A generalization of modal definability Tin Perkov Polytechnic of Zagreb Abstract. Known results on global definability in basic modal logic are generalized in the following sense. A class of Kripke models is usually called modally definable if there is a set of modal formulas such that a class consists exactly of models on which every formula of that set is globally true, i. e. universally quantified standard translations of these formulas to the corresponding first order language are true. Here, the notion of definability is extended to existentially quantified translations of modal formulas a class is called modally -definable if there is a set of modal formulas such that a class consists exactly of models on which every formula of that set is satisfiable. A characterization result is given in usual form, in terms of closure conditions on such classes of models. 1 Introduction One of the ways to measure the expressivity of a language is to establish conditions of definability, which outline the power of a language to describe properties of models. In modal logic, this can be done in a similar way on different levels of semantics. Only the Kripke semantics is considered in this paper, but even so we can speak about local definability on the level of pointed models, global definability for Kripke models (without designated point), or frame definability on the level of Kripke frames, where we demand that a defining formula is valid on a frame, i. e. globally true regardless of a choice of valuation of propositional variables. For the sake of simplicity, only the basic propositional modal language, which has one modal operator, is considered in this paper. As for the semantics, the global level of models is considered, with fixed valuation, but without designated point. Throughout the paper, notation and terminology follows [1], so some of the most basic definitions and results are omitted or only briefly reviewed. A Kripke frame for the basic modal language is a pair F = (W, R), where W is a non-empty set, and R a binary relation on W. A Kripke model based on a frame F is M = (W, R, V ), where V is a function called valuation, which maps every propositional letter p to a subset V (p) W. The truth of a formula is defined locally and inductively as usual, and denoted M, w ϕ. Namely, for the modal operator we have that a formula of a form ϕ is true in w W if M, u ϕ for some u such that wru. We say that a formula is globally true on M if it is true in every w W, and we denote this by M ϕ. On the other hand, a formula is called satisfiable if it is true in some w W.
A generalization of modal definability 129 A class K of Kripke models is modally definable if there is a set S of formulas such that K consists exactly of models on which every formula of S is globally true, i. e. K = {M : M S}. A way to answer to the question of expressiveness of a language is to establish model-theoretical closure conditions that are necessary and sufficient for a class of models to be modally definable. On the level of frames such conditions are given by the famous Goldblatt-Thomason theorem, and for pointed models we have de Rijke characterization (detailed proofs of both are given in [1]). For the global level of models we have the following characterization, which is really the starting point of this paper. Theorem (de Rijke-Sturm). A class K of models is globally definable by a set of modal formulas if and only if K is closed under surjective bisimulations, disjoint unions and ultraproducts, and K is closed under ultrapowers. A class K of models is globally definable by means of a single modal formula if and only if K is closed under surjective bisimulations and disjoint unions, and both K and K are closed under ultraproducts. Here, K denotes the complement of K, that is, the class of all models that are not in K. All of the truth-preserving model constructions used in the theorem bisimulations, disjoint unions and ultraproducts are described in detail in [1], and ultraproducts on even deeper level in [2], since they are fundamental for the first-order model theory. In fact, de Rijke-Sturm results are proved using correspondence between the basic modal language and the first-order language with one binary relation symbol R and a predicate symbol P for each propositional letter p of the basic modal language. It is clear that a Kripke model can be viewed as a model for this first-order language. Correspondence is naturally established by the standard translation, a function that maps every modal formula ϕ to the first-order formula ST x (ϕ), which is defined as follows: ST x (p) = P x, for each propositional letter p ST x ( ) = x x ST x ( ϕ) = ST x (ϕ) ST x (ϕ ψ) = ST x (ϕ) ST x (ψ) ST x ( ϕ) = y(xry ST y (ϕ)) The basic property of the standard translation is that M ϕ if and only if M = xst x (ϕ). Therefore it is clear that every modally definable class of Kripke models is elementary, i. e. definable by a set of first-order formulas. These formulas are of the form xst x (ϕ). This is the point where the idea of this paper emerges. Why not consider the classes definable by formulas of the form xst x (ϕ) also modally definable? For example, the formula defines the class of models such that each point has an R-successor, but its complement, i. e. the class of models such that there is a point which has no successors, is not modally definable, since it is not closed under surjective bisimulations. On the other hand, this class consists exactly of
