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Linear Logic for Taxonomical Networks and Database Updates Preliminary Report
Electronic Notes in Theoretical Computer Science 3 (1996)
URL: http://www.elsevier.nl/locate/entcs/volume3.html
14 pages
Linear Logic for Taxonomical Networks and Database Updates Preliminary Report Christophe Fouquere
LIPN-CNRS URA 1507 Universite Paris-Nord 93430 Villetaneuse, France Jacqueline Vauzeilles
LIPN-CNRS URA 1507 Universite Paris-Nord 93430 Villetaneuse, France
Abstract
The aim of this paper is to propose a logical way to handle uncertain knowledge and change. Databases, diagnostic, planication, taxonomy are some of the domains concerned by this problem. This paper focuses on the means Linear Logic oers to represent taxonomical networks and to perform updates of databases containing incomplete information. The two problems are rst expressed in graph theory: a taxonomical network is a structure for representing knowledge as a graph whose vertices are concepts, and edges are relations between concepts a database is specied by facts, deduction rules, i.e. edges between literals, and update constraints. Their formalization in Linear Logic is performed in a very similar way.
1 Introduction
The aim of this paper is to propose Linear Logic for the representation of uncertain knowledge and change: basically, some knowledge (inference or fact), legitimate in most of the situations, has to be cancelled or forgotten in some special cases. Databases, diagnostic, planication, taxonomy are some of the domains concerned by this problem. Since 1970, a lot of formalisms and systems have been developed for formalizing these problems. Following previous work 2,4,5], we show how Linear Logic (LL) can adequately represent taxo-
c 1996 Elsevier Science B. V.
Fouquere and Vauzeilles
nomical networks with default knowledge, and updates in databases containing incomplete information. In the rst part, we prove that LL can correctly modelize taxonomical networks. A taxonomical network is a structure for representing knowledge as a directed graph whose vertices are concepts, and edges are relations between concepts. We consider three kinds of edges: strict, default and exception edges. These edges precise the way properties (hereafter concept names) are inherited. A default edge between A and B means that A is generally a B , or A inherits the property B . An exception edge between A and B is a direct inhibition of a default edge between A and B , namely A is not a B or A has not the property B , whatever the number or kind of default paths between A and B . A strict edge between A and B states that A is a B (or A has the property B ) whatever the number or kind of exception paths between A and B . We dene the notion of compatible vertices of the graph w.r.t. a set of nodes called the facts. A set of compatible vertices is a maximal set of vertices s.t. exception statements are satised, i.e. each vertex in this set can be inherited by one of the facts of the graph w.r.t. the meanings of the edges. The reader may nd in 4] a proof of equivalences between this description in graph theory, a representation in a fragment of LL and a representation in Reiter's Default Logic. In short, sets of compatible vertices are exactly provable sequents whose right hand side is the conjunction of the variables representing the vertices. In this paper, we propose a new formalization of these networks in such a way that the meaning of sequents becomes straightforward. Furthermore, this induces a representation schema for a whole class of networks, including disjunctive networks. We show in the second part that this schema extends also to database updates. The results presented here are an adaptation in the formalism used for taxonomical networks of previous results of Bidoit, Cerrito and Froidevaux 1,2]. A database can also be viewed as a set of initial facts (which can be positive or negative) and a graph of whose vertices are facts and edges precise the way deductions can be performed. Edges are of two kinds: strict or default edges. Since we want to deal with \consistent" databases (i.e. databases in which it cannot be deduced a fact and its negation) intuitively it supposes that there exists an exception edge between vertices A and :A. For each database, the notions of static semantics and update semantics w.r.t. the insertion of a new fact, are dened through deductions in Linear Logic. Then the database resulting from an update is dened syntactically and it is shown that the static semantics of the update database coincides with the update semantics of the database.
2
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2 Taxonomical Networks This part is a more general formulation of previous results 4,5], which was rst presented in 3]. We rst give a few examples to show graphically what we expect of such taxonomical networks: the dashed (resp. bold) arrow indicates an exception (resp. a strict edge) while default edges are drawn as simple arrows.
Example 2.1
Molluscs are generally Shell-bearers, Cephalopods are generally Molluscs but exceptions to Shell-bearers, Nautili are generally Cephalopods but are Shell-bearers. Shell-bearers are Invertebrate. We want to deduce that Nautili are Cephalopods, Molluscs, Shellbearers and Invertebrate.
Example 2.2
The two exceptions (double arrow on a dashed line) dene an even cycle, giving rise to two interpretations, A being true. Let A be true on one hand one can infer B , C on the other hand one can infer B , C . 0
0
Example 2.3
A is generally a B , B is generally a C and C is an exception to be a B .
We want to conclude that there is no solution to this problem, interpreting results when A is true.
Invertebrate Shell-bearers
Molluscs Cephalopods
Nautili C
C'
B
B' A C B A
2.1 Taxonomical Networks as graphs The graphs we modelize are directed nite graphs with three kinds of edges. Furthermore, we forbid cycles in the graph restricted to strict and default edges. However, the reader may nd in 3] a way to avoid such a restriction.
Denition 2.4
A taxonomic graph is a directed graph G = hD ! ! 9 9 Ki such that: D is a nite set of vertices ! ! 9 9 K are three irre exive relations on D if (a b) 2 ! (resp. (a b) 2 ! (a b) 2 9 9 K) it is said that there exists a strict (resp. a default, an exception) edge between a and b hD ! !i is acyclic 3
Fouquere and Vauzeilles
A Taxonomical Network (TN) is dened as a pair N = hF Gi where G = hD ! ! 9 9 Ki is a taxonomic graph and F , the set of facts, is a subset of D.
