule system, then 1. E X constructed via the countable forward chaining construction with respect to <, where < is any well-ordering of nmon(Y) of order type w, is an extension of Y.
A context
for belief revision
311
2, (Completeness of the construction) Every extension of Y is of the form E-C for a suitably chosen ordering < of nmon(Y) of order type o where E-X is constructed via the countable forward chaining construction. Proof. We note that if 4 is a well-ordering of nmon(Y) of order type w, the countable forward chaining algorithm is just the first o steps of the forward chaining algorithm. Thus
to prove
1, we must
show that if we construct
E + with respect
the forward
chaining algorithm, then E, -X = En< for all A 2 w. In fact, we need only show that E’=E+ o+ Now suppose that w I
.
El,
r/(,+
1) =
“‘,
%l:B1,
‘..,
Bk
$
is defined. Thus tli, . . . . CC,,CZE>and fir, . . . . &, $$ E>. Moreover, rgC,+ll = rq for of rules determined by <. But since some q where {rnjnEO is the ordering E$, there must be some s such that aI,. ., a,~Eb. Hence for all t >, s, EZ = un,, PI, . ..> hr ICI$E,< so that r4 is candidate to be rfC1)for all t > s. Since the function e(t) is one-to-one, it easily follows that there would have to be some finite t such that r4 = r/(,). Thus rfcw+ 1) must not be defined and hence E: = E <. Next we consider the proof of 2. Note that if we apply the proof of Theorem 4.3 to F in this case the most natural thing to do is to order the rules of NG(F, 9) first, say NG(F, 9) = {so, si, . ..}. and then follow this ordering by a listing all the rules of nmon(Y) - NG(F, Y) = (to, tl, . ..}. Now, if NG(F, 9) is finite, then our listing of rules determines a well-ordering < of order type o in which the proof of Theorem 4.3 shows that F = E <. If NG(F, 9) is infinite, then our listing of rules determines a well-ordering < of order type < o + cc).It then follows from the proof of Theorem 4.3 that
and that E;
= F. The key point
to note is that for any
is not in NG(F, 9’), it must be the case that {/zI~,...,~~}~F#QI or a,} $ F. But since F = Uiew Ei<, it also follows that either {@l,. . . . (a) for some i, {P1,...,Pk)r\Ei~ #@or (b) for all j, (~(1, . . . . a,,} $ EJ. In case (a) if we insert r between sk and ski 1 where k = t(i), then this change will have no effect on the construction of the Ei+‘s. That is, the construction of E-C up to stage i can depend only on s,,, , S/(i) and hence we will get the same sets, Ej< forj < i, for any ordering which starts out s O, . . . . S/(i). Thus if we take the ordering s,,, . . . . Sl(i), , fik > n EiS # 8, r is not a candidate to be rfCkJfor r, s 1+/(i), . . . , then because {/Ii, any k > i and hence the insertion of r does not effect the rest of the construction of E <. which
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V. W. Marek et al.
In case (b), we can insert r anywhere in the initial o part of the list and it will have no effect on the construction of the Ei<‘s for i~o because the premises of r are never contained
in such Ei<‘S. In this way, we can see that it is possible
to interleave
all the
r’s in nmon(Y)
ordering
- NG(F, 9’) into the basic ordering sO, sr, . . . so as to create an of order type o but with out changing the sequence E,<, El<, . . . Thus it will
still be the case that F = E$ Theorem Theorem
= Uiiw
4.6 follows immediately
0
from the following
Suppose
(Semi-monotonicity).
D c nmon(Y).
EC<.
9’ = (U, N)
result. is an FC-normal
NRS.
Let
Then
1. 9” = (U, man(Y)
u D) is FC-normal
2. if E’ is an extension
(a) E’cEand (b) NG(E’, 9’)
of 9’,
c NG(E,
NRS, and
then there is an extension
E of Y such that
9).