130 Tin Perkov models on which the formula, i. e., is satisfiable. Following definition enables us to consider any such example also modally definable in a broader sense. Definition. A class K of Kripke models is called modally -definable if there is a set S of modal formulas such that K consists exactly of models on which every formula from S is satisfiable. Note that it is not required that all formulas of S are satisfied at the same point of a model it suffices that each formula of S is satisfied at some point. It seems natural to generalize the notion of modal definability on the global level of models such that it includes -definable classes, together with modally definable classes in usual sense. A prospect of further generalization is outlined in the concluding section of this paper. The following characterization, proof of which is the main result of this paper, holds. Theorem. Let K be a class of Kripke models. 1. K is -definable by a single modal formula if and only if K is closed under total bisimulations and ultraproducts, and K is closed under disjoint unions and ultraproducts. 2. K is -definable by a finite set of modal formulas if and only if K is closed under total bisimulations and ultraproducts, and K is closed under ultraproducts. 3. K is -definable by a set of modal formulas if and only if K is closed under total bisimulations and ultraproducts, and K is closed under ultrapowers. 2 Modal -definability by a finite set of formulas The proof of the statement (1) of the main theorem is trivial, for a case of single formula is exactly dual to the notion of global modal definability by a single formula. Note the obvious fact that any class of models is closed under surjective bisimulations if and only if its complement is closed under total bisimulations. This fact is used for the purpose of minimizing the number of conditions imposed on the complement in characterizations. Proof (of 1). The statement follows from the fact that K is -definable by a single formula if and only if K is globally definable by a single formula. This is a consequence of the fact that ϕ is globally true on a model M if and only if ϕ is not satisfiable on M. Proofs of other statements, however, are not so trivial, since the notion of -definability by a set of formulas is not exactly dual to the notion of global definability. Clearly, a class is globally definable by a set of formulas if and only if its complement consists exactly of models on which a negation of some formula from that set is satisfiable. To say that a class is -definable, means that each formula from the defining set is satisfiable.
A generalization of modal definability 131 The following observation, given by de Rijke and Sturm in the concluding remarks of [4], is used in the proof of the second statement of the theorem. Proposition 1. A class K of models is -definable, i. e. definable by a set of formulas of the form xst x (ϕ 1 ) xst x (ϕ n ) for some modal formulas ϕ 1,..., ϕ n, if and only if K is closed under surjective bisimulations and ultraproducts, and K is closed under ultrapowers. Proof. Omitted fairly standard argument is used. Proof (of 2). Let S = {ϕ 1,..., ϕ n } be a set of modal formulas and let M be a model. Assume that there is a formula from S that is not satisfiable in M. Clearly, negation of that formula is globally true on M, so we have M = xst x ( ϕ 1 ) xst x ( ϕ n ). It easily follows that a class K is -definable by a finite set of formulas if and only if K is -definable by a single formula. By an easy compactness argument from Proposition 1 it follows that a class K is -definable by a single formula if and only if K is closed under surjective bisimulations and ultraproducts, and K is also closed under ultraproducts. The claim immediately follows. Before turning to the infinite case, note that we have got the following preservation result as a consequence of previously established facts. Corollary. A first-order sentence α (over the appropriate alphabet) is equivalent to a formula xst x (ϕ 1 ) xst x (ϕ n ) for some modal formulas ϕ 1,..., ϕ n if and only if it is preserved under total bisimulations. 3 Modal -definability by an infinite set of formulas In this section the infinite disjunctions are used, hopefully on the intuitively clear level which does not call for the proper introduction to the infinitary languages. Results could have been stated and proved without the use of infinite formulas this usage is just for the purpose of clearer statements and easier observation of analogies to the finite case. A -formula is a countably infinite disjunction of the form xst x (ϕ 1 ) xst x (ϕ 2 ) xst x (ϕ 3 )..., for some modal formulas ϕ k, k N. Of course, such formula is true if and only if there exists k N such that xst x (ϕ k ) is true. Saying that a class K of Kripke models is -definable means that there is a set of -formulas such that K consists exactly of models on which all of the formulas from that set are true. Since we work in the language with countably many propositional variables, it is clear, by analogy to the finite case, that a class K of models is -definable if and only if its complement is -definable by a single formula. Proposition 2. A class K of models is -definable if and only if K is closed under surjective bisimulations and ultrapowers, and K is closed under ultrapowers.
