Dimiter Vakarelov
Dynamic Modalities Dedicated to the memory of Leo Esakia
Abstract. A new modal logic containing four dynamic modalities with the following informal reading is introduced: ∀ – always necessary, ∃ – sometimes necessary, and their duals – ♦∀ – always possibly, and ♦∃ – sometimes possibly. We present a complete axiomatization with respect to the intended formal semantics and prove decidability via fmp. Keywords: modal logic, dynamic modalities, completeness theorem, canonical construction, filtration.
Introduction In this paper we introduce a new modal logic called LDM (Logic with Dynamic Modalities). It contains four modalities of dynamic nature with the following informal readings: – ∀ – always necessary, necessary in all situations, – ∃ – sometimes necessary, necessary in some situations, and their duals: – ♦∀ – always possibly, possibly in all situations, and – ♦∃ – sometimes possibly, possibly in some situations. Let us note that such modalities are frequently used in ordinary language. For instance, when the doctor prescribes some medicines, for some of them he says that they should be taken every day (“always necessary”) and for others, that they should be taken only if some special symptoms occur (“sometimes necessary”). Of course some other names for these modalities are possible, for instance ∀ — “absolute necessity”, “strong necessity”, and ∃ — “weak necessity” and similar names for their duals. We call such modalities “dynamic” because they are characteristics of changing necessity and possibility, so in this sense LDM is related to propositional dynamic logic (PDL), but at the same time it is quite different from it. Temporal logics with modalities having similar informal interpretation but with different formal semantics, have been studied semantically by Vladimir Rybakov [4]. One can find also
Special issue dedicated to the memory of Leo Esakia Edited by L. Beklemishev, G. Bezhanishvili, D. Mundici and Y. Venema
Studia Logica (2012) 100: 385–397 DOI: 10.1007/s11225-012-9383-1
© Springer 2012
386
D. Vakarelov
some formal similarity of LDM with the so called Rare-logics studied by Demri and Orlowska in [3]. The rest of the paper is organized as follows. Section 1 is devoted to the syntax and semantics of LDM. Here we discuss with more details its relation to PDL. In Section 2 we present a finite Hilbert style axiomatization of LDM and prove a completeness theorem with respect to the formal semantics. In Section 3 we prove decidability of LDM and Section 4 is for some concluding remarks and open problems. All notions from modal logic not mentioned in the paper can be found in [1, 2] which we refer to as basic books for modal logic.
1.
Syntax and semantics
The language of LDM is an extension of the language of propositional logic with four unary modalities ∀ , ∃ , ♦∀ , ♦∃ with the standard definition of a formula. We consider standard notations for propositional connectives: ¬, ∧, ∨, ⇒, ⇔, ⊥, . The semantics is based on frames (called also LDM-frames) which are structures in the following form: (W, {Ri : i ∈ S}), where W = ∅, S = ∅ and for each i ∈ S, Ri is a binary relation in W . The elements of W are called, as usual, possible worlds, and the elements of S are considered as situations. So, for each situation i ∈ S we associate a binary relation Ri ⊆ W 2 which describes the local necessity and possibility related to i. Note that we do not have in the language formal operators corresponding to these relations simply because we allow their number (finite or infinite) to change from frame to frame. A mapping v(x, p) which assigns to each propositional variable p and x ∈ W the values 0 (falsity) and 1 (truth) is called a valuation and M = (W, {Ri : i ∈ S}, v) is called a model. The extension of v to arbitrary formulas is by induction on the complexity of formulas. The truth conditions of non-modal connectives in a model M are as in the ordinary Kripke semantics and for the modal connectives they are as follows: v(x, ∀ A) = 1 iff (∀i ∈ S)(∀y ∈ W )(xRi y → v(y, A) = 1), v(x, ∃ A) = 1 iff (∃i ∈ S)(∀y ∈ W )(xRi y → v(y, A) = 1), and dually for ♦∀ and ♦∃ : v(x, ♦∀ A) = 1 iff (∀i ∈ S)(∃y ∈ W )(xRi y and v(y, A) = 1), v(x, ♦∃ A) = 1 iff (∃i ∈ S)(∃y ∈ W )(xRi y and v(y, A) = 1). We adopt the standard definitions of a formula to be true in a model, in a frame, and in a class of frames.