Proof. The fact that Y’ is an FC-normal definitions. For part 2, let p = cardinality
NRS is an immediate consequence of our of NG(E’, 9”) and choose a well-ordering
of NG(E’, Y’), {r,: UE~}. Th en extend this well-ordering to a well-ordering {r,: C(EY} of nmon(S). It follows that if E < 1s constructed via our forward chaining algorithm with respect to the well ordering < determined by (r, : Amy}, then proof of Theorem 4.3 shows E’ = E,+ so that E’ z E+. It remains to be proved that NG(E’, 9’) G NG(E<, 9’). Now suppose r=
Then since hence since
al,
..‘,
%:Pl, . . . . Pk
9’).
{c1i, . . . . cl,,} c E’ E E+ and {pi,..., jjk} n E’ = 8. But 4.3, E’ = c1,,,( {cln(r): reNG(E’, by the FC-normality of r, E’ u {$, pi} IS not consistent E + is consistent, E’ u {$, pi} $ E X for any i = 1,. . . ,
E’ = E,<. By Theorem
3..., k and reNG(E<,
i=l
ENG(E’,
*
9’).
q
We prove now the result on the orthogonality Theorem 4.7 (Orthogonality with respect to a consistency ofY,
note that Con(E’) holds 9”))). Thus $EE’ and for any i = 1, . . . . k. But k. Hence Bile s for all
of Extensions). property
of extensions.
Let Y = (U, N)
be an FC-normal
NRS
Con. Then, if E and F are two distinct extensions
EvF$Con.
Proof. By Theorem 4.3, E = uneO E$ where {Ees}a.O, by the forward chaining construction relative to some well t( be the least ordinal such that Ed cF but E,2, $F. an a since otherwise E s F and then by the minimality
is the sequence constructed ordering < of nmon(S). Let Note there must be such of extensions, E = F. Thus
A contextfor belief relrision
313
the rule Xl, r/c,+
..‘>
G:/jl,...,
Pk
1) = +
8 # {PI, . . . . fik}, {PI, . . . . ljk} n E,< = 8 and {cxl, . . . . c(,} c Ed, u {I/J}). Since E,3 1 $ F, it must be that $$F. But this means that /3iEF for some i since otherwise Y/(,+ Ij ENG(F, Y) which would imply that $EF
is
such
that
E,5 1 = cl,,,,(E,<
because F is an extension. By the FC-normality of r/(,+ 1j, E,< u {t,b, pi} is not 0 consistent. But since Eb u {lc/, Bi} c E u F, E u F is also not consistent. Theorem 4.8. Suppose Y = (U, N ) is an FC-normal property
Con such that cl,,,({cln(r):r~nmon(Y)})
NRS with respect to consistency is in Con. Then Y
has a unique
extension.
Proof. For a contradiction, assume P’ has two distinct extentions, El and E2. Then by our proof of Theorem 4.3, Ei = cl,,,( {cln(r): rENG(Ei, Y)}) for i = 1,2. But then Thus E, u E2 is contained in a consistent for i = 1, 2, Ei s cl,,, {cln(r) : rEnmon(Y)}. set so that E, u E2 is consistent, contradicting Theorem 4.7. 0 Theorem 4.9. Let 9 = (U, N) property
be an FC-normal
Con. Then VE U is an element
a consistent
proof scheme with respect
NRS
with respect to a consistency
of some extension
of Y
if and only if cp has
to Con.
Proof. Clearly if PEE where E is an extension, then Con(E) by Corollary 4.4. Thus since fl~C,(fl), there is a consistent minimal proof scheme for cp. Conversely suppose p = ( (cpO, r,,, G,), , (cp,, r,, G,)) is a consistent minimal < i, < m be set of all i < m such that riEnmon(Y). proof scheme for p. Let 0 < iI < ... Now well-order nmon(S) so that ri, . . . , rik are the first k elements in the list. Then if we construct an extension via our forward chaining construction, it is easy to show by induction on k that BEEP. Hence BEE< which is an extension. 0 Theorem 4.10. Suppose Y = (U, N) Suppose further
is an FC-normal
that E; and E; are distinct extensions
distinct extensions
NRS and that D c nmon(Y). of (U, D u man(9)).