The main aim of this section is the denition of compatible vertices of a TN informally a subset E of vertices of a TN N is constituted of compatible vertices if, for each vertex , there exists a path from a vertex of to such that each edge is a strict or a default edge (a strictly default path) and if there does not exist a vertex of E with an exception edge between and (except if there exists a strictly default path between an element of and whose last edge is a strict edge). First inductive denitions of correct path, default correct path, and strictly correct path are given. Denition 2.5 Let N = h Gi be a TN. A sequence 1 of vertices of D is a correct path (resp. default correct path, resp. strictly correct path) in N if and only if one of the following two conditions holds: = 1 and 1 2 , 1 and on the one hand 1 1 is a default correct path, or 1 1 is a correct path and there exists a strictly correct path 1 such that = 1 on the other hand ( 1 ) 2 ! or ( 1 ) 2 ! or ( 1 ) 2 9 9 K (resp. ( 1 ) 2 ! or ( 1 ) 2 !, resp. ( 1 ) 2 !) A sequence 1 of vertices of G is a strictly default path if and only if 2 and either = 1 or for each (2 ) we have ( 1 ) 2 ! or 1 ( 1 ) 2 !. Remark 2.6 Each strictly correct path is a default correct path, and each default correct path is a correct path. Moreover each strictly default path is a default correct path. Example 2.7 The relations ! ! and 9 9 K are respectively represented by bold, simple and dashed arrows. (ex. 2.1 cont.): The TN N1 is represented by the graph of example 2.1: (the following abbreviations are used: and stand respectively for Nautili, Cephalopods, Molluscs, Shell-bearers and Invertebrate) D1 = f g 1 = f g. For instance is a default correct path (so a correct path) then is a strictly correct path (so a default correct path and a correct path) moreover and are strictly default paths. is a correct path but not a default path since is a strictly correct path it can be concluded that is a strictly correct path (also a default correct path and a correct path), but is not a strictly default path. 4 a
F
b
b
a
F
F
n
a
a
a
a : : : an
F
n >
a : : : an
a : : : an
b : : : bq
bq
an
an
an
an
an
an
an
an
an
an
an
an
an
a : : : an
a
F
ai
n
i
i
n
ai
ai
ai
N C M S
N C M S I
F
I
N
N C M S
N C M S I
N C M S I
N S I
N C S
N S
N C S I
N C S I
Fouquere and Vauzeilles
(ex. 2.2 cont.): The TN N2 is represented by the graph of example 2.2: D2 = fA B B C C g F2 = fAg. For instance A B C and A B C are default correct paths so are correct paths A B B and A B B are correct paths but are not default correct paths then A B B C and A B B B C are not correct paths. (ex. 2.3 cont.): The TN N3 is represented by the graph of example 2.3: D3 = fA B C g F3 = fAg. For instance A B C is a default correct path and a strictly default path A B C B is a correct path but not a default correct path A B C B C is not a correct path. 0
0
0
0
0
0
0
0
0
hD ! ! i be a TN a TN M hF G i with G hE ! ! i is a (subTN) N complete if and only if G is a subgraph of G such that: if a 2 E then there exists a correct path a : : : an in M (and in N ) such that a an if a 2 E then either fb a b 2 ! g fc a c 2 ! g and fb a b 2 ! g fc a c 2 ! g and fb a b 2 g fc a c 2 g or fb a b 2 ! g fb a b 2 ! g fb a b 2 g a TN M hF G i with G hE ! ! i is N -completely correct if and only if M is N -complete and if: each path in M is correct if a 2 E and fb a b 2 ! g fb a b 2 ! g fb a b 2 g then either fb a b 2 ! g fb a b 2 ! g fb a b 2 g or there exists a correct path a : : : an a in M such that an an 2
Denition 2.8 Let N = hF G i with G N
=
M
N
=
=
M
N
M
M
M
N
9 9 KN
9 9 KM
N
1
=
(
(
)
)
=
=
M
(
N
(
)
=
M
(
M
)
)
N
)
(
M
(
(
)
=
(
)
9 9 KN
M
=
(
)
9 9 KM
=
M
=
(
(
N
9 9 KM
)
M
=
)
M
(
N
M
=
=
(
=
1
=
9 9 KM
)
)
=
M
(
)
)
=
9 9 KM
9 9 KN
(
=
1
)
and there does not exist a strictly correct path b1 : : : bq such that bq = a. 9 9 KM
Example 2.9
(ex. 2.1 cont.): N1 is the only N1 -completely correct subTN of N1. (ex. 2.2 cont.): There are the two completely correct subTNs of N2 . C
C'
B
B
B' A
B' A
(ex. 2.3 cont.): The TN N3 does not admit a completely correct subTN: 5
Fouquere and Vauzeilles
C
the following subTNs are not comB B pletely correct subTNs of N : A A N is not a completely correct subTN of itself because the path A B C B C is not correct. Denition 2.10 The vertices b : : : b of a net N = hF Gi are said to be compatible if and only if there exists a completely correct subTN M = hF G i of N such that b : : : b are vertices of G and for each i (1 i n), if there exists a correct path in M whose end is b and whose last edge is an 3
3
n
1
M
1
n
M
i
exception edge then there exists a strictly correct path whose end is bi .
Example 2.11
(ex. 2.1 cont.): For instance, Nautili, Cephalopods and Shell-bearers are compatible vertices of N1. (ex. 2.2 cont.): For instance, A B and C are compatible vertices of N2 , while B and B are not compatible vertices of N2 . Note that B and B belong to the two completely correct subTNs of N2 but, in the rst completely correct subTN of N2 , A B B is a correct path whose last edge is an exception edge and in the second completely correct subTN of N2, A B B is a correct path whose last edge is an exception edge. (ex. 2.3 cont.): Since N3 does not admit a completely correct subTN, there do not exist compatible vertices of N3 . The three previous examples prove that a TN can admit zero, one or many completely correct subgraphs.
0
0
0
0
2.2 Taxonomical Networks in Linear Logic We show in this section that Linear Logic can formalize taxonomical networks with strict, default and exception edges. We consider the fragment of (intuitionistic) Linear Logic including the multiplicative constant 1, the multiplicative connectives , the additive connective & , and the exponential connective ! the properties of the connectives of this fragment of Linear Logic are essential in our representation: the properties of the constant \1" (which is a neutral element for the multiplicative conjunction \") can be used to cancel information (we use (A 1) to cancel A) while those of the exponential connective \!" can, using !(A 1), cancel all occurrences of A. Let N = hF Gi with G = hD ! ! 9 9 K i be a TN. The language LN is the linear language constructed from a set V = DfA A 2 DgfA A 2 Dg fA A 2 Dg fA A 2 Dg of propositional variables and including the connectives & ! and the constant 1. 6 N
N
N
D
Fouquere and Vauzeilles
Denition 2.12 We set:
if fB (A B ) 2 !g = fC (A C ) 2 9 9 Kg = fD (A D) 2 !g = then (A) A 1 else (A) A B1 : : : Bm C1 : : : Cp D1 : : : Dn with
fBi i 2 m g fB A B 2 !g fCi i 2 p g fC A C 2 g fDi i 2 n g fD A D 2 !g A A A A A A A A
G
D
1
] =
(
)
1
] =
(
)
1
] =
(
)
2D
!( )
2D
!(
Example 2.13
(ex. 2.1 cont.).
N
S
C C
is provable.
()
(
& 1))
99 K
!(
) & !(
A !(A 1))]
is dened with the formulas:
M S M S S I I . The following sequent N ` N S I C M.