132 Tin Perkov Proof. The necessity of all conditions stated in the proposition for a class to be -definable is easily verified. For the converse, assume that K fulfils all of the closure conditions stated above. Let S be the set of all -formulas that are true on all models in K. Clearly, K lies in the class defined by S, so to prove that S defines K it remains to show that any model on which all of the formulas from S are true is in fact in K. So, let M be a model such that M = S. Let Σ denote the set of all formulas satisfiable in M. There is a model N in K such that each formula from Σ is satisfiable in N. For if not, we have that -formula σ Σ ( xst x ( σ)) is true on all models in K, so it is in S, hence M σ for some σ Σ, which is a contradiction. It is a well-known fact of classical model theory that the ultrapower U N over a countably incomplete ultrafilter U is an ω-saturated elementary extension of N (see [2]), which has a consequence that every formula from Σ is satisfiable in U N. By assumption, U N is in K, and we can assume without loss of generality that M is also saturated. The reader familiar with modal logic knows that the modal equivalence between points of saturated models is in fact a bisimulation. Thus we have a bisimulation from U N to M. To prove that this bisimulation is surjective, let w be any point from M and define Σ w = {σ : M, w σ}. Clearly, Σ w Σ, so we already have that each formula from Σ w is satisfiable in U N. Now, since Σ w is closed under conjunctions, it is finitely satisfiable in U N, which is ω-saturated, thus Σ w is satisfiable in U N. This means that there is an element in U N modally equivalent with w, as desired. This shows that M is also in K, which concludes the proof. Proof (of 3). Due to the previous observations, it suffices to show that K is - definable by a single formula if and only if it is closed under surjective bisimulations and ultrapowers, and K is closed under ultraproducts. Again, necessity is easily verified by definitions. For the converse, let K be a class of models such that all of the above closure conditions hold. It follows from Proposition 2 that K is -definable. Let S be a set of all -formulas that are true on all models from K. Suppose that there is no single -formula α such that K is actually defined by α. So, for any α S there is a model M α in K such that M α = α. For each member xst x (α i ) of the disjunction α such that M α = xst x (α i ), define I αi = {σ S : M σ = xst x (α i )}. Clearly, the family of all these subsets of S has the finite intersection property. So, it can be extended to an ultrafilter U over S. (This is one of the basic properties of ultrafilters, which can be recalled using [1] or [2].) Since K is closed under ultraproducts by assumption, we have that U M σ is also in K. But, for any I αi we have I αi U, so (by Loś fundamental theorem on ultraproducts see [2]) we have U M σ = xst x (α i ), thus U M σ = α for all α S. Since S defines K, U M σ is in K, which is a contradiction.
A generalization of modal definability 133 4 Concluding remarks On the level of Kripke frames, a notion somewhat similar to -definability is the negative definability, defined by Venema in [5], and characterized for some special cases by Hollenberg in [3]. In cited papers, a class of frames is called negatively definable if there is a set of formulas such that a class consists exactly of frames such that any formula of that set is refutable in all points, under some valuation. There is a difference, for satisfiability on the level of frames here means that formula is satisfiable everywhere (by some valuation), while on the level of models it means that it is satisfiable somewhere (with fixed valuation). To generalize the perspective of expressivity of modal logic, it may be worthwhile to try to define more analogous notion of satisfiability and -definability for the level of frames and to give characterization of such definability. Also, similarly to the notion of ±-definability in [3], the notion of generalized modal definability could be defined by saying that a class of models is modally definable if there is a pair (S 1, S 2 ) of sets of formulas such that a class consists exactly of models on which every formula from S 1 is globally true and every formula from S 2 is satisfiable. A model-theoretical characterization of this kind of definability, which is a proper generalization of usual modal definability and -definability, will be presented in a near future paper. References 1. J. Blackburn, M. de Rijke, Y. Venema, Modal Logic, Springer-Verlag, 2003. 2. C.C. Chang, H.J. Keisler, Model Theory, Elsevier, 1990. 3. M. Hollenberg, Characterizations of Negative Definability in Modal Logic, Studia Logica, 60: 357-386, 1998. 4. M. de Rijke, H. Sturm, Global Definability in Basic Modal Logic, in: H. Wansing (ed.), Essays on Non-classical Logic, World Scientific Publishers, 2001. 5. Y. Venema, Derivation rules as anti-axioms in modal logic, Journal of Symbolic Logic, 58: 1003-1034, 1993.