The complexity of predicate default logic over a countable domain

Annals of Pure and Applied Logic 120 (2003) 151 – 163

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The complexity of predicate default logic over a countable domain Robert Saxon Milnikel ∗ Department of Mathematics, Wellesley College, 106 Central St., Wellesley, MA, 02481, USA Received 31 August 2000; received in revised form 18 October 2001 Communicated by S.N. Artemov

Abstract Lifschitz introduced the notion of de0ning extensions of predicate default theories not as absolute, but relative to a speci0ed domain. We look speci0cally at default theories over a countable domain and show the set of default theories which possess an !-extension is 21 -complete. That the set is in 21 is shown by writing a nearly circumscriptive formula whose !-models correspond to the !-extensions of a given default theory; similarly, 21 -hardness is established by a method for translating formulas into default theories in such a way that !-models of the circumscriptive formula correspond to !-extensions of the default theory. (That the set of circumscriptive c 2002 Elsevier formulas which have !-models is 21 -complete was established by Schlipf.)
Science B.V. All rights reserved. MSC: 68Q15; 03B60; 68T27 Keywords: Default logic; Circumscription; Computability theoretic complexity

1. Introduction The study of nonmonotonic logics started as an attempt to capture complicated ideas—systems of belief and their consequences—with simple rules that seemed to have much in common with classical logical rules. As various systems for nonmonotonic reasoning—default logic [22], circumscription [17], autoepistemic logic [21], modal nonmonotonic logics [18], stable model logic programming [3], and nonmonotonic rule systems [13]—were studied, two truths emerged: These systems had enough in common that straightforward translations among all them—with the exception of ∗

Present address: Department of Mathematics, Kenyon College, Gambier, OH 43022, USA. E-mail address: [email protected] (R.S. Milnikel).

c 2002 Elsevier Science B.V. All rights reserved. 0168-0072/03/$ - see front matter  PII: S 0 1 6 8 - 0 0 7 2 ( 0 2 ) 0 0 0 6 4 - 7

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circumscription—were discovered; and they were all tremendously more complicated and expressive than had 0rst been imagined. We will concentrate on one of these systems, Reiter’s default logic, and its connection with one other, McCarthy’s circumscription. In brief, circumscription attempts to 0nd a model of a 0rst-order sentence A(P) with a distinguished unary relation P such that the interpretation of P is minimized. In [24], the set of formulas A(P) possessing such a model (with a countably in0nite domain) was shown to be 21 -complete. Default logic has a much more syntactic feel. It extends propositional or 0rst-order logic with rules of the form ’1 ; : : : ; ’m : M 1 ; : : : ; M

n

;

with the (informal) interpretation: “If the ’’s have been shown to be true, and the ’s are consistent, conclude ”. What it means to be a model of a collection of default rules (a “default theory”) will be de0ned precisely in Section 2, but for now we will simply mention that such models are called “extensions”. In [14], the set of default theories possessing an extension was shown to be 11 -complete. We will show that if one extends default logic to a predicate language, expanding the underlying logic to an !-logic, the set of default theories possessing an extension leaps to being 21 -complete, the same complexity as the analogous question for circumscription. In [10], Lifschitz established that the approach to predicate default logic which most closely matches most people’s intuition is to de0ne extensions relative to a 0xed domain. His approach to default logic generalized work of Guerreiro and Casanova [6] and was semantic in nature. Kaminski [7] and Kaminski et al. [8] brought Lifschitz’ ideas back into Reiter’s original syntactic framework by showing that the 0xed domain could be speci0ed by means of a rule of inference. (This would be a familiar !-rule for countable domains.) To show that the existence of extensions in predicate default logic with a !-rules is in the class 21 , given a predicate default theory (D; W ), we will present a 21 formula closely related to circumscription which says “(D; W ) has an extension”. To show that the problem is 21 hard, we will start with a circumscriptive formula and write a default theory with !-rules whose extensions are in one-to-one correspondence with the !models of the circumscriptive formula. Since Schlipf [24] showed that the existence of !-models of circumscriptive formulas is 21 -complete, this will be suJcient. In Section 2, we will review the basic de0nitions of circumscription and default logic. In Section 3, we will show how to modify the de0nitions of default logic to work over an !-logic base. In Section 4, we will show how this !-based default logic is interrelated with circumscription. In Section 5, we will draw some conclusions about the computablility theoretic complexity of default logic with !-rules based on the relationship with circumscription. 2. Circumscription and predicate default logic: basic denitions and results We will draw notation and terminology for circumscription from [11] and for default logic from [16]. For background information on complexity theory, see [23].