Dynamic Modalities
387
The formal semantics shows that we may take the pair ∀ , ∃ as primitives and the other two to be defined: ♦∃ A =def ¬∀ ¬A, and ♦∀ A =def ¬∃ ¬A, or vice versa. The following lemma will be of later use. Lemma 1.1. Let (W, {Ri : i ∈ S}, v) be a model and define R = i∈S Ri . Then the following equivalence holds for any x ∈ W and any formula A, which shows that ∀ is a normal box modality: v(x, ∀ A) = 1 iff (∀y ∈ W )(xRy → v(y, A) = 1). Proof. Direct verification. Now we present another form of the semantics which can explain its relation to the semantics of normal modal logics. Let W = (W, {Ri : i ∈ S}) be a frame. We define the modal algebra of sets over W with box and diamond modal operations corresponding to each Ri denoted by [Ri ] and < Ri >. We recall the definitions: [Ri ]A = {x ∈ W : (∀y ∈ W )(xRi y → y ∈ A)} and < Ri > A = {x ∈ W : (∃y ∈ W )(xRi y and y ∈ A)}. If v is a valuation in W then for each formula A define the set v(A) = {x ∈ W : v(x, A) = 1}. Now the formal semantics of LDM can be reformulated as follows: v(x, ∀ A) = 1 iff for all i ∈ S, x ∈ [Ri ]v(A) iff x ∈ i∈S [Ri ]v(A), v(x, ∃ A) = 1 iff for some i ∈ S, x ∈ [Ri ]v(A) iff x ∈ i∈S [Ri ]v(A), v(x, ♦∀ A) = 1 iff for all i ∈ S, x ∈< Ri > v(A) iff x ∈ i∈S < Ri > v(A), v(x, ♦∃ A) = 1 iff for some i ∈ S, x ∈< Ri > v(A) iff x ∈ i∈S < Ri > v(A),
This form of the semantics shows that ∀ -modality can be considered as a conjunction (finite or infinite) on the frame level of the normal box modalities [Ri ]v(A), and similarly, the ∃ -modality as a disjunction of these modalities. Similar representations as conjunctions and disjunctions we have for the diamond modalities ♦∀ and ♦∃ . Taking some advantage from Propositional Dynamic Logic (PDL) we may propose another intuitive meaning of the modalities of LDM. To this end we will recall some details about the intuition attached to PDL formulas. PDL considers the set W from the model (W, {Ri : i ∈ S}, v) as a set of data and the relation Ri , i ∈ S, as input-output relation of some program (denoted by i). So S is a set of programs (in general non-deterministic) acting on the elements of W . The formulas in PDL can be considered as properties (monadic predicates) of the elements of W . Namely, given a valuation v,
388
D. Vakarelov
then every formula A defines a monadic predicate in W as follows: “x has the property A” iff v(x, A) = 1. Then the PDL modalities [i] and < i > can be considered as predicate transformers with the following meaning taken from the formal semantics of PDL: x possesses the property [i]A iff all outputs of the program i applied to x have the property A. and similarly for < i > A: x possesses the property < i > A iff some outputs of the program i applied to x have the property A. The above interpretations suggest the following informal readings [i]A: “always after i, A”, and for < i > A: “sometimes after i, A”. Having in mind all this, we may attach a PDL-like meaning to the LDM modalities. Each of the four modalities ∀ , ∃ , ♦∀ , ♦∃ correspond not to a single program i but to a set S of programs. So, in this relation more appropriate are the following PDL-like notations: [S]∀ , [S]∃ , < S >∀ , < S >∃ . These notations are more useful if we consider polymodal extensions of our language with many sets of programs S. Now the formula [S]∀ A can be considered as a predicate transformer with the following sense: x possesses the property [S]∀ A iff all outputs of any program from S have the property A, In a similar way we can interpret the remaining formulas, for instance [S]∃ A: x possesses the property [S]∃ A iff all outputs of some programs from S have the property A, All this suggests the following informal reading of the LDM formulas: [S]∀ A: “always after all programs from S, A”, [S]∃ A: “always after some programs from S, A”, < S >∀ A: “sometimes after all programs from S, A”, < S >∃ A: “sometimes after some programs from S, A”.
2.