Then Y has
El and E2 such that E; c El and E; c E2.
Proof. By Theorem 4.9, we know that there are extensions of 9, El and E2, such that E; s E, and E; E E2. But then the orthogonality of extensions for (U, D u mon(9’)) ensures E; u E; is not consistent. Hence E, u E, is not consistent so that E,#&.
0
8. Recursive FC-normal
In this section,
NRS systems and the complexity of their extensions
we shall give all the proofs of theorems
stated in Section
6.
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V.W. Marek et al.
Our first result will show that under the assumption of FC-normality that once again even recursive NRS systems are guaranteed to have at least one relatively well behaved extension. Theorem 6.8. Suppose that Y = (V, N) is a recursive nonmonotonic rule system and 9 is FC-normal. Then Y has an extension E such that E is r.e. in 0’ and hence E ~~011. Proof. It is easy to see that since N is recursive, man(9) is also recursive. It then easily follows that if X E U and X is r.e., then clmon(X) is also r.e. In fact, there is a recursive function f such that for all e, WfceJ= clmon(W,) where W, is the eth r.e. set = {x : q,(x) 1 }. We shall show that for any recursive well-ordering < of nmon(Y) of order type o, E < is r.e. in 0’ where E X is constructed via the countable forward chaining algorithm with respect to <. That is, i determines an effective listing of nmon(S), rO, rlt . . . . Then the countable forward chaining construction relative to i constructs a sequence of sets, E> -c E> c .... It is straightforward to prove by induction that E,< is r.e. for all e. That is, suppose E,< = W,,, for some e,. Es=E’ Then either that OlI E,-:zE,& SO E,?1 = &A-@ u {cln(ri,.+lJ!)j) = ~~~~~(W,,u {cln(r(,,+ 1j))) which is clearly r.e. Now there are two cases. Either there is an n such that E,% = E,$1 in which case E,+ = E < so that E < is r.e. Otherwise, E,< # E,-$ 1 for all n and hence rfln+ r) is defined for all n. In this case, given an 0’-oracle, we can test for membership in E,,+ and hence it is easy to see that given aI, rk
=
. ..>
%:A, ,.., Pt
E nmon(P),
*
we can test whether cL1,. . ., QzE,,‘, PI, . . ., /ilk, I)$ E,,<. Thus given an O’-oracle, we can effectively find r((“+ Ij. However, gtven ry(,,+ 1l and an r.e. index e, such that Wen= E,<, then we can effectively produce an index r.e. index e,,, such that Thus there is an 0’-effective sequence We,+ 1 = 4,,,05~ u (cWrt,+lJ)) = En?+,. is r.e. in 0’ in this case. e0, el, . . . such that E,> = Wen for all n. Hence E < = un Eb Thus in either case, E X will be r.e. in 0’. 0 Note that Theorem 6.8 is in great contrast to the case of arbitrary recursive NRS Y = (U, N ) where, for example as in Theorem 6.7(a), there exists a recursive NRS 9’ such that Y’ has an extension but 9” has no hyperarithmetic extension. Theorem 6.10. Let Y = (V, N) b e a recursive rule system such that 9’ is Fe-normal and nmon(Y) is finite, then every extension of Y is r.e. Proof. This theorem follows immediately from our argument in Theorem 6.8 once we observe that if nmon(Y) is finite, then there is some finite stage such that E,< equals E < for any well-ordering < of nmon(Y). 0
A context for belief reuision
Another situation where a recursive FC-normal have an r.e. extension is when Y is monotonically finite, then 9 = (U, N)
is guaranteed
315
NRS Y = (U, N) is guaranteed to decidable. In particular if man(Y) is
to have an r.e. extension.
Theorem 6.11. Let Y = (U, N) be a recursive rule system such that Y is FC-normal and monotonically decidable. Then Y has an extension which is r.e.