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2.1. Circumscription There are several varieties of circumscription, with propositional, predicate, formula, and prioritized circumscription being the most prominent. In propositional circumscription, a proposition is true only if it cannot be false; in predicate circumscription, one predicate is minimized while all others are held constant; in formula circumscription, one predicate is minimized while the other speci0ed predicates are allowed to vary; in prioritized circumscription, several predicates are minimized, with the minimization of some taking precedence over the minimization of others, and again some predicates are allowed to vary freely. Propositional circumscription and predicate circumscription are special cases of formula circumcscription. Lifschitz [9] showed that prioritized circumscription is no stronger than formula circumscription, so we will limit ourselves to a careful de0nition of formula circumscription. We will start with a 0rst-order language L with 0nite signature, a unary relation symbol P ∈L, and a tuple of other function or relation symbols Z = (Z1 ; : : : ; Zm ). We have a formula A(P; Z) in L, and we wish to consider only models of A in which the interpretation of P is minimal as we vary the interpretations of the Zi ’s and hold all other function and relation symbols constant. To be more speci0c, given M1 |= A and M2 |= A, we will say that M1 6P; Z M2 if |M1 | = |M2 |, M1 [C] = M2 [C] for symbols C in L\{P; Z1 ; : : : ; Zm }, and M1 [P] ⊆ M2 [P]. We will be concerned with the existence of 6P; Z -minimal models of A(P; Z). Note that to say “M is a 6P; Z -minimal model of A(P; Z)” is a 11 formula: M is a P; Z 6 -minimal model of A(P; Z) if it satis0es A(P; Z) ∧ ∀p; z(p & P → ¬A(p; z)). (The formula p & P is obviously expressible in L.) The formula A(P; Z) ∧ ∀p; z(p & P → ¬A (p; z)) will often be abbreviated CIRC[A; P; Z]. When the arities of Z1 ; : : : ; Zm and P are all 0, we are in the case of propositional circumscription. When Z is empty, we are in the case of predicate circumscription and will omit Z, writing CIRC[A; P]. 2.2. Propositional default logic Default logic was introduced by Reiter in [22]. There is an abundance of material about default logic, including proofs of all results in this section, to be found in [16] and in many other reference works. Denition 2.1. A default d in propositional language L is a triple P; J; where P and J are 0nite sets of formulas from L and is a formula in L, called the prerequisites, justi3cations, and conclusion of the default, respectively. When P = {’1 ; : : : ; ’m } and J = { 1 ; : : : ; n }, P; J; is usually written ’1 ; : : : ; ’m : M 1 ; : : : ; M

n

:

A pair (D; W ) where D is a collection of default rules and W is a set of formulas of L is called a default theory. W is usually referred to as the set of facts of (D; W ). Assuming that the set of atoms U of L is computable, we can identify it with !. We will call a default theory (D; W ) computable when both D, considered as a

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collection of triples, and W are computable under some reasonable encoding of the formulas of L. Denition 2.2. A theory T ⊆ L is called deductively closed with respect to context S in a default theory (D; W ) if and only if W ⊆ T , T is closed under propositional consequence, and furthermore, for every rule ’1 ; : : : ; ’m : M 1 ; : : : ; M if ’1 ∈T; : : : ; ’m ∈T; ¬

1

n

∈ D;

∈= S; : : : ; and ¬

n

∈= S then ∈T .

