Uniform interpolation by resolution in modal logic

Transcription

1 Uniform interolation by resolution in modal logic Andreas Herzig and Jérôme Mengin 1 Abstract. The roblem of comuting a uniform interolant of a given formula on a sublanguage is known in Artificial Intelligence as variable forgetting. In roositional logic, there are well known methods for erforming variable forgetting. Variable forgetting is more involved in modal logics, because one must forget a variable not in one world, but in several worlds. It has been shown that modal logic K has the uniform interolation roerty, and a method has recently been roosed for forgetting variables in a modal formula (of mu-calculus) given in disjunctive normal form. However, there are cases where information comes naturally in a more conjunctive form. In this aer, we roose a method, based on an extension of resolution to modal logics, to erform variable forgetting for formulae in conjunctive normal form, in the modal logic K. 1 Introduction An interolant of logical formulae φ and ψ such that φ = ψ in a logic L is a formula χ that contains only variables that aear in both φ and ψ, and such that φ = χ and χ = ψ. A uniform interolant of φ with resect to a sublanguage L of the language of φ is a formula χ L entailed by φ that can act as an interolant for any ψ L : if φ = ψ, then χ = ψ. In other words, χ behaves like φ when L is concerned, in the sense that ψ has the same L -logical consequences as φ [4]. When the language L is defined as the set of formulae that contain no variable of a given set P, the roblem of comuting a uniform interolant of φ on L is known in Artificial Intelligence as variable forgetting. In roositional logic, there are well known methods for erforming variable forgetting [8, 9]. Variable forgetting is more involved in modal logics, because one must forget a variable not in one world, but in several worlds. It has been shown that modal logic K has the uniform interolation roerty [11, 6] (examles of logics that do not have the uniform interolation roerty include classical first order logic and S4 [7]). [4] roose a simle method for forgetting variables in a modal formula (of µ-calculus) given in disjunctive normal form. However, there are cases where information comes naturally in a more conjunctive form. In this aer, we roose a method, based on an extension of resolution to modal logics, to erform variable forgetting for formulae in conjunctive normal form, in the modal logic K. In the next section, we briefly recall the syntax and semantics of K, and the definitions of disjunctive and conjunctive normal forms in this logic. In section 3, we recall Enjalbert and Fariñas resolution system for K. In section 4, we exlain how their resolution system can be used to erform variable forgetting in that logic. 1 Institut de Recherche en Informatique de Toulouse. herzig,mengin@irit.fr 2 The logic K The language of K, over a set of roositional variables P, is the smallest set L of formulae that contains P and is closed under conjunction, negation and necessity. A ointed model for K is a tule m = (W, R, I, w), where W is a non-emty set, R is a binary relation over W, I assigns to every P and every w W a value I(, w ) {true, false}. Satisfaction of formula φ by model m = (W, R, I, w) is defined by induction as follows: m = if and only if I(, w) = true, for every P; m = φ if and only if m = φ; m = φ ψ if and only if m = φ and m = ψ; m = φ if and only if (W, R, I, w ) = φ for every w such that wrw. W A set of formulae S is said to entail formula ψ, written S = ψ, if for every ointed model m that satisfies every formula in S, m satisfies ψ too. In this case, we will also say that ψ is a logical consequence of S. We will write S = ψ when this is not the case. Given a finite set of formulae {φ 1,..., φ n}, we will sometimes write φ 1,..., φ n = ψ instead of {φ 1,..., φ n} = ψ. As usual, two other connectors and are introduced and defined as abbreviations: φ ψ ( φ ψ) and φ φ. For that local definition of logical consequence, the deduction theorem holds for finite S: S = φ if and only if {} = S φ. There are various ways to define conjunctive and disjunctive normal forms in modal logic (see e.g. [3, 4, 5]). In this aer, we will consider normal forms close to the ones defined by Enjalbert and Fariñas. Literals, clauses and terms can be defined using a grammar in BNF: LitC ::= Clause CNF LitT ::= DNF Term Clause ::= LitC Clause Clause Term ::= LitT Term Term CNF ::= Clause CNF, CNF DNF ::= Term DNF DNF According to this definition, clauses are disjunctions of literals of tye C (for Clause ): such a literal is either a roositional literal, or a clause behind a, or a set/conjunction of clauses behind a. A CNF is then a set/conjunction of clauses. This corresonds exactly to the definition of clauses used by Enjalbert and Fariñas. A term is the dual of a clause: it is a conjunction of literals of tye T, where such a literal is either a roositional literal, or a term behind a, or a disjunction of terms behind a. Every formula of K as an equivalent DNF and an equivalent CNF.

