A STRATEGY FOR FREE-VARIABLE TABLEAUX

Transcription

1 A STRATEGY FOR FREE-VARIABLE TABLEAUX FOR VARIANTS OF QUANTIFIED MODAL LOGICS Virginie Thion Université de Paris-Sud, L.R.I. Virginie.Thion@lri.fr May 2002 Abstract In previous papers [CMC00, CMC01], M. Cialdea Mayer and S. Cerrito have proposed a proof procedure for some variants of first order modal logics called ; the proposed calculi are free variable tableaux methods. Essentially, a first order modal interpretation is a set of classical interpretations equipped with a binary relation, the accessibility relation [Kri63]. However several choices can be made with respect to semantics, concerning the object domains, the designation of terms, the existence of objects [CMC00, CMC01]. So, several variants of quantified modal logics (QMLs) are possible, just by choosing different combinations of the cases considered above. We call them DDE variants (Domains/Designation/Existence variants) of QML. Mayer and Cerrito s system is modular w.r.t. DDE variants. We have implemented (see [Thi01b]) in Objective Caml : the prototype is accessible on line at thion/proto. The role of this paper is to clarify the proof-search strategy underlying our prototype, and to prove its soundness and completeness. Résumé Dans de précédents papiers [CMC00, CMC01], M. Cialdea Mayer et S. Cerrito ont proposé une procédure de preuve, pour des variantes des logiques modales, appelée. Les calculs proposés sont des méthodes par tableaux à variables libres. Une interprétation modale du premier ordre peut être vue comme un ensemble d interprétations de la logique classique du premier ordre, équipé d une relation binaire appelée relation d accessibilité [Kri63]. Toutefois, dans le cadre de cette sémantique, plusieurs choix peuvent être faits concernant les domaines des mondes, l interprétation des termes et l existence des objets [CMC00, CMC01]. On peut donc définir plusieurs variantes des logiques modales quantifiées (QMLs) en choisissant des combinaisons différentes des cas précédemment considérés; nous les appelons variantes DDE (Domaine/Designation/Existence) de QML. Le système de M. Cialdea Mayer et S. Cerrito est modulaire par rapport aux variantes DDE. Nous avons implanté le système (voir [Thi01b]) en Objective Caml : le prototype est accessible en ligne à l adresse suivante : thion/proto. Cet article décrit la stratégie de recherche de preuve ayant permis d implanter le prototype. Il contient également la preuve de la correction et de la complétude de la stratégie. Keywords Proof-search strategie, quantified modal logics, tableaux. Mots clef Startégie de recherche de preuve, logiques modales quantifiées, tableaux. 1

2 1 Introduction : The free variable tableau systems The proof systems are calculi in the free-variable tableau style ([Fit88]) treating different DDE variants of first order modal logics. 1.1 DDE variants The notion of DDE variants is described in the papers [CMC00, CMC01]. Different hypotheses can be considered on first order modal semantics, concerning the object domains, the designation of terms and the existence of objects : The object domains (relation between the domains of the different worlds) : constant domains : All worlds have the same domain. cumulative domains : If the world w is accessible from the world w then the domain of w is included in the domain of w. variable domains : Nothing is supposed concerning the domain of worlds. The designation of terms : rigid designation : The denotation of a function symbol is the same in every world. non rigid designation : Nothing is supposed concerning the denotation of terms. The existence of terms : local terms : For any ground term, its extension belongs to every domain. non local terms : The extension of a ground term does not have to belong to every domain. By choosing different combinations of these variants, several quantified modal logics (QML) are defined : They are called DDE (Domain/ Designation/ Existence) variants of QML. 1.2 The tableau system!#"%$ The tableau system is defined in [CMC00, CMC01]. We recall it briefly here. Parameters The system & deals with special variables called parameters. Parameters appear in terms. Definition 1 (modal substitution) A modal substitution is a function ')(* where ' is a set of parameters and is a set of terms. According to the DEE variant, there can be different restrictions on legal substitutions (We will do not recall these restrictions here, see [CMC00, CMC01]). Tableaux are refutation systems. In order to prove a formula, one tries to refute +, producing a contradiction. This is done by finding a modal substitution that closes every leaf of the tableau (by containing two complementary literals for each leaf). This procedure involves expanding + (by applying expansion rules) until complementary literals are found. In a tableau proof, such a construction takes the form of a tree. If all the branches of the tree can be closed by a same substitution then the tree is closed and so + is refutable. 2