This says that T contains the facts W , is closed in the base logic, and when a rule of D is consistent with the context S, it is applied. If we de0ne (D; W ) (S) to be ∩ {T |T is deductively closed with respect to S in (D; W )}, the following lemma is quite straightforward: Lemma 2.3. (D; W ) (S) is itself deductively closed with respect to S in (D; W ). One of the most useful things to note about (D; W ) (S) is that if one transforms default theory (D; W ) to a collection of monotone rules 
 ’1 ; : : : ; ’m  ’1 ; : : : ; ’m : M 1 ; : : : ; M n ∈ D and ¬ ; : : : ; ¬ ∈ = S ; DS = 1 n  then (D; W ) (S) is precisely the collection of formulas of L which have derivations from axioms W using standard propositional logic augmented by the rules of DS . The fundamental notion in default logic is that of an extension of a default theory. Denition 2.4. Theory S is an extension of default theory (D; W ) if S = (D; W ) (S). From the fact that S ⊆ (D; W ) (S), we see that every element in an extension S has a proof from W in propositional logic augmented by DS . From the fact that (D; W ) (S) ⊆S, we know that an extension S is deductively closed with respect to itself in (D; W ). 3. Predicate default logic with !-rules While Reiter’s approach to default logic, along with some variations and re0nements, worked well for propositional languages and for predicate systems in which all defaults contained only closed formulas, the proper way in which to handle defaults containing open formulas was much debated for many years. In [10], Lifschitz noted that the usual practice of treating open variables as nothing more than metavariables for closed terms of the language produced results far weaker than intuition would suggest were reasonable. He proposed that the extensions of a default theory be dependent on a speci0c universe. Given a default theory, the extensions of that theory over in0nite domains might look diNerent than the extensions over

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a domain of cardinality 5, which in turn might be diNerent from the extensions over a domain of cardinality 1. His approach was based on a semantic de0nition of extension outlined in [6], and involved adding to the language a set of names for all elements of the intended domain. While his approach makes quite a bit of sense, it also has repercussions for theories in which all the rules are composed of closed formulas. In some cases, Lifschitz’ approach and Reiter’s produce diNerent sets of consequences for closed default theories. (See [10] for examples.) However, there are at least two major classes of theories for which they concur: those theories (D; W ) for which every (classical) model of W has the same 0nite cardinality; and those theories (D; W ) for which every (classical) model of W is in0nite. We will be concerned entirely with the latter case. Kaminski et al. [8] worked from Lifschitz’ de0nition of extension and showed that Reiter’s syntactic approach to open defaults could be made to conform to Lifschitz’ in the case of a 0xed universe U , even a potentially in0nite one, by the addition to the base logic of Carnap rules of the form {’(u)}u∈U : (∀x)’(x) Of course when U is !, these are the familiar !-rules from !-logic. (This assumes that all elements of U have names in L. For a detailed discussion of the relation of this assumption to the use of Herbrand models, see [8].) All of the above-mentioned authors worked in more generality than we will require here, and the interested reader is encouraged to read their treatments of the topic. We will work in a language L for predicate logic with equality that has some set of closed terms {Pn|n∈!}. Besides the standard logical rules and axioms for equality, we will include in our base logic axioms nP = mP for each n; m ∈! with n = m, as well as !-rules : : : A(n) P ::: ; ∀xA(x) where A is a formula of L and n ranges over !. Over this base logic, of course, all models of any set W of facts will be in0nite. We are now in a position to de0ne defaults, default theories, and extensions in the setting of a predicate logic with !-rules. Denition 3.1. A default d in predicate language L is a triple P; J; where P and J are 0nite sets of formulas from L and is a formula in L, called the prerequisites, justi3cations, and conclusion of the default, respectively. As before, when P = {’1 ; : : : ; ’m } and J = { 1 ; : : : ; n }, P; J; will be written ’1 ; : : : ; ’m : M 1 ; : : : ; M

n

:

Again, a default theory will be a pair (D; W ) consisting of a set of defaults and a set of facts. The meaning of computable is unchanged.

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So far, we are in very familiar territory, but we now have to decide what is to be done with defaults which have formulas with free variables. The simple answer is that because with our !-rules we have speci0ed that our universe is to be !, we can substitute nP for each n ∈! for each free variable x in a given default and be certain to have covered all possibilities. The only constraint is that we must do so uniformly within each default rule. Denition 3.2. Given a default d=

’1 ; : : : ; ’m : M 1 ; : : : ; M

n

;

a tuple of variables x = (x1 ; : : : ; xk ) so that all free variables appearing in {’1 ; : : : ; ’m ; 1 ; : : : ; n ; } are among those in x, and a tuple Pi = (i1 ; : : : ; ik ) where i1 ; : : : ; ik ∈!, de0ne a closed substitution instance of d to be the default P = d(i)