1
G D
It can be read: Let be the formula describing the properties of the various kind of links, instantiated on the set of nodes D, let be the formula describing the graph G , let N be the only fact (i.e. N is a sure knowledge and N is a given formula), then the conjunction N S I C M can be derived, i.e. fN S I C M g is a set of compatible vertices. Remark 2.14 The assertion of N implies the access (via N ) to S and C and the possibility to reach N . The access to C implies the access to C (since C is not asserted), and so the access (via C ) to M and S , and the possibility to reach C . The access to M implies the access to M (since M is not asserted), and so the access (via M ) to S , and the possibility to reach M . The access to S implies the deletion of S , and the access to S implies (via S ) the access to I , and the possibility to reach S . The access to I implies the possibility to reach I . Note that the sequent C ` C M is provable while the sequent C ` C M S is not provable: the assertion of C implies the access (via C ) to M and S , and the possibility to reach C the access to M implies the access to M (since M is not asserted), and so the access (via M ) to S , and the possibility to reach M since S is asserted, S has to be deleted and so S cannot be reached (via S ). (ex. 2.2 cont.): A ` B B B ` B C B ` B C C ` 1 C ` 1. The two following sequents are provable: D
G
G
D
G
D
0
0
7
0
0
0
Fouquere and Vauzeilles
A ` A B C , A ` A B C (ex. 2.3 cont.): A ` B B ` C C ` B . There is no provable sequent A ` M (M being a product of some elements of fA B C g). The following theorem gives the equivalence between compatible sets of nodes and provable sequents whose right hand side is a conjunction of propositional variables. Theorem 2.15 The vertices B1 : : : Bn of the TN N = hF Gi are compatible if and only if the sequent F ` B1 : : : Bn is provable (with F = D1 : : : Dr if F = fD1 : : : Dr g). Proof. see 3] Note that a representation in classical logic cannot give the expected results as classical logic is unable to deal adequately with nonmonotonicity. The fact that Linear Logic manages variables as resources is fundamental: an exception on a node is expressed as the retrieval (A 1) of the corresponding variable. Note nally that we separate in the prerequisites of the sequent the facts (a sequence of A formulas), from the description of the network ( ), from the description of the meaning of the links ( ). This way of representing taxonomic networks and the meaning of relations between nodes can be generalized in fact to disjunctive networks 3]. 0
G
D
G
G
0
D
D
G
G
D
D
3 Database Updates
Updates are fundamental in database systems. However, their dynamic e ect w.r.t. the database cannot be expressed in classical logic. We show in this section that the twofold point of view of Linear Logic, i.e. dynamic and static, helps to represent this process. This is an adaptation of 1,2] in a formalism close to the one used in the previous section. It is worthy to note that if the initial database is \consistent", it must remain \consistent" after the insertion of a new fact. All facts are represented by atomic or negation of atomic formulae. A fact can be true, false or unknown. The deductions in the database can be performed in two ways represented by two kinds of edges of a graph: intuitively, the rule A!B means that if A is derived, then B is necessarily deduced while the meaning of the rule A!B is that if A is derived then B occurs unless its deduction contradicts other information. If the database contains the rule A!B , if A holds and if the database is updated by the insertion of :B then A has to be deleted, while if :B holds and if the database is updated by the insertion of A then :B has to be deleted. Let P be a set of propositional variables. We set L = P fp p 2 Pg (p is a notation that will allow to interpret the negation of p). A literal is an element of L. 8
Fouquere and Vauzeilles
Denition 3.1
A database graph over L is a directed graph G = hD ! !i such that: the set D of vertices is included in L ! ! are two irre exive relations on D if (a b) 2 ! (resp. (a b) 2 !) it is said that there exists a strict (resp. a default) edge between a and b and that a is the source of the edge and that b is its target for any vertices a and b, there is not simultaneously (a b) 2 ! and (a b) 2 ! no vertex is the target of a default edge and the source of a strict edge hD ! !i is acyclic. Let G = hD ! !i be a database graph. Let G = hD ri be the non-directed graph such that (a b) 2 r i (a b) 2 ! or (b a) 2 ! or (a b) 2 ! or (b a) 2 !. G is safe i there are no vertices a and :a connected by a path in G . A Database (DB ) is dened as a pair DB = hF Gi where G = hD ! !i is a safe database graph and F , the set of facts, is a subset of D. 0
0
3.1 Databases in Linear Logic Let DB = hF Gi with G = hD ! !i be a database we set A = fA A 2 D or A 2 Dg and D = A fA A 2 Ag for each A 2 A we set A = A. We set vs = D fA A 2 Dg fA A 2 Dg fA A 2 Dg fA A 2 Dg and vu = fAi A 2 Dg fAins A 2 Dg fAd A 2 Dg fAdel A 2 Dg. Ls (resp. Lu) is the linear language constructed from the set vs (resp vs vu) of propositional variables and including the connectives & ! and the constant 1. Static Rules For each L 2 D we set: if fB (L B ) 2 !g = fC (L C ) 2 !g = then stat(L) L 1 else stat(L) L B1 : : : Bm C1 : : : Cn with fBk k 2 1 m]g = fB (L B ) 2 !g fCk k 2 1 n]g = fC (L C ) 2 !g
Update Rules Insertion Rules For each L 2 D we set: if fC (L C ) 2 !g = then ins(L) Li 1 else ins(L) Li C1ins : : : Cnins with fCk k 2 1 n]g = fC (L C ) 2 !g Deletion Rules For each L 2 D we set: if fC (C L) 2 !g = then del(L) Ld 1 del else del(L) Ld Adel 1 : : : Ar with fAk k 2 1 r]g = fA (A L) 2 !g. Example 3.2
9
Fouquere and Vauzeilles
L
Static Rules
Update Rules
B
A
C
D
E
:L
L A B A 1 B 1 C L D 1 E 1 L 1 A 1 B 1 C 1 D 1 E 1
Insertion rules Li Ains, Ai 1, B i 1, C i 1, Di 1, E i 1, Li 1, Ai 1, B i 1, C i 1, Di 1, E i 1, Deletion Rules Ld 1, Ad Ldel , B d 1, C d 1, Dd 1, E d 1, Ld 1, Ad 1, B d 1, C d 1, Dd 1, E d 1
As with the logical representation of taxonomical networks, we describe the database graph with one formula G , the meaning of the various kinds of relations by another formula D , and the set of facts F by a sequence of formulas F . We give now the denitions of G and D . The following subsections are devoted to dene the static model of a DB, i.e. what can be deduced with a Database when no changes occur, and to dene the updates of a Database. We end by proving that there is an equivalence between logical updates (i.e. provable sequents whose right hand side is a product of literals, when one adds an insertion literal as a prerequisite) and graph updates (i.e. graph modications by inserting a new fact w.r.t. the meaning of edges). G is a product of all static and update rules, i.e. the logical counterpart of the database graph. D is a product of the logical interpretation of the various kinds of edges and the insertion of a literal. These logical interpretations are obviously to be considered over the set of variables. We distinguish the insertion of a variable (Lins), the default/exception edges (L /L), the strict edges and the deletion of a variable (L , Ldel ). To insert a variable L implies the fact that this variable is sure (L ), and that we have to update the knowledge w.r.t. the insertion rule of L (Li) and the deletion rule of its negation (Ldel ). Default/exception edges are used in an exclusive way (hence \&" between the two interpretations), and an exception (L) deletes each default occurrence (L 1), otherwise the default implies the literal to be sure (L ). The fact that a literal A and its negation A are mutually exclusive is expressed by the conjunctive connective \&" with the same kind of interpretation w.r.t. sure knowledge and deletion (formula ). The fact that a knowledge is sure A implies this knowledge (but we can forget this piece of information: A&1), what is graphically implied by this literal (hence A? ), and that there is an exception on the negation (A). The deletion of its negation (Adel ) implies to 10
Fouquere and Vauzeilles
re the corresponding deletion rule (Ad) and the deletion of each occurrence of this negation (A 1). Denition 3.3 We set: G L2D !stat(L) !ins(L) !del(L) (A) !(A A (A & 1) A ) !(Adel Ad !(A 1)) D A2A ( (A) & (A)) ins L Li Ldel ) L2D !(L L ) & !(L !(L 1))]!(L 3.2 Static models A static model is a maximal consistent set of variables that can be deduced by a DB. A set is consistent if it does not contain a variable and its negation. Deductibility is dened in terms of provable sequents, however it follows from graph deductibility w.r.t. the meaning of edges, in the same spirit of what has been done in the previous section.