3 V Definition 2 (Negation normal form (NNF)) A formula is in NNF if the scope of any occurrence of the negation connective is atomic. Given any formula,, there exists an equivalent formula B equivalent in NNF and using only connectives in -.+0/210/23%4. Thus, we can limit ourselves to consider sentences over -.+0/210/23%4 in NNF The node level In 5 tableaux, a node is annotated by a number called the node level. The level indicates the number of worlds accessed since the initial world. The level 6 indicates that the node s formulae are associated with the initial world; the level 7 indicates that the node s formulae are associated to the worlds which are accessible from the initial world; the level 8 indicates that the node s formulae are associated to worlds which are accessible from the worlds which are accessible from the initial world, etc The annotation of a functional symbol In 59, a functional symbol can be annotated with an integer. This integer is called the annotation of the functional symbol. The notation,:, where, is a formula and indicates that all functional symbols appearing in, and outside the scope of any modal operator are annotated with the and annotation ;. The notation A :, where ACB=-,ED#/2FGF F#/,IH4 is a set of formulae with J ; in denotes -, :D / F FGFG/G, :H The initial node The initial node is 6LKANM 6 is the level of the initial node and A is the set of initial active formulae. A M = A : for some ; which depends on the logic Expansion rules The expansion rules of 52 are : O ;PK,L10Q&/RA ;SK,T/UQ/RA ;PK,L30Q&/RA ;PK,T/RA ;SKQ&/RA W X and W2Y W X WZY ;SKR[NA/UA\ ;I]^6_KA :Z` D ;PK[N,/0A ;=K, : /0[N,T/UA 3

4 + B v ab acb and aed acd ;PK f_,/%[ga/ra\ ;I]6hK, :` D /RA :Z` D ;SKif_,T/%[gA/RA\ ;I]^6_K, :Z` D /RA :Z` D /0[NA Note : A can be empty. j ;PKikml0,T/0A ;=K,^npo :rq lest/ukml0,t/0a where o is a new parameter ;PKUw2l0,T/0A ;PK,^nyx :cz o D : / { {}{ / o#~ : q lest/ua where x : is a new Skolem function i.e. a symbol of level ; in in -.w2l0,l4ƒ A, and o D : /G{ {}{ / o#~ : are all the parameters with level ; in w2l0, that does not occur A rule can be applied to a node only if the node matches with the rule premise. Moreover, the application of a a or a W depends on the propositional base of the logic as shown by the following table : W and a applications Logic K D K4 T S4 W - WZX - W Y W2Y a a b a b a d a b a d Some rules ( W X, ab and acd ) cause a level change. These rules are called dynamic rules. The other rules are called static rules. By applying a dynamic rule, one goes from a node ; KUA to a node ; ]ˆ6 KRAE\. This indicates that we start to explore the worlds which are accessible from the current world (; ). Let s remark that dynamic rules are non reversible i.e. their application can cause a loss of information Example z +Š[0kmlu z l Œ 3 z kml [R z l Œ be a formula where l is a variable symbol and is a Let predicate symbol. We would like to know whether is a theorem of the logic with cumulative domains. Ž As the tableau method is a refutation system, we have to find a closed tableau for +. z [%kmlu z l Œ 1 z wzlef_+r z l Take a look at this tableau rooted at + : 4

5 D j v D D D 6 K z [0kmlu z l Œ 1 z w2lef +u z l Œ O 6K z [%kmlu z l Œ /w2lef +u z l 6 K z [%kmlu z l Œ /%f +r z ab 7 Kgkmlu z l / +r z 7 K z o2 D /kmlu z l / where the underlined formula in each node is the next formula to be treated and the framed literals are the two elements of the branch contradiction which permits to close the branch. Thanks to the substitution - above) is found, so we can conclude that +u z q o2 D 4, we can close the tableau. A proof (the closed tree is a theorem. With the other DDE variant corresponding to variable domains, the substitution of the parameter o of level 7 by the constant of level 6 would not be possible : the substitution would be illegal. Indeed, the object denoted by the the world 6 may not belong to the domain of the world 7. In our prototype, the user indicates the sentence to prove, the propositional base (according to the accessibility relation), chooses the DDE variant and two bounds, respectively on the maximal number of explored worlds in a branch and the number of looping rule (see section 2.2.2) applications to a same formula for any given node level ;. Otherwise, the proof-search is completely automatic. In the next section, we describe the exploration strategy. Some motivations have brought us to this strategy, they will be presented in the third section. The last section outlines the proof of the soundness and completeness of the strategy. 2 The exploration strategy Each node in a tableau tree is a set of formulae so the choice of the next rule to apply is non deterministic. Hence, it is crucial to define a fair and complete strategy of rule applications. 2.1 How to explore and expand the tree efficiently? Different properties are associated to expansion rules. We have classified rules as follows. A rule is: reversible if its application does not lead to a loss of information. Otherwise, the rule is non-reversible. looping if it can be activated an unlimited number of times by the same formula (same = structurally equal). A looping rule is reversible A typical looping rule is the T Lš9 œ}, handling the quantifier ž 5