P : : : ; ’m (i) P : M 1 (i); P : : : ; M n (i) P ’1 (i); ; P (i)

where for a formula  in L with free variables among those in x, (Pi) is the result of replacing variable xj from x by the corresponding ij from Pi. Denition 3.3. Given a default d, let d! be the set  of all closed substitution instances of d. Given a collection of defaults D, let D! = d∈D d! . Now we are once again on familiar ground. We can treat (D! ; W ) exactly like we did D in the propositional case. Denition 3.4. A theory T ⊆ L is called !-deductively closed with respect to context S in a default theory (D; W ) if and only if W ⊆ T , T is closed under the rules of our base logic (including the !-rules), and furthermore, for every rule ’1 ; : : : ; ’m : M 1 ; : : : ; M if ’1 ∈T; : : : ; ’m ∈T; ¬

1

n

∈ D! ;

∈= S; : : : ; and ¬

= n ∈S

then ∈T .

!(D; W ) (S), and DS have essentially the same de0nition as in the propositional case, and the same results hold. We will call a set S which has the property that S = !(D; W ) (S) an !-extension. While the proof of Lemma 2.3 remains unchanged, proofs from W in our base logic augmented by DS are now in0nitary instead of 0nite. The proof that DS is adequate to generate !(D; W ) (S) from W is just as straightforward as before, but involves trans0nite induction on ordinals rather than induction on the integers. (For material on in0nitary proofs, including the relation between proofs as !-branching trees and proofs with computable ordinal length, see [4].)

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4. Predicate default logic and circumscription The relationship between default logic and circumscription has been much written about over the last 20 years. Most of the attention in the direction of expressing defaults with circumscription has focused on 0nite cases, either propositional or with explicit 0nite closed-world=unique name assumptions. The !-rules we have added to default logic function as a closed-world assumption in the countable case. In the other direction, both Lifschitz [10] and Etherington [2] note that there is a fairly simple way to capture CIRC[A; P; Z] in a predicate default logic, as long as Z contains all functions, constants, and relations other than P which appear in A. While their diNerent approaches require diNerent sets of facts W , both use the same single default : M ¬P(x)=¬P(x) to minimize P. In this section, we will 0rst show how, given a formula A, predicate P, and a tuple Z of symbols of L, one can construct a default theory whose !-extensions are precisely the !-models of CIRC[A; P; Z] without the assumption that all symbols in A are either P or in Z. Then we will show that the !-extensions of a given predicate default theory can be captured by a nearly circumscriptive formula. Theorem 4.1. Given a formula A in a 3rst-order language L, a unary relation P, and a set Z = {Z1 ; : : : ; Zm } of symbols of L, one can 3nd a predicate default theory (D A; P; Z ; W ) such that there is a one-to-one correspondence between the !-models M of CIRC[A; P; Z] and !-extensions of (D A; P; Z ; W ). Proof. For the sake of simplicity, and without loss of generality, let us assume that L has no function symbols except constants. Let R1 ; : : : ; Rm , c1 ; : : : ; cn , P be the collection of symbols of L appearing in A. We will de0ne (D A; P; Z ; W ) in the language L! ∪ P ∈!} ∪ {p; z1 ; : : : ; zm }. (We assume that p; z1 ; : : : ; zm ∈= L.) {p; z1 ; : : : ; zm } = L ∪ {k|k The idea will be to use defaults to choose random values for the relations and constants, and then rely on the base logic to prove A and the minimality of P among relations satisfying A. We will abbreviate the circumscriptive formula A(P; Z) ∧ [((∀x)(p(x) → P(x)) ∧ (∃x) (P(x) ∧ ¬p(x))) → ¬A(p; z)] by (P; Z; p; z). Let W = ∅ and let D A; P; Z consist of the following collections of defaults: (1) : MR(x)=R(x) for each relation symbol R appearing in A, where x is of the proper arity, (2) : M ¬R(x)=¬R(x) for each relation symbol R appearing in A, where x is of the proper arity, P = kP for each constant c appearing in A and each k ∈!, and (3) : Mc = k=c (4) : M ¬(P; Z; p; z)=(P; Z; p; z). The 0rst three rules choose values for every relation and constant symbol in A. We can rely on the strength of predicate logic augmented by !-rules to prove (P; Z; p; z) precisely if it is true of the chosen P and Z and every p and z. The adequacy of !-logic to prove exactly the true 11 formulas is proven with great care and detail in Chapter 6 of [4]. (Although we do not have set quanti0ers, the fact that no default