Denition 3.4
A set of literals is consistent i it does not contain A and A. A consistent set m of literals is a static model of a database DB
= hF G i
if: ` M is provable (with F = D : : : D if F = fD1 : : : Dr g G D F 1 r and M being equal to the product of the elements of m) m is maximal. A database is consistent if it has (at least) one static model. Example 3.5 (ex. 3.2 cont.) The assertion of L implies the access (via L) to A and B , the possibility to access to L, the exception on L (i.e. the possibility to erase L ). Let F = fL C g we have the following informal proof of the sequent G D L C ` A B L C : (i) L C (ii) L (L & 1) L C (C & 1) C (iii) L L L C C C (iv) A B L L L C C (v) A B L!(L 1) L C C (vi) A B L C C (vii) A (A & 1) A B (B & 1) B L C C (viii) A (A & 1) A B (B & 1) B L C C (ix) A B L C 11
Fouquere and Vauzeilles
A static model of hF Gi is fA B L C g. Example 3.6 The following two DB examples have peculiarities showing the interest of such a formalization. We let the reader prove the results. Let DB1 = hF Gi where F = fA C g and G = fA!B C !B g. The database has two static models: fA B C g and fA B C g. Let DB2 = hF Gi where F = fA Bg and G = fA!B C !B g. The database has a unique static model: fA B g.
3.3 Update at the Syntax Level Let DB = hF Gi be a database. We dene the relation ! (resp. ! ) as the transitive closure of ! (resp. !). We set, for each A 2 D, ANC (A) = fB (B A) 2 ! g and CONS(A) = fB (A B) 2 ! g. The update of a Database w.r.t. the insertion of a fact is dened in graph theory in the following way. The result of updating DB by inserting L, is the database DB = hRes(F G Lins) Gi where Res(F G Lins) = fLg (F Reject(G Lins)) and Reject(G Lins ) = fA A 2 D and 9B 2 CONS (L) and A 2 ANC (B )g. Example 3.7 (ex. 3.2 and 3.5 cont.) By inserting B , the updated database is hF Gi with F = fL C Bg. It has a unique static model: fA B C Lg. By inserting A, the updated database is hF Gi with F = fA C g. It has a unique static model: fA C Lg.
0
0
0
0
0
3.4 Update Models A consistent set m of literals is an update model of a database hF Gi w.r.t. the insertion of L if: (i) F Lins ` M is provable (F = D1 : : : Dr if F = fD1 : : : Dr g and M being equal to the product of the elements of m). (ii) m is maximal. Remark 3.8 The insertion of L (assertion of Lins) implies the assertion of L , the (forward) propagation of the insertion (via Li) from L to A if (L A) 2 ! (assertion ofdelAins) and the deletion of L (via Ldel). The deletion of A (assertion of A ) implies the possibility to erase A , prohibiting the assertion of A and the (backward) propagation of the deletion from A to L if (L A) 2 ! (via Ad ). Example 3.9 (ex. 3.2 and 3.5 cont.) G
D
12
Fouquere and Vauzeilles
the sequent G D L C B ins ` A B C L is provable so fA B C Lg is an update model of hF Gi w.r.t. the insertion of B . the sequent G D L C Ains ` A C L is provable so fA C Lg is an update model of hF Gi w.r.t. the insertion of A. The following theorem states that a database built from scratch with successive updates is equivalent to a database with the set of facts being given at once. Theorem 3.10 Let G be a safe graph of rules, and let hF0 Gi : : : hFk Gi be a sequence of databases such that F0 = and for each i 2 N , Fi+1 = Res(Fi G Li+1). Then, the following properties hold: hFi Gi has (at least) one update model w.r.t. the insertion of Li+1 hFi+1 Gi is consistent. The static models of hFi+1 Gi are exactly the update models of hFi Gi w.r.t. the insertion of Li+1 . Proof. adaptation of the proof in 2].
4
Conclusion
In this paper, a formalization of two problems requiring uncertain knowledge and change has been proposed. This formalization has been realized in a fragment of propositional linear logic, by using proper axioms which represent respectively the taxonomical network or the database. It should be of some interest to know in which extent these problems can be modelized without using proper axioms, in Light Linear Logic (Light Naive set theory with xpoints). References 1] N. Bidoit, S. Cerrito, and Ch. Froidevaux. Consistency preserving updates. Proceedings of the Post-ILPS'94 Workshop on Uncertainty in Databases and Deductive Systems. (Available as Technical Report of the department of Computer Science, Concordia University). 2] N. Bidoit, S. Cerrito, and Ch. Froidevaux. A linear logic approach to consistency preserving updates. Journal of Logic and Computation, 1996. 3] C. Fouquere. Th ese d'habilitation a diriger des recherches. PhD thesis, Universite Paris 13, 1995. 4] C. Fouquere and J. Vauzeilles. Linear logic and exceptions. Journal of Logic and Computation, 4(6):859{876, 1994.
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5] C. Fouquere and J. Vauzeilles. Inheritance with exceptions: an attempt at formalization with linear connectives in unied logic. In London Mathematical Society, editor, Advances in Linear Logic, Lecture Note Series, 222, pages 167{ 196. J.Y. Girard and Y. Lafont and L. Regnier, 1995.
