On Decidability of Intuitionistic Modal Logics

Transcription

1 On Decidability of Intuitionistic Modal Logics Natasha lechina School of Computer Science and IT University of Nottingham Dmitry Shkatov School of Computer Science and IT University of Nottingham bstract We prove a general decidability result for a class of intuitionistic modal logics. The proof is a slight modification of the Ganzinger, Meyer and Veanes [6] result on decidability of the two variable monadic guarded fragment of first order logic with constraints on the guard relations expressible in monadic second order logic. 1 Introduction Intuitionistic modal logics arise naturally in many areas, in particular in computer science. Intuitionistic logic is a natural underlying logic when dealing with functions and types, and modalities may correspond to various computational phenomena like termination etc., see for example [8,3,2]. There is a large literature on intuitionistic modal logics, for example [7,12,13,10,4,14,15,16]. Formal properties of various flavours of intuitionistic modal logics have been studied extensively. In this paper, we are interested in general decidability results for intuitionistic modal logics. In [15,16] general results on decidability and finite modal property of intuitionistic modal logic have been proved using an embedding of intuitionistic modal logics with n modalities in classical modal logics with n +1modalities. Those results are typically of the form Suppose that an intermediate logic Int +Γis decidable. Then intuitionistic modal logics IntK Γ, IntK Γ p p and IntK Γ (where IntK is intuitionistic logic with axioms and rules for modal system K added) are decidable [16], Theorem 13. In this paper we present a result of the form If an intuitionistic modal logic L is defined as a set of formulas valid in a class of Kripke models satisfying conditions

2 C on the intuitionistic and modal accessibility relations, and conditions in C are of a certain form, then L is decidable. The proof is a slight generalisation of the result in [6] and uses a translation into the two variable monadic guarded fragment of first order logic. Unfortunately, the decidability proof does not give a good decision procedure since it proceeds by reduction to satisfiability of formulas of SkS (monadic second-order theory of trees with constant branching factor k, [11]) where decision procedure has non-elementary complexity. The reason that we concentrate on applying the result of [6] to intuitionistic modal logic rather than to classical modal logic is that while applicability to systems such as classical S4 or S5 is immediatelyobvious (in fact, explicitlystated in [6]), some additional work needs to be done to apply it to the intuitionistic modal logic. 2 Intuitionistic modal logics Intuitionistic modal languages are obtained by adding either or both of the unary connectives (necessity) and (possibility) to the language of propositional intuitionistic logic, which contains a stock of propositional parameters ar = {p 1,p 2,...}, a unary connective, and binary connectives,, and. nalogously to and, in intuitionistic logic and are not required to be dual. Semantics of intuitionistic modal logics extends Kripke semantics for intuitionistic propositional logic. n intuitionistic Kripke model is a structure M = (W, R,V) such that (i) W, (ii) R is a reflexive and transitive binary relation on W, and (iii) V is a function from ar into the powerset of W such that, for all w W and p ar, ifw V (p) and wrv, thenv V (p) (condition we will refer to as upward persistency for propositional variables). Elements of W are called nodes. Truth at a node is defined as follows ( and stand for classical implication and negation, respectively): M,w = p M,w = ϕ iff w V (p); iff v(r(w, v) (M,v = ϕ)); M,w = ϕ ψ iff M,w = ϕ and M,w = ψ; M,w = ϕ ψ iff M,w = ϕ or M,w = ψ; M,w = ϕ ψ iff v(r(w, v) ( (M,v = ϕ) or M,v = ψ); To accommodate formulas of the form ϕ and ϕ, intuitionistic Kripke models are augmented with binary relations R and R. There is no single accepted way of defining the meaning of and in intuitionistic logic. The following clauses 2