Proof. We shall show exactly as in Theorem 6.8 that for any recursive well-ordering via the countable < of nmon(Y) of order type o, E + is r.e. where Es is constructed forward chaining algorithm with respect to 4. Again, there are two cases. If there is some n such that E,.? = EL 1, then E,,, = E < and E < is r.e. Otherwise, E,< # E,? 1 for all n and hence r/(,+1I is defined for all n. In this case, it is easy to prove by induction that E2 = clmon((c(rf(Ij), . . . , c(rf(,))}) for all n 3 1. Because 9’ is monotonically decidable, it follows that E, -X is recursive for all n. Indeed there is a recursive function of cl,,,(DJ. Thus given function f such that qfckj is characteristic rfcnI}, we can effectively find k, such that Dkn = (c(rf(lI), . . . . c(rg(,))} and (r41j, . . . . then (arty,,) will be the characteristic function of E,,<. But then we can use (P/(~,) to effectively find rpcn+ 1J. Thus we can effectively find the sequence of rules rytlj, rfc2), . . . and since E < = cl,,,( {c(rf(lj), c(r/(2j), . ..}). Es is r.e. q
Next we consider the case where Y = (U, N) is a finite FC-normal nonmonotonic rule system, i.e. we assume that both U and N are finite. In this case, we shall see that our forward chaining algorithm runs in polynomial time. For complexity considerations, we shall assume that the elements of U are coded by strings over some finite alphabet C. Thus every aeU will have some length which we denote by 11 a I/. Next, for a rule
r=
aI, . . . . a,:bI,
. . . . b,
C
we define
Finally,
for a set Q of rules, we define
IIQII = c IId. rsQ
Theorem 6.12. Suppose Y = (V, N) is a Jinite FC-normal nonmonotonic rule system and < is some well-ordering of nmon(S). Then E-X as constructed via our forward chaining algorithm can be computed in time 0( IIman(Y) II - II nmon(Y) (I +
IInmoW)l12).
316
V. W. Marek et al.
Proof. First observe that our forward
chaining
algorithm
will stop after stage n where
n = )nmon(Y)I since 8(k) is a one-to-one function, i.e., at then end of stage n, there will be no possrble candidate for r((,+ r) sothatE,$=Ez,=E<. Next consider a stage k + 1 in the forward chaining construction. Given Ek<, we must make a pass through whether maintaining
the rules in order to check for each rule r = ‘I”‘.’ a”;b”‘.” bm
(aI, . . . . a,) s E,,< and
for some constant
{b,, . . . . b,, c} n Ek< = 8. Notice
that
at a cost of
this check in C
11steps
C and we call such a check a rule check. Now assuming
that we
an appropriate
data structure
we can perform
the rules in order, if r is the first rule such that {al, . . . , a,} G E,,? and b,, c} n Ek< = 8, then r = rk+lr E& 1 = cl,,,({c} u E,$). Moreover, if 1, . . . . {b (b,, . . . . b,, c> n Ek. # 8, then we know that r can never be a candidate to rj for any j > k so that we can just mark rule r and never consider it again. Of course, we also mark rk+ 1 at stage k + 1 if it is defined so that at each stage will mark at least one rEnmon(Y). Moreover, if rk+r is not defined, then we can stop since then we know Ek+ = E<. process
It follows that at stage k + 1, we need to look at most Inmon(Y) I - k rules and hence perform at most Inmon(Y)I - k rule checks. Since the construction must stop at requires at most stage Inmon(Y)I, it follows that the entire construction (a) of computing (b) )nmon(Y)I operations of c&,,,,(0). (4 the computation
cI,,,,(Ek< u {c}), and