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URL: http://www.elsevier.nl/locate/entcs/volume3.html
14 pages
Linear Logic for Taxonomical Networks and Database Updates Preliminary Report Christophe Fouquere
LIPN-CNRS URA 1507 Universite Paris-Nord 93430 Villetaneuse, France Jacqueline Vauzeilles
LIPN-CNRS URA 1507 Universite Paris-Nord 93430 Villetaneuse, France
Abstract
The aim of this paper is to propose a logical way to handle uncertain knowledge and change. Databases, diagnostic, planication, taxonomy are some of the domains concerned by this problem. This paper focuses on the means Linear Logic oers to represent taxonomical networks and to perform updates of databases containing incomplete information. The two problems are rst expressed in graph theory: a taxonomical network is a structure for representing knowledge as a graph whose vertices are concepts, and edges are relations between concepts a database is specied by facts, deduction rules, i.e. edges between literals, and update constraints. Their formalization in Linear Logic is performed in a very similar way.
1 Introduction
The aim of this paper is to propose Linear Logic for the representation of uncertain knowledge and change: basically, some knowledge (inference or fact), legitimate in most of the situations, has to be cancelled or forgotten in some special cases. Databases, diagnostic, planication, taxonomy are some of the domains concerned by this problem. Since 1970, a lot of formalisms and systems have been developed for formalizing these problems. Following previous work 2,4,5], we show how Linear Logic (LL) can adequately represent taxo-
c 1996 Elsevier Science B. V.
Fouquere and Vauzeilles
nomical networks with default knowledge, and updates in databases containing incomplete information. In the rst part, we prove that LL can correctly modelize taxonomical networks. A taxonomical network is a structure for representing knowledge as a directed graph whose vertices are concepts, and edges are relations between concepts. We consider three kinds of edges: strict, default and exception edges. These edges precise the way properties (hereafter concept names) are inherited. A default edge between A and B means that A is generally a B , or A inherits the property B . An exception edge between A and B is a direct inhibition of a default edge between A and B , namely A is not a B or A has not the property B , whatever the number or kind of default paths between A and B . A strict edge between A and B states that A is a B (or A has the property B ) whatever the number or kind of exception paths between A and B . We dene the notion of compatible vertices of the graph w.r.t. a set of nodes called the facts. A set of compatible vertices is a maximal set of vertices s.t. exception statements are satised, i.e. each vertex in this set can be inherited by one of the facts of the graph w.r.t. the meanings of the edges. The reader may nd in 4] a proof of equivalences between this description in graph theory, a representation in a fragment of LL and a representation in Reiter's Default Logic. In short, sets of compatible vertices are exactly provable sequents whose right hand side is the conjunction of the variables representing the vertices. In this paper, we propose a new formalization of these networks in such a way that the meaning of sequents becomes straightforward. Furthermore, this induces a representation schema for a whole class of networks, including disjunctive networks. We show in the second part that this schema extends also to database updates. The results presented here are an adaptation in the formalism used for taxonomical networks of previous results of Bidoit, Cerrito and Froidevaux 1,2]. A database can also be viewed as a set of initial facts (which can be positive or negative) and a graph of whose vertices are facts and edges precise the way deductions can be performed. Edges are of two kinds: strict or default edges. Since we want to deal with \consistent" databases (i.e. databases in which it cannot be deduced a fact and its negation) intuitively it supposes that there exists an exception edge between vertices A and :A. For each database, the notions of static semantics and update semantics w.r.t. the insertion of a new fact, are dened through deductions in Linear Logic. Then the database resulting from an update is dened syntactically and it is shown that the static semantics of the update database coincides with the update semantics of the database.
2
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2 Taxonomical Networks This part is a more general formulation of previous results 4,5], which was rst presented in 3]. We rst give a few examples to show graphically what we expect of such taxonomical networks: the dashed (resp. bold) arrow indicates an exception (resp. a strict edge) while default edges are drawn as simple arrows.
Example 2.1
Molluscs are generally Shell-bearers, Cephalopods are generally Molluscs but exceptions to Shell-bearers, Nautili are generally Cephalopods but are Shell-bearers. Shell-bearers are Invertebrate. We want to deduce that Nautili are Cephalopods, Molluscs, Shellbearers and Invertebrate.
Example 2.2
The two exceptions (double arrow on a dashed line) dene an even cycle, giving rise to two interpretations, A being true. Let A be true on one hand one can infer B , C on the other hand one can infer B , C . 0
0
Example 2.3
A is generally a B , B is generally a C and C is an exception to be a B .
We want to conclude that there is no solution to this problem, interpreting results when A is true.
Invertebrate Shell-bearers
Molluscs Cephalopods
Nautili C
C'
B
B' A C B A
2.1 Taxonomical Networks as graphs The graphs we modelize are directed nite graphs with three kinds of edges. Furthermore, we forbid cycles in the graph restricted to strict and default edges. However, the reader may nd in 3] a way to avoid such a restriction.
Denition 2.4
A taxonomic graph is a directed graph G = hD ! ! 9 9 Ki such that: D is a nite set of vertices ! ! 9 9 K are three irre exive relations on D if (a b) 2 ! (resp. (a b) 2 ! (a b) 2 9 9 K) it is said that there exists a strict (resp. a default, an exception) edge between a and b hD ! !i is acyclic 3
Fouquere and Vauzeilles
A Taxonomical Network (TN) is dened as a pair N = hF Gi where G = hD ! ! 9 9 Ki is a taxonomic graph and F , the set of facts, is a subset of D.