3 are encountered in the literature (see chapter 3 of [13] for a comprehensive survey): ( 1 ) M,w = ϕ iff v(wr v M,v = ϕ) ( 2 ) M,w = ϕ iff v(wrv u(vr u M,u = ϕ)) ( 1 ) M,w = ϕ iff v(wr v M,v = ϕ) ( 2 ) M,w = ϕ iff v(wrv u(vr u M,u = ϕ)) Observe that definition ( 2 ) gives rise to a modality which does not distribute over disjunction (hence to a non-normal modal logic). On top of the requirement that R is reflexive and transitive, some additional conditions are usually imposed on R, R,andR. s a rule, these conditions specify the way R, R,andR interact. For example, the following conditions usually accompany truth clauses ( 1 ) and ( 1 ) (see [14]): (1) R R R= R (2) R R 1 R= R 1 nother condition occurring in the literature (see for example [5]) stipulates that (3) R R In section 4 we will define a large class of conditions on R, R, R including reflexivity, transitivity, and conditions (1) - (3) above, which following [6] we call mso closure conditions or closure constraints. We will show that intuitionistic modal logics defined using those conditions are decidable. 3 Embedding into two-variable monadic fragment Our decidability proof hinges on the result of [6] that a monadic two-variable guarded fragment GFmon 2 of classical first-order logic, where in addition, some relations satisfy conditions that can be expressed as monadic second-order definable closure constraints, is decidable. For our purposes, we need a slightly more general result, since the decidability proof of [6] does not accommodate conditions involving several relations (other than equality) that is such conditions as (1), (2), and (3) which are crucial for intuitionistic modal logics. Our proof consists of two parts. First, we show that every intuitionistic modal logic L defined semantically with any of the truth clauses ( 1 ) ( 2 ) can be translated into GFmon. 2 Secondly, we show that if L is in addition the logic of aclassm of Kripke models defined by an acyclic set of mso definable closure conditions on relations R, R,andR (some terminology will be defined later) then L is decidable, since GFmon 2 over such models is (the last result being an easy generalisation of that of [6]). We start by defining GFmon. 2 In the following definitions, FV(ϕ) stands for the set of free variables of ϕ, x stands for a sequence of variables. We assume a first order languagewhich contains predicate letters of arbitrary arity, includingequality 3

4 =, and no constants or functional symbols. Definition 3.1 The guarded fragment GF of first-order logic is the smallest set that contains all first-order atoms and is closed under boolean connectives and the following rule: if ρ is an atom, ϕ GF,andx FV(ϕ) FV(ρ),then x(ρ ϕ) and x(ρ ϕ) GF (in such a case ρ is called a guard). Definition 3.2 The monadic two-variable guarded fragment GFmon 2 is a subset of GF containing formulas ϕ such that (i) ϕ has no more than two variables (free or bound), and (ii) all non-unary predicate letters of ϕ occur in guards. Next, we show that intuitionistic modal logics can be translated into GFmon 2 without equality. To that end, we define, by mutual recursion, two translations, τ x and τ y, so that a first-order formula τ v (ϕ)(v {x, y}) contains a sole free variable v, which intuitively stands for the world at which ϕ is being evaluated in the Kripke model. τ x is defined by τ x (p) := (x) τ x ( ϕ) := y(r(x, y) τ y (ϕ)) τ x (ϕ ψ) :=τ x (ϕ) τ x (ψ) τ x (ϕ ψ) :=τ x (ϕ) τ x (ψ) τ x (ϕ ψ) := y(r(x, y) ( τ y (ϕ) τ y (ψ))) τ x ( ϕ) := y(r(x, y) x(r (y, x) τ x (ϕ))) τ x ( ϕ) := y(r(x, y) x(r (y, x) τ x (ϕ))) τ y is defined analogously, switching the roles of x and y. This translation assumes modal truth clauses ( 2 ) and ( 2 ). Clauses for ( 1 ) and ( 1 ) are even simpler (and familiar from classical modal logic): τ x( ϕ) := y(r (x, y) τ y(ϕ)) τ x ( ϕ) := y(r (x, y) τ y (ϕ)) Theorem 3.3 Let φ be an intuitionistic modal formula and M be a class of models of intuitionistic modal logic. Let M M. Then, M,w = φ iff M = τ x (φ)[w] (where M is taken as a model of first order logic with R, R, R interpreting R, R,R ). From the theorem it follows that if satisfiability problem of GFmon 2 over M is decidable, then satisfiability problem of intuitionistic modal logic over M is decidable. It is well known that the guarded fragment is decidable over the class of all first order models ([1]). Decidability of GFmon 2 over models with reflexive transitive guards is proved in [6]. From this and from the fact that upward persistency for propositional variables occurring in φ is expressible in GFmon 2 it follows immediately that basic intuitionistic modal logic (with no conditions connecting R, R, 4