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provides information about p or z (they were symbols not in L) means that we have a de facto universal quanti0er on p and z, and of course A is 0rst-order, so (P; Z; p; z) is de facto 11 .) The 0nal default in D A; P; Z prevents us from having an !-extension given our choice of R1 ; : : : ; Rm , c1 ; : : : ; cn , P unless (P; Z; p; z) holds for all choices of p and z. (In general, a default : M ¬ = prevents extensions which do not contain . If there is no proof of independent of this default, then the default is applied, providing a proof of . But since this default was the only way to prove and ¬ is not consistent with any context containing , the default cannot be applied!) The one-to-one correspondence between !-models of CIRC[A; P; Z] and !-extensions of (D A; P; Z ; W ) is clear. Each !-extension of (D A; P; Z ; W ) is, essentially, the consequences in !-logic of the diagram of an !-model of CIRC[A; P; Z]. Thus, given an !-extension of (D A; P; Z ; W ), 0nding an !-model of CIRC[A; P; Z] is simple and effective. Given an !-model of CIRC[A; P; Z], one also needs a 11 complete oracle to determine a corresponding !-extension of (D A; P; Z ; W ), because provability in !-logic is 11 -complete. Our other goal in this section is to show that given a predicate default theory (D; W ) we can write a nearly circumscriptive formula whose !-models are the !-extensions of (D; W ). To begin, we will need a couple of easy lemmas. Lemma 4.2. If (D; W ) is a predicate default theory, and S; T ⊆ L are sets of formulas, then there if a 3rst-order formula (D; W; T; S) saying “T is !-deductively closed in (D; W ) with context S”. Proof. We will work in a language L whose individuals correspond through some encoding to the formulas of L. We will rely on the computable nature of the axioms of predicate logic to 0nd a formula saying “(the code of) is (the code of) an axiom”. (For example, we could use the language of number theory for L and include in  the axioms of primitive recursive arithmetic and a #0 formula for determining whether formulas are axioms.) For the sake of intuition and readability, we will write “’ ∈T ” for “T (p)” where T is a unary relation in L and p is the code in L of the L-formula ’. It is simple to 0nd formulas saying: • for every formula , if ∈W then ∈T , • for every axiom of predicate logic, ∈T , • for every pair for formulas ’ and , if ’ ∈T and ’ → ∈T , then ∈T , • for every formula ’(x), if ’(iP)∈T for all i ∈!, then (∀x)’(x) ∈T , and • for every default ’1 ; : : : ; ’m : M 1 ; : : : ; M n if d ∈D and ’1 ; : : : ; ’m ∈T and ¬ 1 ; : : : ; ¬ d=

n

∈= S, then ∈T .

Taken together, these formulas will say “T is !-deductively closed in (D; W ) with context S”.