Now if 11nmon(Y)/I = klnmon(Y)( for some k, then the rule checks could require 0( I/nmon(Y) II”) steps. Next consider the computations cl,,,(A) which are required for (b) and (c) above. We claim all this can be done 0( I(man(Y) II - Ij nmon(Y) 11)steps. Since in our construction all the elements of EkX must appear in one of the rules, we can assume IIA I) < IIman(Y) /I + IInmon(Y) 11.Now we can first make a pass through all the rules of nmon(Y) to get a list of all the elements of U which occur in one of the rules, Call this set V. Another pass through the rules will allow us to set up a system of pointers from each CE V to the set of rules rEmon such that c occurs in the set of premises of r. We can also mark which c are in A. All this will require CI (II monW4P)II + IInmofl(Y)II1d CI (II man(Y) ((* ((nmon(Y) II) steps. Now for each CEA, use the pointers from c to the rules remon to update each r by marking each premise of r in A. Now if a rule rGmon(.Y) has all of its premises marked, we mark the conclusion of r, i.e., we add the c/n(r) to cl,,,(A) and use the pointers from cln(r) to rules in man(Y) to further update the premises of each rule by marking h(r). We continue in this fashion until there are no more rules to update in which case A together with the marked conclusions will form the &,,(A). Now assuming that updates can be performed in constant time, each rule remon can require at most I\r )I updates in this process since once all the premises of a rule have been marked we no longer have to consider it. Thus we require at most IIman(Y) 11updates so the entire process takes at most 0( (Iman(Y) 11) steps. Thus to compute the monotonic
317
A context for belief revision
closures
required
0( ((man(Y)
in (b) and
I(- ((nmon(Y)
(c) above
(() steps.
We pause to make one further
takes
0((1
+ Inmon(Y)l)
observation
about
any highly recursive
NRS Y = (U, N) for which the underlying consistency property decidable. That is, if ,Y is highly recursive, then we can effectively minimal
proof
schemes
for any
XEU. If Con is finitely
decidable,
effectively tell if any of the proof schemes for x are consistent. a consistent proof scheme if and only if x is in some extension we can effectively decide whether following
* 11man(Y)
11)d
q
x in some extension
FC-normal
Con is finitely find the set of then
we can
By Theorem 4.9, x has of Y’. Thus in this case,
of Y. Thus we have proved the
result.
Theorem 6.13. Let 9’ = (U, N) be a highly recursive rule system such that 9’ is FC-normal with respect to a decidable consistency property Con. Then {uEU: u is in some extension qf Y is recursive.
We end this section with a result that shows that recursive least as expressive
as highly recursive
FC-normal
NRS’s are at
NRS.
Theorem 6.14. Let T be a recursive tree in 2’” such that [T] # 8. Then there is an FC-normal recursive NRS Y =
(2) i for all cx~T which are terminal G! (3) L& -cc-o, a-1: (4) ~ E tl :
(5) a^l
a-0
=&
nodes.
for all CXE T such that M is not a terminal
for all C(ET such that CIis not a terminal a:a-1 cc^o
node.
node.
for all cr~T such that cz is not a terminal
node.
318
V. W. Marek et al.
(6) s
for all EE T such that M is not a terminal
s
(7) F
for all CZT - (8) and all @LJ.
(8) ;
for all CET and /I I= CL
CX, lx-o: (9) a^l
c1,a-1: cr^o
node.
for all C(ET such that c1is not a terminal
node.
It is easy to see that P’ = (U, N ) is a recursive nonmonotonic rule system. Given a path rc=(rc(O),rr(l),...) through T, let E,= {a:ct is a node on rr}u{E:a is not a node on rc>. We claim that E is an extension of Y if and only if E = E, for some m[T]. It is easy to see that E, is an extension for every XE[T]. That is, (1) shows that QkCE,(@) and repeated use of the rules in (5) will allow us to show that rc(n))~C~$) for all n. Then the rules in (6) will allow us to show that for every MO), ’‘. 2 4 (40), . . ., n(n - l), &n))~C~,(ti) 6(n) =
Now
suppose
0
if n(n) = 1,
i 1
if n(n) = 0.