Axiomatization
We propose the following axiom system for LDM: Axiom schemes for LDM. (I) Axiom schemes for classical propositional logic. (II) Axiom schemes for ∀ and ∃ :
389
Dynamic Modalities
(K∀ ) ∀ (A ⇒ B) ⇒ (∀ A ⇒ ∀ B), (M ono∃ ) ∀ (A ⇒ B) ⇒ (∃ A ⇒ ∃ B), (∀ → ∃ ) ∀ A → ∃ A. Rules of inference: Modus ponens (MP): A,A⇒B , B A ∀ Necessitation for (N): ∀ A .
The operations ♦∀ and ♦∃ are taken by the definitions given above. The notation LDM will be used also for the just introduced axiomatic system. Let us note that the axiom (K∀ ) and the rule (N) has been taken because ∀ is a normal box modality. This fact will be used later on in the canonical constructions needed for the completeness theorem.
Lemma 2.1. [Soundness Lemma] If A is a theorem of LDM then A is true in the class of all LDM-frames. Proof. Direct verification. Lemma 2.2. (i) The following rules are provable in LDM: (ia) (MONO ∀ ) (MONO ∃ )
A⇒B , ∀ A⇒∀ B
A⇒B , ∃ A⇒∃ B
(MONO ♦∃ )
(MONO ♦∀ )
A⇒B ♦∃ A⇒♦∃ B
A⇒B . ♦∀ A⇒♦∀ B
(ib) The rule of replacement of equivalents. (ii) Examples of theorems and non-theorems (iia) The formula A = ∃ (p ⇒ q) ⇒ (∃ p ⇒ ∃ q) is not theorem of LDM (and hence ∃ is not normal modality). (iib) The formulas ∃ and ∀ A ∧ ∃ B ⇒ ∃ (A ∧ B) are theorems of LDM. Proof. (i) (ia) The rules (MONO ∀ ) and (MONO ♦∃ ) follow from the fact that ∀ and ♦∃ are normal modalities. The rule (MONO ∃ ) follows from the axiom (M ono∃ ) and the rule (N). The rule (MONO ♦∀ ) can be derived from the rule (M ono∃ ) and the definition of ♦∀ . (ib) The rule of replacement of equivalents follows from the rule of replacement of equivalents for the propositional calculus and the rules mentioned in (ia). (iia) Let x, y, z be different points and let W = {x, y, z}, we have two relations R1 and R2 in W defined by: xR1 y, xR2 z, p is true only at y and q ia false in all points. Then A is falsified at x. (iib) The first formula can be obtained by modus ponens from the theorem ∀ and axiom (∀ → ∃ ) ∀ → ∃ . For the second formula
390
D. Vakarelov
we proceed as follows. First by application of the rule (MONO ∀ ) to the classical theorem A ⇒ (B ⇒ A ∧ B) we obtain ∀ A ⇒ ∀ (B ⇒ A ∧ B). Then by the axiom ∀ (B ⇒ A ∧ B)) ⇒ (∃ B ⇒ ∃ (A ∧ B)) we obtain ∀ A ⇒ (∃ B ⇒ ∃ (A ∧ B)) which is equivalent to the required formula.
3.