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Lemma 4.3. If S is !-deductively closed in (D; W ) with context S, and no T & S is !-deductively closed in (D; W ) with context S, then S = !(D; W ) (S). Proof. Since S is !-deductively closed in (D; W ) with context S, !(D; W ) (S) ⊆ S. By (the analog of) Lemma 2.3, we know that !(D; W ) (S) is !-deductively closed in (D; W ) with context S, so it is impossible that !(D;W ) (S) & S, and thus !(D; W ) (S) = S. The same lemmas and the same proofs hold for standard default logic, so the discussion below could apply equally well to standard default logic. From these lemmas, we can see that (D; W; S; S) ∧ ∀T (T & S → ¬(D; W; T; S)) says “S is a minimal set !-deductively closed in (D; W ) with respect to S”, and hence “S is an !-extension of (D; W )”. The form of this minimality condition calls out for a connection to circumscription, and one is easily found. However, the tempting obvious translation into the language of circumscription does not work: CIRC[(D; W; S; S); S] says “S is !-deductively closed with respect to itself in (D; W ), and no subset of S is !-deductively closed with respect to itself in (D; W )”. This is not equivalent to saying that no subset of S is !-deductively closed with respect to context S in (D; W ). Example 4.4. If D = {: M =⊥} and W = ∅, then (D; W ) has no extensions. This default is similar to the 0nal one in our embedding of CIRC[A; P; Z], and for the same reasons insists that there be an independent derivation of ⊥ from the rest of (D; W ). Since the rest of (D; W ) is empty, this is impossible. The only case in which a set S of formulas is deductively closed in (D; W ) using itself as context is when S consists of all formulas, since a deductively closed S has to contain ⊥ and be closed in the base logic. Thus, the set of all formulas is the minimal set deductively closed in (D; W ) using itself as context. However, if one uses the set of all formulas as context, any set deductively closed in the base logic will be deductively closed in (D; W ) with respect to that context. What we want to do is minimize one of the occurrences of S in (D; W; S; S) while holding the other steady as S. This is not something that formula circumscription—or any other variety of circumscription of which I am aware—is capable of expressing directly. However, one can come close with the formula [(∀x)T (x) ↔ S(x)] ∧ CIRC[(D; W; T; S); T ]: Something comparable to this formula was generated by F. Lin and Y. Shoham in the context of logic programs and the modalities of their logic GK [12]. 1 The fact that they and I arrived at this formula and not a purely circumscriptive one is one indication that this may well be the most natural way to translate defaults, logic programs, etc., into something very close to circumscription. 1

Thanks to V. Lifschitz for pointing out this related work.

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From the comments and lemmas above, the following theorem emerges: Theorem 4.5. Let predicate default theory (D; W ) be given, and let (D; W; S; T ) be as above. If M is an L structure, then M |= [(∀x)T (x) ↔ S(x)] ∧ CIRC[(D; W; T; S); T ] if and only if the set of formulas corresponding to M [S] is an !-extension of (D; W ). 5. Computability theoretic corollaries We have shown that the !-models of a circumscriptive formula can be expressed as the !-extensions of a predicate default theory, and that we can capture the !-extensions of a predicate default theory with a nearly circumscriptive formula. What does this tell us about the computability theoretic complexity of predicate default logic with !-rules? Quite a bit, it turns out. Let us 0rst recall some results about standard default logic (propositional or predicate) and circumscription. The following results were proved in [14] for nonmonotone rule systems, but they carry over to standard default logic quite straightforwardly. Theorem 5.1. For any computable default theory (D; W ), there is a computable tree $(D;W ) ⊆ !¡! such that there is an e:ective, one-to-one degree-preserving correspondence between the set of all extensions of (D; W ) and [$(D;W ) ], the set of all in3nite paths through $(D;W ) . Theorem 5.2. Let $ be any computable tree. There is a default theory (D$ ; W$ ) such that there is an e:ective one-to-one degree-preserving correspondence between [$] and the set of all extensions of (D$ ; W$ ). There are a host of results about computable trees and their paths, for which we refer the reader to [1]. The pair in which we are interested state that the set of trees which have no paths is 11 -complete and that the set of trees which have at least one path is 11 -complete. These lead us to a pair of corollaries. Corollary 5.3. The set of computable default theories (without !-rules) which have at least one extension is 11 -complete. Corollary 5.4. The set of computable default theories (without !-rules) which have no extensions is 11 -complete. While for a propositional formula, the existence of a model minimizing a single proposition is decidable (a cursory inspection shows that one exists if the circumscriptive formula is satis0able), the question of the existence of in0nite models in the cases of predicate and formula circumscription is much more diJcult.