5 and R ) is decidable. The purpose of this paper is to generalise the result of [6] to include classes of models defined using conditions involving interaction between R, R and R. 4 Closure conditions In this section we define the form of conditions which yield decidable intuitionistic modal logics. Definition 4.1 Let W be a non-empty set. unary function C on the powerset of W n is a simple closure operator if, for all, W n, (i) C() (C is increasing), (ii) implies C() C( ) (C is monotone) (iii) C() =C(C()) (C is idempotent). n m+1-ary function C on the powerset of W n is a parametrised closure operator if, given any choice of m relations 1,..., m W n, it gives rise to a unary function C 1,..., m (parametrised by 1,..., m ) that is a simple closure operator on the powerset of W n.(c 1,..., m may be viewed as the curried version of C.) Example 4.2 reflexive, transitive closure operator for binary relations TC() is a simple closure operator. Example 4.3 function Incl () = is a closure operator parametrised by. With fixed, it gives rise to a simple closure operator. Definition 4.4 condition on relation is a simple closure condition if it can be expressed in the form C() =, wherec is a simple closure operator. condition on relation is a parametrised closure condition if it can be expressed in the form C 1,..., m () =,wherec is a parametrised closure operator. Example 4.5 Reflexivity-and-transitivity is a simple closure condition, since it can be expressed in the form TC() =. Example 4.6 Condition is a closure condition on parametrised by, since it can be stated as Incl () =. Given a set of closure conditions on a collection of relations S, wewantto preclude circularity while closing off relations in S. Definition 4.7 Let S be a set of relations and C a set of closure conditions on relations in S. Let us say for, S, that depends on if C contains a parametrised condition of the form C 1,...,,... m () =. set of closure conditions C is acyclic, if its associated depends on relation is acyclic. Furthermore, we are not interested in arbitrary closure operators, but only in those definable in monadic second-order logic. Let ϕ(x 1,...,x n ) M stand for the set of n-tuples satisfying ϕ in model M. 5

6 Definition 4.8 closure operator C 1,..., m on n-ary relations is mso (-definable), if there exists a monadic second-order formula C 1,..., m with predicate parameters 1,..., m and, such that, for any model M and any n-ary formula φ, C 1,..., m ( φ M )= C 1,..., m (φ/ )) M. Example 4.9 Closure operator TC is definable by the mso formula TC (z 1,z 2 )= X(X(z 1 ) x, y(x(x) (x, y) X(y)) X(z 2 )) Example 4.10 Closure operator Incl is definable by the mso (in fact, first-order) formula Incl (z 1,z 2 )= (z 1,z 2 ) (z 1,z 2 ) Theorem 4.11 Let φ GFmon 2 and C be an acyclic set of mso closure conditions on relations in φ so that at most one closure condition is associated with each relation. It is decidable whether φ is satisfiable in a model satisfying C. roof. See appendix. Theorems 3.3 and 4.11 give us our main theorem: Theorem 4.12 Let M be a class of intuitionistic modal models defined by an acyclic set of mso closure conditions on R, R, and R so that at most one closure condition is associated with each relation, and let φ be an intuitionistic modal formula. Then, it is decidable whether φ is satisfiable in M. 5 Examples In this section, we state several decidability results just to illustrate our approach. The first example is by no means a surprise, although we doubt if anyone has proved this for all possible combinations of truth definitions for modalities. Essentially this is decidability of several flavours of basic intuitionistic modal logic (no conditions on the modal accessibility relation). Example 5.1 n intuitionistic modal logic L with two modalities and,defined by a class of models where R R 1 R= R 1 R R R= R and employing any of the truth definitions for modalities ( 1 ), ( 2 ), ( 1 ), ( 2 ) (in any combination, e.g. ( 1 ) with ( 2 ); possibly with more modalities provided all truth definitions are translatable in GFmon 2 ), is decidable. roof. The class of models of L is defined by the following closure conditions on R, R and R: (i) R is reflexive and transitive; (ii) R R 1 R= R 1 6

7 (iii) R R R= R There is clearly at most one condition for each of the relations R, R and R,and the set of conditions is acyclic. We have shown in Examples 4.2 and 4.5 that the condition on R is a closure condition and in Example 4.9 that it is mso-definable. It remains to be shown that conditions on R and R correspond to an mso-definable closure conditions. Consider a function Comp () =. ssume that is reflexive and transitive. Then by the reflexivity of ; is obviously monotone in ; andcomp is idempotent because of the transitivity of. This proves that Comp is a closure operator provided that is reflexive and transitive. Conditions of the form = can be expressed as closure conditions: Comp () =. In particular, our condition on R can be expressed as Comp R (R )=R and the condition on R can be expressed as Comp R (R 1 )=R 1. The condition is also mso-definable; in fact, it is definable by a first order formula: Comp (z 1,z 2 )= x y( (z 1,x) (x, y) (y, z 2 )) We have shown that the class of models of L conforms to the conditions of Theorem 4.12 which proves that L is decidable. The next example looks like a known result (decidability of LL [5]), but for a slightly different logic (without fallible worlds): Example 5.2 n intuitionistic modal logic L with one modality, definedbya class of models where R is reflexive and transitive; R R and employing the truth definition ( 2 ) for the modality, is decidable. roof. The class of models of L is defined by the following closure conditions: (i) TC(R )=R ; (ii) TC(R) =R; (iii) Incl R (R) =R (see Examples 4.3 and 4.6). This set of conditions is acyclic and each condition is mso definable. However there are two constraints associated with R: it is required to be closed both with respect to TC and to Incl R. To satisfy the conditions of Theorem 4.12 we need to combine them into one mso definable closure condition. Luckily, TC Incl is a closure operator with the property that for any relation, TC(Incl ()) = TC() = and Incl () = 1 1 In general, for any two closure operators C 1 and C 2 there is a closure operator C 3 such that, for 7