MET and
where
LY is not
on
rr. Then
there
is an
n such
that
MO), . . . > n(n - l), 6(n)) L cc Hence repeated use of the rules in (3) will allow us to show cl~C,~(@). Thus E, E C,(@). A straightforward proof by induction on the length of a derivation will show that C&J) c E,. Hence E, = C,r(8) and E, is an extension of Y. Next assume that E is an extension . First we claim that E # U. For if E = U, then rule cannot be used to derive C”(0) = &o.(0) since every strictly nonmonotonic elements of C,(0). Since [T] # 8, there is some path ICE[ T]. By our previous remarks, E, is an extension. But clearly E, # U and &,,,($!I) s E,. Thus C,(0) c E, # U so that U is not an extension. Once we know that U is not an extension, the rules in (7) show that for any CIET - (8}, E can contain at most one of a and E. Now assume by induction on n that there is a node b,, = (p(O), . . . , /?(n - 1)) of length n such that fi, is on some infinite path through T and for all MET with 1CI1< n, C(EE if and only if a c fl,, and iie:E if and only if c( & p,,. Now the rules in (3) ensure that if YET is such that IyI > n and y does not extend b,,, then TEE. Thus the only possible nodes, y of length n + 1 such that yeE are y = Pn^O and y = j$,^l. Because we are assuming that fi,, is on an infinite path through T, we know that at least one of /&^O and /I,,^1 must be on an infinite path through T. Now suppose Pn^O is not on an infinite path through T. Then A, = {ye:q J fin-O& VET} is finite by Konig’s lemma. Note that if qcAO and q is a terminal node, then ij~E by (2). But then it is easy to see that since T is a (0,2)-tree,
A context for belief revision
repeated
319
use of the rules of (4) will allow us to show GEE for all SEA,,. In particular,
P,,*OEE and hence by rule (9)
Thus we have shown that if j?,,^Ois not on an infinite path through T, then P,,*l EE, /&*l is on an infinite path through T, and for all UE T such that 1~) d n + 1, VEE if and only if g & /3,,*1 and FREEif and only if ye& /3,-l. Thus our inductive
hypothesis
n + 1 with /3,,+i = /$,^l. A similar argument will show that our inductive holdsatn+ 1 with/?,+, = /?,,^Oif pn^l is not on an infinite path through can assume that both @n-0 and /?,,^l lie on infinite rules
paths through
holds at hypothesis T. Thus we
T. Now consider
the
from (5). Note that R. shows Pn^l EE if /?/O$ E and RI shows /$^OEE if /$^l #E. Thus at least one of Bn*Oand /$ll must be in E. We claim that it cannot be the case that both PnAOand Bn^l are in E. For if /$*O, /?,,ll EE, then consider the minimal proof scheme P = ((Q,
TO, G,),
. . . .
of shortest possible length such that G, n E = 8 and c(, = P,,^O. Note that any rule Y with conclusion ,Bn^Ois either RI or comes from (7) or (8). Now I, # RI since RI is blocked for E if { P,,*O,/3,*1) E E. Since U # E, we cannot use any rule from (7). Thus
where yz$,^O. It follows that y = C(ifor some i. It cannot otherwise the proof scheme (ff02 ro> Cm>,...> (ai, ri, Gi),
P’ =
fin-o, A,
be that i < m - 1 since
Gi
”
>>
would violate our choice of p since it could be refined to a minimal proof scheme p” of length < the length of p such that cln(p”) = /I.^0 and supp(p”) n E = 0. Thus y = tl,_ i. But then rm- 1 = f where y c 6 from (8) or r,_1
=
cc:cC(l - i) ff*i
where a^i = y from (5). If rm_ 1 = $, then 6 = Cljfor some j < m - 1 and as before, the proof scheme <~o,ro,Go),.~.,
>
320
would
V. W. Marek et al.
violate
our choice of p. If
rm-l =
cc:cc^(l - i)
then either (i) az/3,^0
’ and c( = Cljfor some j < m - 1 or (ii) a = /InA and PnAO= C(jfor
some j < m - 1. In either case, we would violate our choice of p. Thus there can be no such p and hence /In*0 4 C,(0) if {/In-O, p,*l} z E. Hence if (p”*O, /?,,^l} G E, E # C,(0) which contradicts the fact that E is an extension. Thus we must conclude one of B,,lO and /I,,^1 is in E. Now if P,,^OEE, then the rule
from (9) shows BnAlEE and our induction If /?,^l E E, then the rule
hypothesis
that exactly
holds at n + 1 with fi,+ i = fi,^O.