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Theorem 5.5 (Schlipf [24]). The set of formulas A(P) (or A(P; Z)) such that there exists a 6P -minimal (or 6P; Z -minimal) !-model of A(P) (or A(P; Z)) is 21 -complete. Schlipf’s proof that the set is in 21 is quite short and straightforward. To assert the existence of interpretations of P; Z and all other symbols in A uses some number of second-order existential quanti0ers. Because saying M |= CIRC[A; P; Z] is 11 , we have some number of existential quanti0ers followed by a 11 formula, giving us a 21 formula. That any 21 set can be captured as the subset of formulas from a given set of formulas which have 6P -minimal !-models is much harder. Schlipf accomplishes this by showing that the true ! in Kripke–Platek set theory can be captured through predicate circumscription, and then is able to treat the second order 21 formula as a formula in an admissible fragment of KP. This proof is far beyond the scope this paper. Let us note that the nearly circumscriptive formula expressing “S is an !-extension of (D; W ), [(∀x)T (x) ↔ S(x)] ∧ CIRC[(D; W; T; S); T ] is 11 , since CIRC[(D; W; T; S); T ] is 11 and of course (∀x)T (x) ↔ S(x) is 0rst order. Thus, the sentence asserting the existence of a model of [(∀x)T (x) ↔ S(x)] ∧ CIRC[(D; W; T; S); T ]; that is the sentence asserting the existence of an !-extension of the predicate default theory (D; W )is 21 in D and W . In light of Theorem 5.5 and because, as we have just seen, the set of computable predicate default theories with !-extensions is in 21 , it is in fact possible, given a computable predicate default theory (D; W ), to write a purely circumscriptive formula which will have an !-model if and only if (D; W ) has an !-extension. However, this formula will be extremely complicated compared to the one above. Another pair conclusions we can draw immediately from Theorem 5.5 and the translations of the previous section is: Corollary 5.6. The set of computable predicate default theories which have at least one !-extension is 21 -complete. Corollary 5.7. The set of computable predicate default theories which have no !extensions is 21 -complete. If we think of extensions as coherent, justi0ed points of view in a framework represented by the rules of D, there are two important sets of formulas we want to take note of: those which can be part of some coherent, justi0ed point of view, and those which must be part of any coherent, justi0ed point of view. If you take as your set of conclusions things which are present in at least one extension, you are reasoning credulously or bravely (both terms are widely used). If, on the other hand, you believe

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only those facts true in all extensions, you are reasoning skeptically or cautiously. (The standard contrasts are credulous vs. skeptical reasoning and brave vs. cautious reasoning.) The following de0nition applies equally well to 0nite propositional default theories and predicate default theories with or without !-rules. Denition 5.8. An formula ’ is in the set of credulous (!-)consequences of a default theory D if there is some (!-)extension S of D such that ’ ∈S. A formula ’ is in the set of skeptical (!-)consequences of a default theory D if ’ ∈S for all (!-)extensions S of D. Because a default theory (D; W ) can easily be modi0ed to a default theory (D ; W ) with the property that the (!-)extensions of (D ; W ) will be precisely the (!-)extensions of (D; W ) which include a given formula ’, the set of credulous consequences of a given default theory is, in general, at the same level of the computability theoretic hierarchy as the problem of testing whether a given theory has an extension. Similarly, one can modify (D; W ) to insist that a given formula ’ be excluded from extensions, and that places skeptical reasoning and the problem of testing for a lack of extensions at the same level of the hierarchy. Thus, while the sets of credulous and skeptical consequences of standard default theories are in general respectively 11 - and 11 -complete, the sets of credulous and skeptical !-consequences of predicate default theories are in general respectively 21 and 21 -complete.

6. Conclusions and further research While the relationship between default logic and circumscription has been much studied in 0nite cases, we have shown that the same ideas which connect them in the 0nite case, closed-world and unique name assumptions, also allow for a connection in the countable case. A corollary of this is that an explicit countable closed-world assumption pushes default logic from the relatively well-understood world of 11 -logic to the much more esoteric world of 21 -logic inhabited also by predicate and formula circumscription. One question which can now be explored is the relationship between stable model logic programming over in0nite domains and abductive logic programming, proved to be D11 -complete by Marek et al. [15]. (D11 is the class of sets which can be de0ned as the diNerence between two 11 sets.) Another avenue for further study is the expression of skeptical reasoning for predicate default logic with !-rules directly in a 21 proof theoretical framework (see [5]) analogous to the analysis of standard 11 skeptical reasoning in [20]. Let me conclude by thanking Denis Hirschfeldt for his careful and insightful reading of an earlier draft of this paper and for much helpful advice. One could not ask for a better colleague.