2 (Uniformly) interolating a DNF is relatively simle, because of two roerties of interolation. The first one is general in logic: if χ is a (uniform) interolant of φ and χ a (uniform) interolant of φ, then χ χ is a (uniform) interolant of φ φ (since if φ φ = ψ, then φ = ψ and φ = ψ, thus χ = ψ and χ = ψ ). The second one concerns the interlay of modal connectives and interolation: a uniform interolant of φ φ φ 1... φ n, where φ does not contain any modality, is simly : χ χ (χ 1 χ )... (χ n χ ), where the χs are interolants of the φs (this is equivalent to the result roved in [4], and used in [10], on interolation for slightly different disjunctive normal forms: a uniform interolant of φ φ 1... φ n (φ 1... φ n) is φ φ 1... φ n (φ 1... φ n)). In the sequel, we roose a method to comute a uniform interolant of a CNF. 3 Resolution in modal logic From a model theoretic oint of view, resolution can be understood as an alication of the general set theoretic roerty: (M N) (M N ) (M M ) N N. In terms of logical formulae, this can be rehrased as: φ C, ψ C = (φ ψ) C C. In the case where ψ = φ, we obtain the usual resolution rule of classical logic: φ C, φ C = C C. But in modal logic, this can be used to roduce other inference rules. In articular, if φ ψ = χ,then φ, ψ = χ and φ, ψ = χ, thus φ C, ψ C = χ C C φ C, ψ C = χ C C Note these deductions can be made at any deth in a modal formula in negation normal form, since if φ = χ, then φ = χ, and (φ ψ) = (φ ψ χ). In order to define a ractical inference system, one has to define recisely which inferences are allowed. Enjalbert and Farinas [5] define a set of inference rules that we recall below. In the sequel, we adot an in line notation for inference rules: A 1,..., A n = B denotes an inference rule whose remisses are A 1,..., A n, and whose consequent is B. Enjalbert and Farinas resolution system for K is defined by a set of conditional, recursive meta -rules; it is the smallest set of inference rules that contains, for P { }: 2 (rule ) C 1, E C 2 = C 1 C 2; (rule ) C 1, C 2 = C 1 C 2; (rule ) C 1 C 1, (C 2, E) C 2 = (C 2, E, C 3) C 1 C 2 if C 1, C 2 = C 3; (rule ) C 1 C 1, C 2 C 2 = C 3 C 1 C 2 if C 1, C 2 = C 3; (rule 2) (C 1, C 2, E) C = (C 1, C 2, E, C 3) C if C 1, C 2 = C 3; (rule 1) (C 1, E) C = (C 1, E, C 2) C if C 1 = C 2; (rule ) C 1 C = C 2 C if C 1 = C 2; 2 The meta -system of [5] is slightly different from the resentation given here, mainly because we do not searate the introduction of from the introduction of modalities. We assume that the consequents are always normalized, that is simlified using the following equivalences: A A B A B, A A,, A. In the descrition above, we have drawn a box around, in each remiss, the literal resolved uon. Let us stress that each of the inference rules of this resolution system is meta -derived from a unique simle (rule ) or (rule ) for some P; if it is derived from some rule, we call the resolved variable. The that indexes the rules denotes this variable or. The height of the derivation of a resolution rule will be the number of meta-rules used to obtain it, minus 1. Examle 1 The following are derivations of valid resolution rules: r, r = r (rule r) (rule ) r, (r ) = r (This derivation is of height 1.), = (rule ) (rule ), = (rule ), s = s (rule ), ( s) = s (This derivation is of height 3.) Definition 1 A deduction by resolution of clause C from clause set S is a sequence of inferences by resolution I 1,..., I n such that for every I i, each remises of I i belongs to S, or is the consequent of rule I j for some j < i. Examle 2 Let S = { r, (r ), ( s), ( s, q)}. There is a deduction by resolution of from S, for examle with the rules I 1 = r, (r ) = r, I 2 =, ( s) = s, and I 3 = s, ( s, q) = s. r (r ) ( s) r s s ( s, q) Enjalbert and Fariñas show that their resolution system is sound and comlete for K with resect to refutation: given clause set S, S = if and only if there is a deduction by resolution of from a subset of S. Moreover, since the modal deth of the consequent of some inference by resolution cannot be greater that the deth of its remises, there can be only a finite set of clauses that can be deduced by resolution from a finite set of clauses S (recall that redundant literals are imlicitly simlified when resolutions are erformed). We will show in the next section that forgetting variables from P in a set of modal clauses S can be done by erforming all ossible resolutions on variables from P, and then eliminating all occurrences of the variables of P. The roerty of resolution that will enable us to do that is that, given a subset of roositional variables P, we can re-arrange inferences so that resolutions on variables from P aear before other resolutions. Formally:

3 Proosition 1 Given a subset of roositional variables P, if there is a deduction by resolution of clause C from clause set S, then there is a deduction by resolution I 1,..., I n of some clause C from S such that C subsumes C and for every i, j, if I i resolves on a variable from P whereas I j resolves on or a variable from P P, then i < j. Note that the resulting clause C may not exactly be the original, it may in fact be stronger. More recisely, we define subsumtion as follows: Definition 2 A clause C subsumes a clause D if every literal of C subsumes some literal of D, and a set of clauses E subsumes a set of clauses F if every clause of F is subsumed by some clause of E. A roositional literal subsumes itself; a literal C subsumes a literal D if C subsumes D; and a literal E subsumes F if E subsumes F. Examle 3 e (r, s, ) q subsumes (e f) (r t, s) q because: e subsumes (e f) because e subsumes e f; (r, s, ) subsumes (r t, s) because r t is subsumed by r and s is subsumed by s. q subsumes q Examle 2 (continued) We can re-arrange resolutions, so that resolutions on r come last: r (r ) ( s) ( s, q) s ( s, q, ) ( s, q,, r) Pro. 1 is an easy consequence of the following lemma: Lemma 1 Given clauses A, B, C, I and F, if there is an - resolution from A, ossibly using side clause B, giving clause I, and a -resolution from I, ossibly using side clause C, giving clause F, then there exist clauses I, F1,..., Fn (for some n 0) and F such that F subsumes F and there is a -resolution from A, ossibly using side clause C, giving clause I, and a sequence of -resolutions from I, ossibly using side clause B, giving successively F1,..., Fn, F. In other words, if A(, B) = I and I(, C) = F, then A(, C) = I, and I (, B) = F1 and F1 (, B) = F2 and... and Fn(, B) = F. Or, with some ictures: if we have (C) A (B) I then we also have: F r (C) A (B) I F 1 F 2 F n F Proof. The roof is rather tedious, because there are quite a few cases to consider. Let us start with the easy case: the second resolution is not on the literal obtained from the first resolution. In this case, A is of the form A = l A l A A, whereas B = l B B and C = l C C (if needed), with l A(, l B) = l I, and l A(, l C) = l F. Then we just have to take I = l A l F A ( C ) and F = F. Pictorially: (before) (after) ( l C C ) ( l C C ) l A l A A ( l B B ) l I l A A ( B ) l I l F A ( B )( C ) l A l A A ( l B B ) l A l F A ( C ) l I l F A ( B )( C ) Notice that there is nothing to change here if l I and/or l F is, that is, if one of the two resolutions is a roositional one, or if it is a one, the clauses I, F and I are just shorter. Let us now consider what haens when the literal resolved uon in the second resolution is the one obtained in the first resolution. In articular, this means that the first resolution is not a roositional one, nor a one (since in these cases, two literals are discarded and not relaced by anything), and thus the second one is not roositional either, since it oerates on a modal literal. If the -resolution involves a -literal, say in clause A, then it can be a 1 or a 2 resolution on A alone, or a -resolution with a side clause B of the form B = B B. Note we cannot have here B =, for this would mean that the literals of A and B that are involved would not be relaced by anything, so the literal of A involved in the -resolution would be another one, a case we have already covered. Thus B, and the resulting literal is a -literal containing one new clause I that is not in the -literal of A. The resulting -literal is then resolved uon in the -resolution, with or without the hel of side-clause C, which must then be of the form C = C C. If the -resolution does not involve I, we can conclude quite easily: in this case, A is of the form A = (E, A, A ) A, where E is a set of clauses and A and A are the clauses involved in the - and -resolution: we have A (, B ) = I and A (, C ) = F, so we also have (E, A, A ) A (, C C ) = (A, A, E, F ) A ( C ) and (A, A, E, F ) A ( C ), B B = (A, A, E, I, F ) A ( B )( C ). Pictorially:

4 (before) (after) ( A, A, E) A ( B B ) (A, A, E, I ) A ( B ) (A, A, E, I, F ) A ( B )( C ) (A, A, E) A ( B B ) ( A, A, E, F ) A ( C ) (A, A, E, I, F ) A ( B )( C ) If the -resolution does involve I, then we ll have to use some inductive argument in order to conclude: A is of the form A = (E, A ) A, and we have A (, B ) = I and I (, C ) = F : ( A, E) A ( B B ) (A, E, I ) A ( B ) (A, E, I, F ) A ( B )( C ) If we have roved that the result of the lemma holds for every resolution whose derivation height is strictly less than that of our current -resolution, then we can assume it holds for A (, B ) = I, in which case there exist I, F 1,..., F n (for some n 0) and F such that F subsumes F and A (, C ) = I and I (, B ) = F 1 and F 1 (, B ) = F 2 and... and F n (, B ) = F. Now, we can use another resolution to obtain I, since we still have A (, B ) = I. Thus we have: ( A, E) A ( B B ) (A, E, I ) A ( B ) ( A, E, I, F 1 ) A ( B )( C ) ( A, E, I, F 1, F 2 ) A ( B )( C ) ( A, E, I, F 1,..., F n, F ) A ( B )( C ) (A, E, I, F 1,..., F n, F, I ) A ( B )( C ) We have used an inductive argument in the receding aragrah. The base case corresonds to an -resolution whose derivation height is zero: these are the roositional resolutions and the resolution, we have already roved that the result holds for these cases. Let us turn now to the case where there is no -literal involved in the -resolution: this resolution is derived using a -rule or a - rule. Then there is a clause A of the form A = A A, and ossibly a side -clause B = B B, such that A (, B ) = I, and thus I = I A( B ). The -resolution involves the literal roduced in the -resolution (we have studied the other case first). Let us start with the case where this -resolution does not involve any -literal: it is a - or a -resolution: there may be a clause C = C C, with I (, C ) = F, and F = F A ( B ) (C ). Using an inductive hyothesis again, we can assume there are clauses I and F such that A (, C ) = I and I (, B ) = F. (For the sake of simlicity, we assume that there are no other intermediate clauses F 1,..., F n ; the roof would be very similar with more intermediate clauses.) In this case, we can choose I = I A ( C ) and F = F A ( B )( C ): (before) (after) A A ( B B ) I A ( B ) F A ( B )( C ) A A ( B B ) I A ( C ) F A ( B )( C ) If the other literal of the -resolution is a -literal, we have C = (C, E) C, and F = (C, E, F ) A ( B )( C ), with I, C = F. Since A (, B ) = I, the same inductive hyothesis again allows us to assume that there are clauses I and F such that A, C = I and I (, B ) = F. In order to get I in the literal after the resolution, we need to erform another -resolution. In the end, we can choose I = ( C, E, I ) A ( B ) and F = (C, E, F, I ) A ( B ) C : (before) ( C, E) C A A ( B B ) I A ( C ) (C, E, F ) A ( B ) C

5 (after) ( C, E) C A A ( B B ) ( C, E, I ) A ( B ) (C, E, F ) A ( B ) C (C, E, F, I ) A ( B ) C (r ) ( s) (r s) ( s, q) s ( s, q, ) ( s, q,, r) Let P = {s,, q}, we now comute Su(P, R). The clauses ( s, q), ( s, q, ) and ( s) are suressed by Su(P, ) because they contain only occurrences of s, and q. The clauses (r ) and (r s) are discarded too, because they do not secify in which case r is true. In the clause ( s, q,, r), one conjunct is ket, r. So R nop = { r}. 4 Uniform interolation by resolution Suose now that we want to comute a uniform interolant for clause set S and a sublanguage L nop defined by a subset of variables P : L nop is the sublangage of L whose formulas have no occurrences of atoms in P. We can roceed as follows: 1. recursively add to S all clauses obtained from S by resolutions on variables from P : this gives a set of clauses S res(p ) ; then 2. suress from S res(p ) all information about variables from P ; formally, we define an oerator Su such that Su(P, C) associates to clause C a clause that forgets what C says about variables from P : if C is of the form C or C for some P, then Su(P, C) = ; otherwise, that is, if no variable of P aears at ground level in C: Su(P, C R 1... R n C 1... C n) = C Su(P, R 1)... Su(P, R n) Su(P, C 1)... Su(P, C n) where for each i, Su(P, R i) = {Su(P, C) C R i}. We also erform the natural simlification: C and R { } R. Let S nop = {Su(P, C) C S res(p ) }. We claim that S nop is a