The main aim of this section is the denition of compatible vertices of a TN informally a subset E of vertices of a TN N is constituted of compatible vertices if, for each vertex , there exists a path from a vertex of to such that each edge is a strict or a default edge (a strictly default path) and if there does not exist a vertex of E with an exception edge between and (except if there exists a strictly default path between an element of and whose last edge is a strict edge). First inductive denitions of correct path, default correct path, and strictly correct path are given. Denition 2.5 Let N = h Gi be a TN. A sequence 1 of vertices of D is a correct path (resp. default correct path, resp. strictly correct path) in N if and only if one of the following two conditions holds: = 1 and 1 2 , 1 and on the one hand 1 1 is a default correct path, or 1 1 is a correct path and there exists a strictly correct path 1 such that = 1 on the other hand ( 1 ) 2 ! or ( 1 ) 2 ! or ( 1 ) 2 9 9 K (resp. ( 1 ) 2 ! or ( 1 ) 2 !, resp. ( 1 ) 2 !) A sequence 1 of vertices of G is a strictly default path if and only if 2 and either = 1 or for each (2 ) we have ( 1 ) 2 ! or 1 ( 1 ) 2 !. Remark 2.6 Each strictly correct path is a default correct path, and each default correct path is a correct path. Moreover each strictly default path is a default correct path. Example 2.7 The relations ! ! and 9 9 K are respectively represented by bold, simple and dashed arrows. (ex. 2.1 cont.): The TN N1 is represented by the graph of example 2.1: (the following abbreviations are used: and stand respectively for Nautili, Cephalopods, Molluscs, Shell-bearers and Invertebrate) D1 = f g 1 = f g. For instance is a default correct path (so a correct path) then is a strictly correct path (so a default correct path and a correct path) moreover and are strictly default paths. is a correct path but not a default path since is a strictly correct path it can be concluded that is a strictly correct path (also a default correct path and a correct path), but is not a strictly default path. 4 a
F
b
b
a
F
F
n
a
a
a
a : : : an
F
n >
a : : : an
a : : : an
b : : : bq
bq
an
an
an
an
an
an
an
an
an
an
an
an
an
a : : : an
a
F
ai
n
i
i
n
ai
ai
ai
N C M S
N C M S I
F
I
N
N C M S
N C M S I
N C M S I
N S I
N C S
N S
N C S I
N C S I
Fouquere and Vauzeilles
(ex. 2.2 cont.): The TN N2 is represented by the graph of example 2.2: D2 = fA B B C C g F2 = fAg. For instance A B C and A B C are default correct paths so are correct paths A B B and A B B are correct paths but are not default correct paths then A B B C and A B B B C are not correct paths. (ex. 2.3 cont.): The TN N3 is represented by the graph of example 2.3: D3 = fA B C g F3 = fAg. For instance A B C is a default correct path and a strictly default path A B C B is a correct path but not a default correct path A B C B C is not a correct path. 0
0
0
0
0
0
0
0
0
hD ! ! i be a TN a TN M hF G i with G hE ! ! i is a (subTN) N complete if and only if G is a subgraph of G such that: if a 2 E then there exists a correct path a : : : an in M (and in N ) such that a an if a 2 E then either fb a b 2 ! g fc a c 2 ! g and fb a b 2 ! g fc a c 2 ! g and fb a b 2 g fc a c 2 g or fb a b 2 ! g fb a b 2 ! g fb a b 2 g a TN M hF G i with G hE ! ! i is N -completely correct if and only if M is N -complete and if: each path in M is correct if a 2 E and fb a b 2 ! g fb a b 2 ! g fb a b 2 g then either fb a b 2 ! g fb a b 2 ! g fb a b 2 g or there exists a correct path a : : : an a in M such that an an 2
Denition 2.8 Let N = hF G i with G N
=
M
N
=
=
M
N
M
M
M
N
9 9 KN
9 9 KM
N
1
=
(
(
)
)
=
=
M
(
N
(
)
=
M
(
M
)
)
N
)
(
M
(
(
)
=
(
)
9 9 KN
M
=
(
)
9 9 KM
=
M
=
(
(
N
9 9 KM
)
M
=
)
M
(
N
M
=
=
(
=
1
=
9 9 KM
)
)
=
M
(
)
)
=
9 9 KM
9 9 KN
(
=
1
)
and there does not exist a strictly correct path b1 : : : bq such that bq = a. 9 9 KM
Example 2.9
(ex. 2.1 cont.): N1 is the only N1 -completely correct subTN of N1. (ex. 2.2 cont.): There are the two completely correct subTNs of N2 . C
C'
B
B
B' A
B' A
(ex. 2.3 cont.): The TN N3 does not admit a completely correct subTN: 5
Fouquere and Vauzeilles
C
the following subTNs are not comB B pletely correct subTNs of N : A A N is not a completely correct subTN of itself because the path A B C B C is not correct. Denition 2.10 The vertices b : : : b of a net N = hF Gi are said to be compatible if and only if there exists a completely correct subTN M = hF G i of N such that b : : : b are vertices of G and for each i (1 i n), if there exists a correct path in M whose end is b and whose last edge is an 3
3
n
1
M
1
n
M
i
exception edge then there exists a strictly correct path whose end is bi .
Example 2.11
(ex. 2.1 cont.): For instance, Nautili, Cephalopods and Shell-bearers are compatible vertices of N1. (ex. 2.2 cont.): For instance, A B and C are compatible vertices of N2 , while B and B are not compatible vertices of N2 . Note that B and B belong to the two completely correct subTNs of N2 but, in the rst completely correct subTN of N2 , A B B is a correct path whose last edge is an exception edge and in the second completely correct subTN of N2, A B B is a correct path whose last edge is an exception edge. (ex. 2.3 cont.): Since N3 does not admit a completely correct subTN, there do not exist compatible vertices of N3 . The three previous examples prove that a TN can admit zero, one or many completely correct subgraphs.
0
0
0
0
2.2 Taxonomical Networks in Linear Logic We show in this section that Linear Logic can formalize taxonomical networks with strict, default and exception edges. We consider the fragment of (intuitionistic) Linear Logic including the multiplicative constant 1, the multiplicative connectives , the additive connective & , and the exponential connective ! the properties of the connectives of this fragment of Linear Logic are essential in our representation: the properties of the constant \1" (which is a neutral element for the multiplicative conjunction \") can be used to cancel information (we use (A 1) to cancel A) while those of the exponential connective \!" can, using !(A 1), cancel all occurrences of A. Let N = hF Gi with G = hD ! ! 9 9 K i be a TN. The language LN is the linear language constructed from a set V = DfA A 2 DgfA A 2 Dg fA A 2 Dg fA A 2 Dg of propositional variables and including the connectives & ! and the constant 1. 6 N
N
N
D
Fouquere and Vauzeilles
Denition 2.12 We set:
if fB (A B ) 2 !g = fC (A C ) 2 9 9 Kg = fD (A D) 2 !g = then (A) A 1 else (A) A B1 : : : Bm C1 : : : Cp D1 : : : Dn with
fBi i 2 m g fB A B 2 !g fCi i 2 p g fC A C 2 g fDi i 2 n g fD A D 2 !g A A A A A A A A
G
D
1
] =
(
)
1
] =
(
)
1
] =
(
)
2D
!( )
2D
!(
Example 2.13
(ex. 2.1 cont.).
N
S
C C
is provable.
()
(
& 1))
99 K
!(
) & !(
A !(A 1))]
is dened with the formulas:
M S M S S I I . The following sequent N ` N S I C M.