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References [1] D. Cenzer, J.B. Remmel, 10 classes in mathematics, in: Y.L. Ershov, S.S. Goncharov, V.W. Marek, A. Nerode, J.B. Remmel (Eds.), Handbook of Recursive Mathematics, Vol. 2, North-Holland, Amsterdam, 1998. [2] D.W. Etherington, Relating default logic and circumscription, in: Proc. 11th Internat. Joint Conf. on Arti0cial Intelligence, Milan, Italy, August 1987. [3] M. Gelfond, V. Lifschitz, The stable semantics for logic programs, in: R. Kowalski, K. Bowen (Eds.), Proc. 5th Ann. Symp. on Logic Programming, MIT Press, Cambridge, MA, 1988. [4] J.-Y. Girard, Proof Theory and Logical Complexity, Vol. 1, Bibliopolis, Napooli, 1987. [5] J.-Y. Girard, Proof Theory and Logical Complexity, Vol. 2, North-Holland, Amsterdam, to appear. [6] R. Guerreiro, M. Casanova, An alternative semantics for default logic, preprint, in: The 3rd Internat. Workshop on Nonmonotonic Reasoning, South Lake Tahoe, 1990. [7] M. Kaminski, A comparative study of open default theories, Arti0cial Intelligence 77 (1995) 285–319. [8] M. Kaminski, J.A. Makowsky, M. Tiomkin, Extensions for open default theories via the domain closure assumption, J. Logic Comput. 8 (1998) 169–187. [9] V. Lifschitz, Computing circumscription, in: Proc. IJCAI, Los Angeles, CA, 1985. [10] V. Lifschitz, On open defaults, in: J. Lloyd (Ed.), Computational Logic. Symposium Proceedings, Springer, Berlin, 1990. [11] V. Lifschitz, Circumscription, in: D.M. Gabbay, C.J. Hogger, J.A. Robinson (Eds.), Handbook of Logic in Arti0cial Intelligence and Logic Programming, Nonmonotonic Reasoning and Uncertain Reasoning, Vol. 3, Clarendon Press, Oxford, 1994. [12] F. Lin, Y. Shoham, A logic of knowledge and justi0ed assumptions, Arti0cial Intelligence 57 (1992) 271–289. [13] W. Marek, A. Nerode, J.B. Remmel, Nonmonotonic rule systems I, Ann. Math. Arti0cial Intelligence 1 (1990) 241–273. [14] W. Marek, A. Nerode, J.B. Remmel, Nonmonotonic rule systems II, Ann. Math. Arti0cial Intelligence 5 (1992) 229–263. [15] W. Marek, A. Nerode, J.B. Remmel, On the complexity of abduction, in: Proc. 11th Ann. IEEE Symp. on Logic in Computer Science, IEEE Computer Society Press, Los Alamitos, CA, 1996. [16] W. Marek, M. TruszczyUnski, Nonmonotonic Logic: Context-dependent Reasoning, Springer, Berlin, 1993. [17] J. McCarthy, Circumscription—a form of nonmonotonic reasoning, Arti0cial Intelligence 13 (1980) 27–39. [18] D. McDermott, J. Doyle, Nonmonotonic logic I, Arti0cial Intelligence 13 (1980) 41–72. [19] R. Milnikel, Nonmonotonic logic: a monotonic approach, Ph.D. Thesis, Cornell University, May 1999 [for further reading]. [20] R. Milnikel, Tableau proofs for skeptical reasoning, in: T. Eiter, G. Gottlob (Eds.), Proc. FLOC ’99 Workshop on Complexity-Theoretic and Recursion-Theoretic Methods in Databases, Arti0cial Intelligence, and Finite Model Theory, Trento, Italy, 1999, pp. 62–71. [21] R.C. Moore, Possible-world semantics for autoepistemic logic, in: R. Reiter (Ed.), Proc. Workshop on Non-Monotonic Reasoning, 1984. Reprinted in: Logic and Representation (CSLI Lecture Notes 39), Stanford Univ. Press, Stanford, CA, 1995, pp. 145–151. [22] R. Reiter, A logic for default reasoning, Arti0cial Intelligence 13 (1980) 81–132. [23] H. Rogers, Theory of Recursive Functions and ENective Computability, McGraw-Hill, New York, 1967. [24] J. Schlipf, Decidability and de0nability with circumscription, Ann. Pure Appl. Logic 35 (1987) 173–191.