Completeness
The completeness theorem for LDM will be done by a special canonical construction. Since ∀ is a normal K-modality we first associate to LDM the canonical model (W c , Rc , v c ). Here W c is the set of all maximal consistent sets of LDM, v c is the canonical valuation: for any Γ ∈ W c and any variable p, v c (Γ, p) = 1 iff p ∈ Γ, Rc is the canonical accessibility relation defined by: for Γ, Δ ∈ W c : ΓRc Δ iff (∀A)(∀ A ∈ Γ → A ∈ Δ). We will use the following fact known from the canonical completeness proof for K: Fact 3.1. For any Γ ∈ W c and any formula A: ∀ A ∈ Γ iff (∀Δ ∈ W c )(ΓRc Δ → A ∈ Δ). The canonical model for LDM will be a modification of (W c , Rc , v c ) by replacing the relation Rc with a suitably defined set {Ri : i ∈ S c }, where all Ri are binary relations in W c ( here S c is the set of canonical situations). In order to see how to define the set {Ri : i ∈ S c } let us say that it should guarantee the proof of the following two statements which in turn are needed for the Truth Lemma for the canonical model: Statement 1. ∀ A ∈ Γ iff (∀i ∈ S c )(∀Δ ∈ W c )(ΓRi Δ → A ∈ Δ), Statement 2. ∃ A ∈ Γ iff (∃i ∈ S c )(∀Δ ∈ W c )(ΓRi Δ → A ∈ Δ). Let I be an indexed set which enumerates all binary relations in W c . The elements of I will be called virtual canonical situations. In order to define the set S c of canonical situations we will try to find a characteristic property of a virtual situation i in order to be a canonical situation. Suppose for a moment that the set S c is already defined and that Statement 1 is true. Suppose also that i is an arbitrary member of S c . Then by Statement 1 we obtain that the following thing should be true: for any formula A and for any Γ, Δ ∈ W c , if ∀ A ∈ Γ and ΓRi Δ then A ∈ Δ. This is equivalent to the following condition (1) Ri ⊆ Rc and we take it as one of the properties for i to be a canonical situation. Another property for i can be obtained by looking at Statement 2, namely
Dynamic Modalities
391
(2) For any formula A and any Γ ∈ W c : ∃ A ∈ Γ → (∃Δ ∈ W c )(ΓRi Δ and A ∈ Δ). Having in mind the above informal reasoning we define: a virtual situation i ∈ I is a canonical situation if the corresponding relation Ri satisfies the conditions (1) and (2). We put S c to be the set of all canonical situations. Now we are ready to prove Statement 1 and Statement 2 reformulated as a separate Proposition. Proposition 3.2. Let B be a formula and Γ ∈ W c . Then: (i) ∀ B ∈ Γ iff (∀i ∈ S c )(∀Δ ∈ W c )(ΓRi Δ → B ∈ Δ), (ii) ∃ B ∈ Γ iff (∃i ∈ S c )(∀Δ ∈ W c )(ΓRi Δ → B ∈ Δ). Proof. (i) (→) Suppose i ∈ S c , Γ, Δ ∈ W c , ∀ B ∈ Γ and ΓRi Δ. Then by (1) we obtain ΓRc Δ and by Fact 3.1 we get B ∈ Δ. (i) (←) Now we will reason by contraposition. Suppose ∀ B ∈ Γ0 and proceed to show that (∃i ∈ S c )(∃Δ0 ∈ W c )(Γ0 Ri Δ0 and A ∈ Δ0 ). Since ∀ B ∈ Γ0 , Γ0 ∈ W c , then by Fact 3.1 there exists Δ0 ∈ W c such that Γ0 Rc Δ0 and B ∈ Δ0 . Let Γ be any element of W c . Suppose that ∃ A be an arbitrary formula such that ∃ A ∈ Γ. Then by axiom (∀ → ∃ ) we obtain that ∀ A ∈ Γ. Then by Fact 3.1 there exists Δ ∈ W c such that ΓRc Δ and A ∈ Δ. We denote one such Δ by Δ(∃ A ∈ Γ). Note that from the notation Δ(∃ A ∈ Γ) we can immediately conclude that ΓRc Δ and A ∈ Δ. Now we put: R = {(Γ0 , Δ0 )} ∪ {(Γ, Δ) : Δ = Δ(∃ A ∈ Γ)) for some formula ∃ A ∈ Γ}. Obviously R is a binary relation in W c so there is i ∈ I such that R = Ri . The above construction guaranties that Ri satisfies (1) and (2), so i ∈ S c . Also we have Γ0 Ri Δ0 , and this finishes this part of the proof. (ii) (→) Suppose Γ0 ∈ W c and ∃ B ∈ Γ0 . In order to construct some i ∈ S c with the required properties we need some preparation. To this end suppose that Γ is an arbitrary element of W c such that ∃ B ∈ Γ. Let A be an arbitrary formula such ∃ A ∈ Γ. Conditions ∃ B ∈ Γ and ∃ A ∈ Γ imply by axiom (M ono∃ ) ∀ (B ⇒ A) ⇒ (∃ B ⇒ ∃ A), that ∀ (B ⇒ A) ∈ Γ. Then by Fact 3.1 there exists a Δ ∈ W c such that ΓRc Δ and B ⇒ A ∈ Δ, consequently B ∈ Δ and A ∈ Δ. Denote one such Δ by Δ(∃ B ∈ Γ, ∃ A ∈ Γ). From this notation we can immediately conclude that ΓRc Δ, B ∈ Δ and A ∈ Δ. Now define R = {(Γ, Δ) : Δ = Δ(∃ B ∈ Γ, ∃ A ∈ Γ)) for some formula ∃ A ∈ Γ} ∪ {(Γ, Δ) : ∃ A ∈ Γ for some formula A and Δ = Δ(∃ A ∈ Γ)}.