1
G D
It can be read: Let be the formula describing the properties of the various kind of links, instantiated on the set of nodes D, let be the formula describing the graph G , let N be the only fact (i.e. N is a sure knowledge and N is a given formula), then the conjunction N S I C M can be derived, i.e. fN S I C M g is a set of compatible vertices. Remark 2.14 The assertion of N implies the access (via N ) to S and C and the possibility to reach N . The access to C implies the access to C (since C is not asserted), and so the access (via C ) to M and S , and the possibility to reach C . The access to M implies the access to M (since M is not asserted), and so the access (via M ) to S , and the possibility to reach M . The access to S implies the deletion of S , and the access to S implies (via S ) the access to I , and the possibility to reach S . The access to I implies the possibility to reach I . Note that the sequent C ` C M is provable while the sequent C ` C M S is not provable: the assertion of C implies the access (via C ) to M and S , and the possibility to reach C the access to M implies the access to M (since M is not asserted), and so the access (via M ) to S , and the possibility to reach M since S is asserted, S has to be deleted and so S cannot be reached (via S ). (ex. 2.2 cont.): A ` B B B ` B C B ` B C C ` 1 C ` 1. The two following sequents are provable: D
G
G
D
G
D
0
0
7
0
0
0
Fouquere and Vauzeilles
A ` A B C , A ` A B C (ex. 2.3 cont.): A ` B B ` C C ` B . There is no provable sequent A ` M (M being a product of some elements of fA B C g). The following theorem gives the equivalence between compatible sets of nodes and provable sequents whose right hand side is a conjunction of propositional variables. Theorem 2.15 The vertices B1 : : : Bn of the TN N = hF Gi are compatible if and only if the sequent F ` B1 : : : Bn is provable (with F = D1 : : : Dr if F = fD1 : : : Dr g). Proof. see 3] Note that a representation in classical logic cannot give the expected results as classical logic is unable to deal adequately with nonmonotonicity. The fact that Linear Logic manages variables as resources is fundamental: an exception on a node is expressed as the retrieval (A 1) of the corresponding variable. Note nally that we separate in the prerequisites of the sequent the facts (a sequence of A formulas), from the description of the network ( ), from the description of the meaning of the links ( ). This way of representing taxonomic networks and the meaning of relations between nodes can be generalized in fact to disjunctive networks 3]. 0
G
D
G
G
0
D
D
G
G
D
D
3 Database Updates
Updates are fundamental in database systems. However, their dynamic e ect w.r.t. the database cannot be expressed in classical logic. We show in this section that the twofold point of view of Linear Logic, i.e. dynamic and static, helps to represent this process. This is an adaptation of 1,2] in a formalism close to the one used in the previous section. It is worthy to note that if the initial database is \consistent", it must remain \consistent" after the insertion of a new fact. All facts are represented by atomic or negation of atomic formulae. A fact can be true, false or unknown. The deductions in the database can be performed in two ways represented by two kinds of edges of a graph: intuitively, the rule A!B means that if A is derived, then B is necessarily deduced while the meaning of the rule A!B is that if A is derived then B occurs unless its deduction contradicts other information. If the database contains the rule A!B , if A holds and if the database is updated by the insertion of :B then A has to be deleted, while if :B holds and if the database is updated by the insertion of A then :B has to be deleted. Let P be a set of propositional variables. We set L = P fp p 2 Pg (p is a notation that will allow to interpret the negation of p). A literal is an element of L. 8
Fouquere and Vauzeilles
Denition 3.1
A database graph over L is a directed graph G = hD ! !i such that: the set D of vertices is included in L ! ! are two irre exive relations on D if (a b) 2 ! (resp. (a b) 2 !) it is said that there exists a strict (resp. a default) edge between a and b and that a is the source of the edge and that b is its target for any vertices a and b, there is not simultaneously (a b) 2 ! and (a b) 2 ! no vertex is the target of a default edge and the source of a strict edge hD ! !i is acyclic. Let G = hD ! !i be a database graph. Let G = hD ri be the non-directed graph such that (a b) 2 r i (a b) 2 ! or (b a) 2 ! or (a b) 2 ! or (b a) 2 !. G is safe i there are no vertices a and :a connected by a path in G . A Database (DB ) is dened as a pair DB = hF Gi where G = hD ! !i is a safe database graph and F , the set of facts, is a subset of D. 0
0
3.1 Databases in Linear Logic Let DB = hF Gi with G = hD ! !i be a database we set A = fA A 2 D or A 2 Dg and D = A fA A 2 Ag for each A 2 A we set A = A. We set vs = D fA A 2 Dg fA A 2 Dg fA A 2 Dg fA A 2 Dg and vu = fAi A 2 Dg fAins A 2 Dg fAd A 2 Dg fAdel A 2 Dg. Ls (resp. Lu) is the linear language constructed from the set vs (resp vs vu) of propositional variables and including the connectives & ! and the constant 1. Static Rules For each L 2 D we set: if fB (L B ) 2 !g = fC (L C ) 2 !g = then stat(L) L 1 else stat(L) L B1 : : : Bm C1 : : : Cn with fBk k 2 1 m]g = fB (L B ) 2 !g fCk k 2 1 n]g = fC (L C ) 2 !g
Update Rules Insertion Rules For each L 2 D we set: if fC (L C ) 2 !g = then ins(L) Li 1 else ins(L) Li C1ins : : : Cnins with fCk k 2 1 n]g = fC (L C ) 2 !g Deletion Rules For each L 2 D we set: if fC (C L) 2 !g = then del(L) Ld 1 del else del(L) Ld Adel 1 : : : Ar with fAk k 2 1 r]g = fA (A L) 2 !g. Example 3.2
9
Fouquere and Vauzeilles
L
Static Rules
Update Rules
B
A
C
D
E
:L
L A B A 1 B 1 C L D 1 E 1 L 1 A 1 B 1 C 1 D 1 E 1
Insertion rules Li Ains, Ai 1, B i 1, C i 1, Di 1, E i 1, Li 1, Ai 1, B i 1, C i 1, Di 1, E i 1, Deletion Rules Ld 1, Ad Ldel , B d 1, C d 1, Dd 1, E d 1, Ld 1, Ad 1, B d 1, C d 1, Dd 1, E d 1
As with the logical representation of taxonomical networks, we describe the database graph with one formula G , the meaning of the various kinds of relations by another formula D , and the set of facts F by a sequence of formulas F . We give now the denitions of G and D . The following subsections are devoted to dene the static model of a DB, i.e. what can be deduced with a Database when no changes occur, and to dene the updates of a Database. We end by proving that there is an equivalence between logical updates (i.e. provable sequents whose right hand side is a product of literals, when one adds an insertion literal as a prerequisite) and graph updates (i.e. graph modications by inserting a new fact w.r.t. the meaning of edges). G is a product of all static and update rules, i.e. the logical counterpart of the database graph. D is a product of the logical interpretation of the various kinds of edges and the insertion of a literal. These logical interpretations are obviously to be considered over the set of variables. We distinguish the insertion of a variable (Lins), the default/exception edges (L /L), the strict edges and the deletion of a variable (L , Ldel ). To insert a variable L implies the fact that this variable is sure (L ), and that we have to update the knowledge w.r.t. the insertion rule of L (Li) and the deletion rule of its negation (Ldel ). Default/exception edges are used in an exclusive way (hence \&" between the two interpretations), and an exception (L) deletes each default occurrence (L 1), otherwise the default implies the literal to be sure (L ). The fact that a literal A and its negation A are mutually exclusive is expressed by the conjunctive connective \&" with the same kind of interpretation w.r.t. sure knowledge and deletion (formula ). The fact that a knowledge is sure A implies this knowledge (but we can forget this piece of information: A&1), what is graphically implied by this literal (hence A? ), and that there is an exception on the negation (A). The deletion of its negation (Adel ) implies to 10
Fouquere and Vauzeilles
re the corresponding deletion rule (Ad) and the deletion of each occurrence of this negation (A 1). Denition 3.3 We set: G L2D !stat(L) !ins(L) !del(L) (A) !(A A (A & 1) A ) !(Adel Ad !(A 1)) D A2A ( (A) & (A)) ins L Li Ldel ) L2D !(L L ) & !(L !(L 1))]!(L 3.2 Static models A static model is a maximal consistent set of variables that can be deduced by a DB. A set is consistent if it does not contain a variable and its negation. Deductibility is dened in terms of provable sequents, however it follows from graph deductibility w.r.t. the meaning of edges, in the same spirit of what has been done in the previous section.