uniform interolant of S on L nop. In order to see this, suose that φ is a formula of L nop such that S = φ: let S be a conjunctive normal form of φ, by refutation comletness of Enjalbert and Fariñ as resolution system, there is a deduction by resolution of from S S. Let us use Pro. 1 and re-arrange this deduction to get a sequence of inferences where all resolutions on variables from P are before the others: it has the form I 1,... I k,..., I n, where inferences I 1,..., I k are on variables from P, and I k+1,..., I n are resolutions on or on other variables. Let C 1,..., C m be the remisses of the inferences I k+1,..., I n that are in S or that are consequents of inferences I 1,..., I k ; and let R be the set of clauses obtained by alying Su(P, ) to C 1,..., C m: R S nop, and there is a deduction by resolution from R S for, since there is a deduction by resolution from {C 1,..., C m} S and clauses in R are simler than C 1,..., C m. Thus S nop S =, hence S nop = φ. Examle 2 (continued) Suose we need to comute a L r interolant of R = { (r ), ( s), ( s, q)} where L r is the language whose only variable is r. We first erform all ossible resolutions that do not involve r: 5 Conclusion The results above show that, although it is more comlicated than in roositional logic, resolution can be used to comute a uniform interolant in modal logic K. In order to have a ractical method, one would need to recise a rocedure to systematically comute P -resolvants of a given set of clauses. An algorithm like [2] s saturation by set could be used, couled with the elimination of subsumed clauses. From an imlementation oint of view, an imortant difference between resolution in roositional logic and in modal logic is the reresentation of clauses: they can be efficiently reresented in a table when there are no modalities. From this oint of view, the aroach of [1] seems romising: they flatten modal formulas, using a naming scheme for the ossible worlds, a little like skolemization in first-order logic, and then erform resolution on flat clauses. In order to roerly use this aroach for interolation, one would need to define a sort of de-skolemization in order to regain modal formulas after interolation has been erformed on flat clauses. REFERENCES [1] Carlos Areces, Maarten de Rijke, and Hans de Nivelle, Resolution in modal, descrition and hybrid logic, Journal of Logic and Comutation, 11(5), , (2001). [2] Philie Besnard, René Quiniou, and Patrice Quinton, A theorem rover for a decidable subset of default logic, in Proceedings of the Third National Conference on Artificial Intelligence, Morgan Kaufmann, (1983). [3] Meghyn Bienvenu, Prime Imlicates and Prime Imlicants in Modal Logic, in Proceedings of the 22nd National Conference on Artifcial Intelligence (AAAI 07), eds., Robert C. Holte and Adele Howe, AAAI Press, (2007). [4] Giovanna D Agostino and Giacomo Lenzi, On modal mu-calculus with exlicit interolants, Journal of Alied Logic, 4(3), , (2006). [5] Patrice Enjalbert and Luis Fariñas del Cerro, Modal resolution in clausal form, Theoretical Comuter Science, 65(1), 1 33, (1989). [6] Silvio Ghilardi and Marek W. Zawadowski, Undefinability of roositional quantifiers in the modal system s4, Studia Logica, 55(2), , (1995). [7] Silvio Ghilardi and Marek W. Zawadowski, Sheaves, Games, and Model Comletions, volume 14 of Trends in Logic, Kluwer, [8] Jürg Kohlas, Serafín Moral, and Rolf Haenni, Proositional Information Systems, Journal of Logic and Comutation, 9(5), , (1999). [9] Jérôme Lang and Pierre Marquis, Resolving inconsistencies by variable forgetting, in Proceedings of the Eights International Conference on Princiles of Knowledge Reresentation and Reasoning (KR-02), eds., Dieter Fensel, Fausto Giunchiglia, Deborah L. McGuinness, and Mary-Anne Williams, Morgan Kaufmann, (2002).

6 [10] Balder ten Cate, Willem Conradie, Maarten Marx, and Yde Venema, Definitorially comlete descrition logics, in Proceedings of the Tenth International Conference on Princiles of Knowledge Reresentation and Reasoning (kr 06), eds., Patrick Doherty, John Mylooulos, and Christoher A. Welty, AAAI Press, (2006). [11] Albert Visser, Bisimulations, model descritions and roositional quantifiers, in Gödel 96: Logical foundations of mathematics, comuter science and hysics Kurt Gödel s legacy, ed., Petr Hájek, A K Peters Ltd, (2001).