392
D. Vakarelov
In the above definition the notation Δ = Δ(∃ A ∈ Γ) is taken from the proof of (i). Obviously R is a binary relation in W c so R = Ri for some i ∈ I. The above construction guarantees that conditions (1) and (2) for Ri are satisfied, so i ∈ S c . For the case ∃ B ∈ Γ0 we can see that for every Δ ∈ W c such that Γ0 Ri Δ (see the first part of the definition of R) we have B ∈ Δ. This concludes this part of the proof. (ii) (←) We will reason in this case by contraposition. Suppose that ∃ B ∈ Γ and proceed to show that (∀i ∈ S c )(∃Δ ∈ W c )ΓRi Δ. But this immediately follows from condition (2). Corollary 3.3. The set S c is not empty. Proof. By an application of Proposition 3.2(ii) to the Theorem ∃ we obtain that there exists i ∈ S c , which implies S c = ∅. Now we define the canonical model — M c = (W c , {Ri : i ∈ S c }, v c ). By Corollary 3.3 we have that the set of canonical situations S c is not empty which ensures that the canonical frame (W c , {Ri : i ∈ S c }) is an LDM-frame. Lemma 3.4 (Truth Lemma for the canonical model). The following is true for any formula A and Γ ∈ W c : v c (Γ, A) = 1 iff A ∈ Γ. Proof. We proceed by induction on the construction of the formula A. If A = p is a propositional variable, then the lemma follows from the definition of the canonical valuation. The case when A is a Boolean combination of formulas is easy. The cases A = ∀ B and A = ∃ B follow directly from Proposition 3.2. Theorem 3.5 (Completeness Theorem). Let Σ be a set of formulas and A be a formula. Then: (i) (Strong form) Σ is consistent ←→ Σ has a model. (ii) (Weak form) A is a theorem ←→ A is true in all frames. Proof. (i) (→) Let Σ be a consistent set of formulas. Then there exists a maximal consistent set Γ containing Σ. By the Truth Lemma for the canonical model 3.4 all formulas from Γ are true at Γ, so the canonical model is a model of Γ and consequently it also is a model of Σ. (i) (←) Let Σ has a model, i.e. there exist a model M = (W, {Ri : i ∈ S}, v) and x ∈ W such that for every formula A ∈ Σ, v(x, A) = 1. Define the set of formulas Γ = {A : v(x, A) = 1}. It is easy to see that Γ is a maximal consistent set of formulas. Since Σ is a subset of Γ, Σ is also consistent.
Dynamic Modalities
393
(ii) (→) This is just the Soundness Lemma 2.1. (ii) (←) We will proceed by contraposition. Suppose that A is not theorem of LDM. Then there exists a maximal consistent set Γ such that A ∈ Γ. Then by the Truth Lemma 3.4 we have v c (Γ, A) = 0, so A is false in the canonical model. Note that (ii) can be derived directly from (i).
4.