6 branching when it produces more than one expansion, hence generates two branches. Only one rule, called V, is branching in the considered system. This rule is reversible. Each rule enjoys one and only one of these four properties: (1) reversible and not looping and not branching, (2) looping, (3) branching, (4) non reversible. This table resumes the rules properties : rule reversible looping branching has the property O Ÿ (1) V Ÿ Ÿ (3) W X (4) W Y Ÿ Ÿ (2) acb (4) a d (4) j Ÿ Ÿ (2) v Ÿ (1) 2.2 Strategy Expansion A block is a sequence of rule applications. A block is built by stages: stage 1: Any reversible not looping and not branching rule is applied as far as possible until no such a rule is applicable. stage 2: For each formula, the corresponding looping rule is applied just once in the block 2. stage 3: Any V 0 is applied as far as possible until no such a rule is applicable. stage 4: Even if several non-reversible rules can be applied, only one non-reversible rule is applied and this ends the construction of the block. Each tableau branch is expanded block by block. Since the only non reversible rules are dynamic, the end of any block corresponds to an access to a new world. The tree exploration is by depth-first search Bounds Two bounds are defined : the first is the maximum number of any looping rule applications to a same formula for a level and the second one is the maximum number of node levels in a branch. These bounds are needed because a branch, in principle, could be expanded infinitely. 2 but see the following discussion of the backtracking mechanism 6

7 2.2.3 Closure test In order to know whether a branch is closed, we have to test if a substitution which closes the branch exists. This test is always done before building a new block (and when the branch can not be expanded anymore) Choice points and backtracking We have to deal with two types of choice points: (type A) which non-reversible rule apply in stage 4? and (type B) which substitution to choose in order to eventually close all the branches of the tree?. A branch cannot be expanded any more when the maximal exploration bound is reached or when no more rule can be applied to it. A branch closure fails when the branch cannot be expanded any more and no substitution can close it. When the current branch closure fails, the applications of reversible rules can not be the cause of failure (by the reversible rule definition). There can be different reasons of failure : more reversible rules have to be applied in stages 2 and 3 3, or a non-reversible rule is a bad choice in a stage 4 of a block (corresponding to a bad choice of type A), or the substitution chosen to close the previous branches fails to close the current branch, corresponding to a bad choice of type B. If the branch closure fails, we backtrack always before stage 4 : this allows us not to loose the previous reversible rule applications and apply a looping rule one more time. This is essential to the completeness of the strategy and allows us to apply the j Z 0 to the same formula times (where is a bound provided by the user) in a given block, as an effect of backtracking. For instance, the formula kml%wz z +u z l 1_ z Œ 1 f z is unsatisfiable w.r.t. varying domains, rigid designation and local terms in the logic but a tableau can not be closed if the j _ 0 is applied just once. Our prototype correctly instanciate twice the universal formula and detects unsatisfiability, provided that is greater or equal to 2. The first choice of backtracking is the most recent choice point of type A. If the current branch can not be closed by backtracking over all the choice points of type A in the branch, then one tries to find another substitution on the previous branch (corresponding to backtrack on a choice of type B). 3 Soundness and completeness proofs of the strategy In this section, we will outline the proof of soundness end completeness of the ideal strategy. By ideal, we mean that no bound is imposed on the number of explored worlds or the number of looping rule applications to a same formula. You can find the complete proof is in [Thi01a] 3.1 Soundness of the strategy Our strategy chooses a particular order for the rules (of 5 ) application so the tableau produced by the strategy is necessarily a tableau in. The proof calculi & are correct and complete (see the proof in [CMC01]) so our strategy is correct. 3.2 Completeness of the strategy Below, we define the formal notion of -saturated branch of a tableau, where The intuition behind this notion is the following : a branch Q is -saturated if and only if, for a given level, static rules are applied as far as possible before applying a dynamic rule, but the j is applied at must times (and the ; Y just once for a formula 4 ). 3 That is, a needs to be applied again to the same formula. 4 In fact, the «G needs not to be applied more than once to a formula for a given node level. 7