p,*1:
shows that Pn^OeE and our induction hypothesis holds at n + 1 with /I, + I = 0,-l. This completes our induction and shows that 0 = /I,, t pi c Bz c . . . determines a path l7 through T and that E = E,. It now follows that the corresponding 7cH E, is our desired effective one-to-one degree-preserving correspondence between Y(Y) and J?(Y). Thus to complete our proof of the theorem, we need only check that Y is FC-normal with respect to some consistency property. We define X _c U to be consistent, Con(X), if and only if X 5 E, for some ne[T]. It is easy to check that properties (l), (2), (4) hold for Con. For property (3), note that if X G E,, then c&,,,,(X) c C,$) = E, since E, is an extension. Thus Con defines a consistency property. Finally we must check that all rules r~nmon(Y) are FC-normal with respect to Con. Note the only rules renmon(Y) are of the form 9 and g from (5). Now clearly (a-0, a-1) is not consistent so that all we need to show is that if Xc U is such that Con(X), cl,,,(X) = X, aeX, a-(1 - i)$X, then X u (u-i} is consistent. Now consider the possibilities for a monotonically closed consistent set X. For some rc = (rc(O), rc(l), . ..)e[T]. X c E, = {a:ct is on rr} u {cC:a is not on rc}. First suppose for infinitely many n, (n(O), . . . . z(n))eX. Then rules of the form (8) show (rc(O), . . . , n(m))EX = cl,,,(X) for every m. Next rules of the form (6) allow us to show (K(O), . . , x(n - l), 1 - x(n))EX = cl,,,(X) for all n and then rules of the form (3) will allow us to show VEX = c&,,,,(X) for all CLnot on rr. That is, X = E, if for infinitely many n, (n(O), . . . . x(n))EX. But in such a case, MEE, = X, a*(1 - i) $ E, = X implies a*iEE, = X so that Con(X u {Eli>) holds. Thus we must assume that there is a largest m, say n, such that (n(O), . . . , n(m))eX. Then rules of the form of (8) will allow us to show that (n(O), . . . . n(m))EX = clmO,,(X) for all m G n. Next rules of the form (6) will allow us to show that for all m < n, (x(O), . . . . x(m - l), 1 - x(m))EX
= cl,,,(X)
and then rules of the form (3) will allow
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A context .for belief revision
us to show that ZEX = c1,,,(X) for all MET such that CIand q are incomparable where n(n)). Next we claim that it must be the case that both q-0 and y-1 lie on V = MO), . *. 2 infinite through
paths
through
T. For if not, suppose
T. Then by K&rig’s Lemma
q^i does not lie on an infinite
path
{/I : PE T & fi J q-i} is finite and we can argue as
above that repeated use of the rules in (2) and (4) will show ~Ai~clmon(8) E X. But then -=7 from (9) shows ~“(1 - i)~cl~~~(X) = X violating our choice of q. Next the rule e suppose
that
q-i does not lie on z and ylA(l - i) lies on 7~. Now
consider
the set
A = (a: c17qAi and E$X}. Note that rules of the form (3) ensure that if EEA and ~“i c /I F ~1, then BEA. That is, if P$A, then VEX and hence EE&,,(X) = X by repeated use of the rules in (3). Thus the set of nodes in A determine a subtree of Trooted a q. We claim that there are no nodes PEA which are terminal with respect to A and hence A is infinite. For suppose BEA is a node which is terminal with respect to A. Thus fl# X and hence fl is not a terminal node of T because otherwise the rules of (2) would ensure P;clmon(8) E X. But then it must be the case that p^O, p1~T and -p-0, pll~X. But then the rule piEm from (4) would show that /?EcZ,,,(X) = X violating our choice of 8. Thus A has no terminal nodes and hence A is infinite since Neil A. But this means that there is a least one path rc* = (x*(O), *(l), . . .) through T such that q-i is on z* and (n*(O), . . . , rc*(k)) # X for all k. Thus X c E, and X c E#. Finally, it is easy to see that the only rules of the form & such that crux, M~$X are such that (a)
Mc n
and
a*(1 -j)
(b)
a = ‘I,
j = i,
(c)
a = I?,
j=l-i,
and
r= ye, ~‘(1 - j) = ~~(1 - i), and
or
a*(1 - j)= q^i.