Decidability
In this Section we will show that LDM possesses fmp applying a suitable modification the method of filtration from normal modal logics. Let M = (W, {Ri : i ∈ S}, v) be a model and Σ be a finite set of formulas closed under subformulas satisfying the following closure conditions: (Σ1) ∀ A ∈ Σ iff ∃ A ∈ Σ, (Σ2) If ∃ B, ∃ A ∈ Σ then ∀ (B ⇒ A) ∈ Σ, (Σ3) ∃ ∈ Σ We construct the filtrated model M = (W , {Ri : i ∈ S }, v ) as follows. Define an equivalence relation ≡ in W by putting; x ≡ y iff (∀A ∈ Σ)(v(X, A) = 1 ↔ v(y, A) = 1) and define W = {|x| : x ∈ W } where |x| is the equivalence class determined by ≡. Since ∀ is a K-modality with corresponding Kripke relation R = i∈S Ri , we take an arbitrary filtration for R and denote it by R . We will use the following facts from the filtration theory for K-modalities: Fakts 4.1. (i) xRy → |x|R |y|, and consequently, (∀i ∈ S)(xRi y → |x|R |y|), (ii) |x|R|y| → (∀∀ A ∈ Σ)(v(x, ∀ A) = 1 → v(y, A) = 1), (iii) v(x, ∀ A) = 1 iff (∀|y| ∈ W )(|x|R|y| → v (|y|, A) = 1). (iv) The cardinality of W satisfies the inequality W ≤ 2Σ . We will use the relation R to define S and the relations Ri in W for i ∈ S . The idea is to simulate the canonical construction from the Completeness Theorem for LDM. To this end let {Ri : i ∈ I} be an enumeration of all relations in W . Let S be the subset of I such that the following conditions are satisfied for the corresponding relations Ri : (1) Ri ⊆ R ,
(2) (∀∃ A ∈ Σ)(∀x ∈ W )(v(x, ∃ A) = 0 → (∃|y| ∈ W )(|x|Ri |y| and v(y, A) = 0).
394
D. Vakarelov
Denote by M = (W , {Ri : i ∈ S }, v ). At this moment we are not sure if this is a model because we do not know if the set S is not empty. We will obtain this as a corollary from the next Lemma. Lemma 4.2 (Filtration Lemma). The following condition holds for any x ∈ W and B ∈ Σ: v(x, B) = 1 iff v (|x|, B) = 1. Proof. The case when B is a variable follows from the definition of v . Boolean combinations are trivial, so we will consider only the cases when B is in the form ∀ B and ∃ B. (a) Case ∀ B ∈ Σ. We have to show the equivalence: v(x, ∀ B) = 1 ←→ v (|x|, ∀ B) = 1. (→) Suppose v(x, ∀ B) = 1 and proceed to show that v (|x|, ∀ B) = 1. To this end suppose that i ∈ S and |x|Ri |y|. Then by (1) we have |x|R |y|. Then by Fact 4.1 (ii) we get v(y, B) = 1 and by the inductive hypothesis for B we obtain v (|y|, B) = 1. This shows that v (|x|, ∀ B) = 1. (←) We will proceed by contraposition. Let ∃ A be any element of this form from Σ and let x be any element of W such that v(x, ∃ A) = 0. Then by (Σ1) ∀ A ∈ Σ and by the axiom (∀ → ∃ ), ∀ A ⇒ ∃ A (which is true in all models) we obtain v(x, ∀ A) = 0. Then by Fact 4.1 (iii) (∃|y| ∈ W )(|x|R |y| such that v (|y|, A) = 0) and by the inductive hypothesis for A, v(y, A) = 0. We denote one such |y| by |y|(v(x, ∃ A) = 0). Note that this notation immediately implies |x|R |y| and v (y, A) = 0. Suppose now v(x0 , ∀ B) = 0, ∀ B ∈ Σ. Then by Fact 4.1 (iii) and the inductive hypothesis for A and B, there exists |y0 | ∈ W such that |x0 |R|y0 | and v (|y0 |B) = 0. We will construct an Ri with i ∈ S such that (∃|y| ∈ W )(|x|Ri |y| and v (|y|, B) = 0). We define: Ri =def {(|x0 |, |y|0 )} ∪ {(|x|, |y|) : for some ∃ A ∈ Σ, v (x, ∃ A) = 0 and |y| = |y|(v (x, ∃ A) = 0)}.