8 so the conditions can be reformulated as (i) TC(R )=R ; (ii) TC(Incl R (R)) = R; and it is straightforward to show that the second condition is mso definable. Finally, a non-example: we failed to reformulate the condition R R R R defining an intuitionistic modal logic in [9] as a closure condition. 6 Conclusions We have described a general method for proving decidability of an intuitionistic modal logic by translating it into monadic GF 2 and showing that conditions on the intuitionistic and modal accessibility relations can be expressed using mso definable closure operators. We illustrate this method by showing that it works for various truth definitions for modalities and various conditions on the intuitionistic and modal accessibility occurring in the literature. Most of the decidability results for particular logics obtained as illustrations of our proof are already known, but we believe that our method can easily yield new results, especially for logics with non-normal modalities defined using the truth definition ( 2 ) which are less well studied. Obviously, the same method works for intuitionistic logic with more than two modalities, provided all truth definition are translatable in GF mon 2. cknowledgement This work was supported by the ESRC grant GR/M98050/01. References [1] ndréka, H., J. van Benthem and I. Németi, Modal languages and bounded fragments of predicate logic, Journal of hilosophical Logic 27 (1998), pp [2] Benton, N., G. Bierman and V. de aiva, Computational types from a logical perspective, Journal of Functional rogramming 8 (1998). [3] Davies, R. and F. fenning, modal analysis of staged computation, in: G. Steele, Jr., editor, roc. of 23rd OL (1996), pp [4] Došen, K., Models for stronger normal intuitionistic modal logics, Studia Logica 44 (1985), pp [5] Fairtlough, M. and M. Mendler, ropositional lax logic, Information and Computation 137 (1997). any relation, C 3 () = iff C 1 () = and C 2 () =. Hence, if a logic can be defined by a finite set of closure conditions on some relation, then it can also be defined by giving a single condition on. 8

9 [6] Ganzinger, H., C. Meyer and M. Veanes, The two-variable guarded fragment with transitive relations, in:roc. 14th IEEE Symposium on Logic in Computer Science (1999), pp [7] Goldblatt, R., Metamathematics of modal logic, Reports on mathematical Logic 6,7 (1976), pp , [8] Moggi, E., Notions of computation and monads, Information and Computation 93 (1991), pp [9] N. lechina and M. Mendler and V. de aiva and E. Ritter, Categorical and Kripke semantics for constructive modal logics,in:roceedings CSL 01 (2001), pp [10] lotkin, G. and C. Stirling, framework for intuitionistic modal logic, in: J. Halper, editor, Theoretical spects of Reasoning about Knowledge, [11] Rabin, M., Decidability of second-order theories and automata on infinite trees, Transactions of the merican Mathematical Society 141 (1969), pp [12] Servi, G. F., On modal logics with intuitionistic base, Studia Logica 27 (1986), pp [13] Simpson,., The roof Theory and Semantics of Intuitionistic Modal Logic, h.d. thesis, University of Edinburgh (1994). [14] Wolter, F. and M. Zakharyaschev, Intuitionistic modal logics, in: Logic in Florence, [15] Wolter, F. and M. Zakharyaschev, On the relation between intuitionistic and classical modal logics, lgebra and Logic (1996). [16] Wolter, F. and M. Zakharyaschev, Intuitionistic modal logics as fragments of classical bimodal logics, in: E. Orlowska, editor, Logic at Work, Kluwer, roof of Theorem 4.11 This appendix contains the proof of theorem 4.11, which is a slight generalisation of the proof of Theorem 4 from [6]. In that proof all relations are assumed to be closed under equivalence (to show decidability of the fragment with equality). Note that closure under equivalence is a special case of a parametrised closure constraint so we do not need to treat it separately here. Let φ GFmon 2 and let C be an acyclic set of mso closure conditions on relations in φ. φ is satisfiable in a model satisfying C iff N, the Skolemised form of φ, is satisfiable in a Herbrand model in which all conditions from C hold. The idea of the decidability proof is to reduce the latter problem to satisfiability of formulas of SkS (mso theory of trees with constant branching factor k), where k is the number of Skolem function symbols in N. We construct an mso formula MSO N,inthe vocabulary of SkS, such that MSO N is satisfiable iff N has a Herbrand model satisfying closure conditions from C. The construction proceeds in three stages. 9