In cases (a) or (b), a^( 1 - j)is on the path 7cand hence X u {cl-(1 - j)} G E, so that Con(X u (a-(1 -j)}) holds. In case (c), a-(1 -j) is on the path rr* and X u (a^(1 - j)} c E,* so that Con(X u {cz^(l -j)}) also holds in this case. Thus we have shown that all rules of nmon(S) are FC-normal with respect to Con. 0 Note that in the case where [T] has no recursive elements, (9~ T: ‘1 is an infinite path through Tj is not recursive and hence we will not be able to effectively decide whether a finite set S G U is consistent. Thus it is crucial that we make no assumptions about the effectiveness of the consistency property.
9. Conclusions In this paper we exhibited and investigated a natural condition which ensures that a logic program always has a stable model or a default theory or truth maintenance system always has an extension. This condition, called in our paper “FC-normality” is an abstraction of Reiter’s notion of normality in default logic coupled with Scott’s notion of Information System. This condition can be formulated in the general setting
V. W. Marek et al.
322
of nonmonotonic rules systems and hence can be immediately translated into other nonmonotonic reasoning formalisms including logic programming with negation as failure, default logic, and truth maintenance systems. We have shown that FC-normal nonmonotonic rule systems FC-normal logic programs, FC-normal logic programs normal truth maintenance systems) have all the desirable
(and, consequently.
with classical negation, FCproperties of Reiter’s normal
default theories such as the existence of extensions, the semimontonicity extensions, and the orthogonality of extensions. Indeed, when FC-normal tonic rule systems default
theories
are translated
into the language
which we call FC-normal
property of nonmono-
of default logic, we get a class of
default theories,
which strictly contain
the
class of normal default theories of Reiter and yet continue to have all the desirable properties of Reiter’s normal default theories. We gave a general construction, which we called the forward chaining construction, which constructed all possible extensions of an FC-normal nonmonotonic rule system based on well orderings of the set of nonomontonic rules of the system. By analyzing the forward chaining construction, we were able to establish bounds on the complexity of reasoning with FC-normal systems. For example, while there are recursive non-monotonic rule systems which have extensions but which have no hyperarithmetic extensions, we showed that every recursive normal nonmonotonic rule systems has an extension which is r.e. in 0’. Moreover, for any finite nonmonotonic rule system, we showed that one can always construct an extension in polynomial time. We hope that the technique presented in this paper can serve as an indication to designers of intelligent systems, in particular to monitoring systems, as well as to logic programmers why some programs behave better than other programs. FC-normal programs offer a “smooth transition” from one belief state to another. In particular modification of belief rules (but not of “hard facts”) can be handled smoothly by such systems. This is one way to maintain programs in a way which permits one to expand the present belief set to a larger one in the presence of new rules. It is expected that a further research into the nature of consistency properties for non-monotonic rule systems and their complexity will produce general techniques for writing normal
and maintaining FC-normal nonmonotonic rule system (and hence FClogic programs, extended normal default logics, and FC-normal truth main-
tenance
systems) for specific applications.
References [l]
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