8 Q Q Q Q B Definition 3 [staticly saturated set of formulae] Let A be a set of annotated modal formulae, labeled by ; and possibly containing parameters. A is staticly saturated if and only if : (1) if,c1ªq < A then,=< A and Q < A (2) if,c3ªq < A then,=< A or Q <?A (3) if wzlr, < A then,^npx : z o :D /G{ {G{G/Go : ~ sª< A where x : is a new Skolem function i.e. a symbol of level ; that does not occur in -.w2l0,l4e A, and o D : / { {}{ /Go ~ : are all the parameters with level ; in -.w2l0,l4. (4) if [g,s< A then, : < A Definition 4 [ -saturated branch] Let be a natural integer. Let Q rooted at A where A is a set of annotated modal formulae. (I) Q (1) Q does not contains any application of dynamic rule is a leaf, corresponding to the root A : be a tableau branch is a -saturated branch if and only if A is a set of statically-saturated formulae (definition 3) or Q (2) Q has the form AUD A.. is a closed leaf. where A D B A anda F FGF A H is -saturated and is a static rule ArH is -saturated if and only if is not a j end A F FGF A H is -saturated or, is a j, ²±C³ and A FGF F A H is 6 -saturated. (II) A dynamic rule is applied in Q AUD. ArH µ9 : A H ` D i.e. Q where A D B A, µ# #: is the first dynamic rule applied in and J* Ž6 has the form. Au : µ is a -saturated branch if and only if A D FGF F A H is a -saturated branch and, if A H ` D is not a closed leaf (i.e. Jº¹ branch. ;» ) then A H ` D FGF F A : µ is a -saturated 8

9 / Á Definition 5 ( -saturated tableau) A tableau is -saturated if all its branches are -saturated. Lemma 1 Let A be a unsatisfiable set of formulae. If there is a closed tableau ¼ for A then there are a h< and a closed tableau ¼ \ for A such that ¼²\ is -saturated. The proof is based on an induction on the size of ¼. By lack of place, we will not develop it here. Lemma 2 Let h< let ¼&\ be a closed and -saturated tableau. For any ¾ m½ ¾ pair of adjacent static rules z r½, the tableau obtained by the permutation of and is a closed and -saturated tableau. By lack of place, we give only a sketch of the proof. We prove case by case that any two adjacent static rules can be permuted without losing the closed and -saturated properties of the tableau. That is, we prove permutation of an O with an O, with a V, with a W Y, etc. Thirteen cases are considered. Theorem 1 (Completeness) Let A be an unsatisfiable set of modal formulae. The strategy finds a closed tree rooted at A. Proof: We just give here a sketch of the proof. Let A be a unsatisfiable set of modal formulae. As is complete [CMC01, CMC00], there is a closed tableau rooted as A. We will call this tableau ¼. According to the lemma 1, there is an integer < tableau ¼\ rooted at A. and there is a closed and -saturated By lemma 2, we can assume, without loss of generality that static rules are applied in ¼ \ in the same order as in the strategy. After at most Q backtracking (where Q depends on ), our strategy either finds ¼À\ or a smaller closed tableau for A. Thus, the strategy finds a closed tableau rooted at A. Acknowledgments The author wishes to thank Serenella Cerrito who is the adviser of her Stage de DEA (first year of graduate studies), and Marta Cialdea Mayer for many helpful discussions. References [CMC00] Marta Cialdea Mayer and Serenella Cerrito. Variants of first-order modal logics. In R. Dyckhoff, editor, Proc. of Tableaux 2000, volume 1847 of LNAI, pages Springer Verlag, [CMC01] Marta Cialdea Mayer and Serenella Cerrito. Ground and free-variable tableaux for variants of quantified modal logic. Studia Logica, special issue on Analytic Tableaux, 69, [Fit88] M. Fitting. First-order modal tableaux. Journal of Automated Reasoning, 4: , [Kri63] S. A. Kripke. Semantical analysis of modal logic Â, normal propositional calculi. Zitschrift für Mathematische Logik und Grundlagen der Mathematik, 9:67 96, [Thi01a] V. Thion. Logiques modales et contraintes dynamiques en bases de données. Technical report, L.R.I. Orsay, France, Rapport de stage de DEA. [Thi01b] Virgini Thion. System description of ÃÄ Å %ÆmÇŒÈtÉ#ÊËÇ : a theorem prover for first order modal logics. In C. Areces and M. de Rijke, editors, Proceedings of Methods for Modalities 2, Amsterdam, The Netherlands, November