10 Stage 1. For each predicate in N, construct formula ϕ, as follows. Let (t 1 ),...,(t m ) be all positive literals of N containing (each t i is either t i,or t i1,t i2, depending on the arity of ). Let t[s] be the result of substituting a term s for the free variable of t (note that positive literals contain at most one free variable). First, for each of the above (t i ), we introduce a new unary second-order variable X (ti ). Second, for a binary predicate, we put ϕ (z 1,z 2 )= m z(x (ti )(z) z 1 = t i1 [z] z 2 = t i2 [z]) i=1 (for unary predicates, just ignore z 2 ). Intuitively, the relation defined by ϕ is the minimal extension of. Next, for each predicate that has a closure condition imposed on it, we define the closure ψ of ϕ with respect to the closure condition on. For each such we have a single closure condition C, which may be parametrised by other predicates. For simplicity, assume that C is parametrised by a single predicate that, in its own turn, has a simple closure condition C. We know, then, that C is definable by an MSO formula C (z 1,z 2 ) containing and C is definable by an MSO formula C (z 1,z 2 ), containing and.first,wedefine the closure of with respect to its simple closure condition: ψ (z 1,z 2 )=C (z 1,z 2 )[ϕ / ] that is, we replace every occurrence of in C (z 1,z 2 ) with ϕ. Next, we define the closure of with respect to its parametrised condition: ψ (z 1,z 2 )=C (z 1,z 2 )[ψ /,ϕ / ] In general, for any acyclic set C of conditions on the collection of relations S, we first define the simple closures, then the closures parametrised by relations with simple closure conditions, etc. The acyclicity of C ensures that this procedure can be carried out. Stage 2. For each clause χ = {ρ 1,...,ρ l } in N (which is thought of as a conjunction of clauses), we construct formula MSO χ, as follows. First, for every literal ρ in χ, we define formula MSO ρ, according to the rule MSO ρ = X ρ (x), if ρ is a non-ground atom containing x zx ρ (z), if ρ is a ground atom ψ (t), if ρ is (t) where ψ is the formula constructed at stage 1. Secondly, we put MSO χ = ρ χ MSO ρ. Stage 3. Lastly, let MSO N = X x χ N MSO χ. Having defined MSO N, we now prove that N has a Herbrand model satisfying the closure conditions in C iff MSO N is satisfiable in a tree. We can assume, 10

11 without loss of generality, that the nodes of a tree are terms of a Herbrand universe of N and the ith successor of node s is node f i (s). Let s denote this tree by T. ( ) ssume that N has a Herbrand model satisfying the closure conditions in C. We want to show that T satisfies MSO N. Fix witnesses for second-order variables X ρ of MSO N as follows: (i) If t i is non-ground, then X (ti ) = {a : = (t i [a])}. (ii) If t i is ground, then X (ti ) is a non-empty set. We know that for each clause χ of N, and each tuple a, = χ(a). This means that for each a, there is a literal ρ in χ such that = ρ(a). We show that for any a and ρ, if = ρ(a), thent = MSO ρ (a). Hence = χ(a) implies T = MSO χ (a). We have three cases to consider, depending on the form of ρ. The first two (non-ground atom (t i ) and ground atom) are exactly the same as in [6]. If ρ is a negative literal (t i ), we need to show that T = ψ (t)(a). It suffices to show that ψ. Indeed, this, together with our assumption that = (t)[a], implies T = ψ (a). First, the definition of T guarantees that ϕ. Hence, by monotonicity of closure operators, C 1 ( ϕ ) C 1 ( ).By definition of ψ, C 1 ( ϕ )= ψ ; furthermore, since satisfies conditions in C, C 1 ( )= ; hence, ψ. ( ) ssume that MSO N is true in T. We have to find a Herbrand model satisfying conditions in C where N is true. Define as follows. The universe of is the set of nodes of T,and = ψ. First, we prove that satisfies closure conditions C. To this end, we have to show that C 1 ( )=. Indeed, C 1 ( )=C 1 ( ψ ) =C ψ 1 (C ψ 1 ( ϕ )) = C ψ 1 ( ϕ )= ψ =. Second, we need to show that satisfies all clauses in N. This part of the proof is exactly the same as in [6]. 11