Denition 3.4
A set of literals is consistent i it does not contain A and A. A consistent set m of literals is a static model of a database DB
= hF G i
if: ` M is provable (with F = D : : : D if F = fD1 : : : Dr g G D F 1 r and M being equal to the product of the elements of m) m is maximal. A database is consistent if it has (at least) one static model. Example 3.5 (ex. 3.2 cont.) The assertion of L implies the access (via L) to A and B , the possibility to access to L, the exception on L (i.e. the possibility to erase L ). Let F = fL C g we have the following informal proof of the sequent G D L C ` A B L C : (i) L C (ii) L (L & 1) L C (C & 1) C (iii) L L L C C C (iv) A B L L L C C (v) A B L!(L 1) L C C (vi) A B L C C (vii) A (A & 1) A B (B & 1) B L C C (viii) A (A & 1) A B (B & 1) B L C C (ix) A B L C 11
Fouquere and Vauzeilles
A static model of hF Gi is fA B L C g. Example 3.6 The following two DB examples have peculiarities showing the interest of such a formalization. We let the reader prove the results. Let DB1 = hF Gi where F = fA C g and G = fA!B C !B g. The database has two static models: fA B C g and fA B C g. Let DB2 = hF Gi where F = fA Bg and G = fA!B C !B g. The database has a unique static model: fA B g.
3.3 Update at the Syntax Level Let DB = hF Gi be a database. We dene the relation ! (resp. ! ) as the transitive closure of ! (resp. !). We set, for each A 2 D, ANC (A) = fB (B A) 2 ! g and CONS(A) = fB (A B) 2 ! g. The update of a Database w.r.t. the insertion of a fact is dened in graph theory in the following way. The result of updating DB by inserting L, is the database DB = hRes(F G Lins) Gi where Res(F G Lins) = fLg (F Reject(G Lins)) and Reject(G Lins ) = fA A 2 D and 9B 2 CONS (L) and A 2 ANC (B )g. Example 3.7 (ex. 3.2 and 3.5 cont.) By inserting B , the updated database is hF Gi with F = fL C Bg. It has a unique static model: fA B C Lg. By inserting A, the updated database is hF Gi with F = fA C g. It has a unique static model: fA C Lg.
0
0
0
0
0
3.4 Update Models A consistent set m of literals is an update model of a database hF Gi w.r.t. the insertion of L if: (i) F Lins ` M is provable (F = D1 : : : Dr if F = fD1 : : : Dr g and M being equal to the product of the elements of m). (ii) m is maximal. Remark 3.8 The insertion of L (assertion of Lins) implies the assertion of L , the (forward) propagation of the insertion (via Li) from L to A if (L A) 2 ! (assertion ofdelAins) and the deletion of L (via Ldel). The deletion of A (assertion of A ) implies the possibility to erase A , prohibiting the assertion of A and the (backward) propagation of the deletion from A to L if (L A) 2 ! (via Ad ). Example 3.9 (ex. 3.2 and 3.5 cont.) G
D
12
Fouquere and Vauzeilles
the sequent G D L C B ins ` A B C L is provable so fA B C Lg is an update model of hF Gi w.r.t. the insertion of B . the sequent G D L C Ains ` A C L is provable so fA C Lg is an update model of hF Gi w.r.t. the insertion of A. The following theorem states that a database built from scratch with successive updates is equivalent to a database with the set of facts being given at once. Theorem 3.10 Let G be a safe graph of rules, and let hF0 Gi : : : hFk Gi be a sequence of databases such that F0 = and for each i 2 N , Fi+1 = Res(Fi G Li+1). Then, the following properties hold: hFi Gi has (at least) one update model w.r.t. the insertion of Li+1 hFi+1 Gi is consistent. The static models of hFi+1 Gi are exactly the update models of hFi Gi w.r.t. the insertion of Li+1 . Proof. adaptation of the proof in 2].
4
Conclusion
In this paper, a formalization of two problems requiring uncertain knowledge and change has been proposed. This formalization has been realized in a fragment of propositional linear logic, by using proper axioms which represent respectively the taxonomical network or the database. It should be of some interest to know in which extent these problems can be modelized without using proper axioms, in Light Linear Logic (Light Naive set theory with xpoints). References 1] N. Bidoit, S. Cerrito, and Ch. Froidevaux. Consistency preserving updates. Proceedings of the Post-ILPS'94 Workshop on Uncertainty in Databases and Deductive Systems. (Available as Technical Report of the department of Computer Science, Concordia University). 2] N. Bidoit, S. Cerrito, and Ch. Froidevaux. A linear logic approach to consistency preserving updates. Journal of Logic and Computation, 1996. 3] C. Fouquere. Th ese d'habilitation a diriger des recherches. PhD thesis, Universite Paris 13, 1995. 4] C. Fouquere and J. Vauzeilles. Linear logic and exceptions. Journal of Logic and Computation, 4(6):859{876, 1994.
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Fouquere and Vauzeilles
5] C. Fouquere and J. Vauzeilles. Inheritance with exceptions: an attempt at formalization with linear connectives in unied logic. In London Mathematical Society, editor, Advances in Linear Logic, Lecture Note Series, 222, pages 167{ 196. J.Y. Girard and Y. Lafont and L. Regnier, 1995.
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