We see by the construction that the conditions (1) and (2) are satisfied, so i ∈ S . Also we have |x0 |Ri |y0 |, which completes the proof of this case. (b) Case ∃ B ∈ Σ. We have to show the equivalence:
v(x, ∃ B) = 1 ←→ v (|x|, ∃ B) = 1. (→) Suppose v(x0 , ∃ B) = 1 and proceed to show that there exists i ∈ S such that for any |y| ∈ W , if |x0 |Ri |y|, then v (|y|, B) = 1. To this end let x be any element of W such that v(x, ∃ B) = 1 and let ∃ A be any formula of this form from Σ such that v(x, ∃ A) = 0. Then by (Σ2) we
Dynamic Modalities
395
obtain ∀ (B ⇒ A) ∈ Σ and by axiom (M ono∃ ), ∀ (B ⇒ A) ⇒ (∃ B ⇒ ∃ A), we obtain v(x, ∀ (B ⇒ A)) = 0. Then by Fact 4.1 (iii) and the inductive hypothesis for A and B, (∃|y| ∈ W )(|x|R |y| and v (|y|, A) = 0 and v (|y|, B) = 1), and hence v(y, A) = 0. We denote one such |y| by |y|(v(x, ∃ A) = 0, v(x, ∃ B) = 1). This notation and the construction implies that |x|R |y|, v (|y|, B) = 1 and v(y, A) = 0. Now define Ri =def {(|x|, |y|) : v(x, ∃ B) = 1 and |y| = |y|(v(x, ∃ A) = 0, v(x, ∃ B) = 1) for some ∃ A ∈ Σ} ∪ {(|x|, |y|) : |y| = |y|(v(x, ∃ A) = 0 for some ∃ A) such that v(x, ∃ A) = 0}. The above construction guarantees that conditions (1) and (2) are fulfilled, so i ∈ S . Moreover, for any |y| ∈ W such that |x0 |Ri |y| we have, again by the construction, that v (|y|, B) = 1. This shows that v (|x0 |, ∃ B) = 1 which finishes the proof of this case. (←) We will reason by contraposition. Suppose that v(x, ∃ B) = 0 and proceed to show that v (|x|, ∃ B) = 0. Conditions ∃ B ∈ Σ and v(x, ∃ B) = 0 imply by (2) that for any i ∈ S there exists |y| ∈ W such that |x|Ri |y| and v(y, B) = 0. Then by the inductive hypothesis for B we obtain that v (|y|, B) = 0. All this shows that v (|x|, ∃ B) = 0. Note that in the above proof there are some cases for which we constructed some relations Ri such that i ∈ S . However, this does not guarantee in general that the set S is not empty - this could be true if these cases indeed can happen. So, we need the following Corollary. Corollary 4.3. The set S is not empty and hence M = (W , {Ri : i ∈ S }, v ) is a model. Proof. By condition (Σ3) ∃ ∈ Σ Then by Lemma 4.2 the following condition holds: (∃i ∈ S )(∀|y| ∈ W )(|x|Ri |y| → v (|y|, ) = 1). This condition is equivalent to ∃i ∈ S which shows that S is a non-empty set. Theorem 4.4 (Finite model property for LDM). (i) LDM possesses fmp. (ii) LDM is decidable. Proof. (i) Let M = (W, {Ri : i ∈ S, v) be a model, A be a formula and v(x0 , A) = 0 for some x0 ∈ W . Let Σ be the smallest finite set of formulas, containing the formula A and closed with respect to conditions (Σ), (Σ2) and (Σ3). It is easy to see that such Σ exists and its cardinality is not greater than 2n+1 where n is the number of subformulas of A. Take the filtration
396
D. Vakarelov
M of M trough Σ. Then by the Fact 4.1 (iv) the cardinality of M is not greater than 22n+1 , and by the Filtration Lemma 4.2 A is falsified in M . (ii) is a corollary from the proof of (i).
5.
Concluding remarks
In this paper we introduced a new modal logic LDM with two basic box modalities ∀ and ∃ and their duals — the diamond modalities ♦∀ and ♦∃ . These modalities have a dynamic nature, because they are interpreted in frames having many accessibility relations, showing that the necessity and the possibility may change within the frame. We provide a complete axiomatization of LDM and decidability via fmp. One can see from the axiomatization that ∀ and ♦∃ are normal modalities, while ∃ and ♦∀ are non-normal monotonic modalities. So LDM has both normal and non-normal monotonic modalities but both kinds of modalities have a kind of relational semantics. The completeness proof goes through a new type canonical construction which may be of some special interest. The idea of this construction comes from the representation theory of dynamic contact algebras given in [5]. The proof that LDM possesses fmp goes through a new kind of filtration which combines the main idea from the canonical construction of LDM and the filtration technique for normal modal logics. The paper can be considered as a starting point in the study of such modalities and LDM is just the minimal logic in this class. So all standard questions for a given class of logics, like definability, completeness, complexity, possible relations to known classes of logics, are open. Acknowledgements. The author was supported by the contract DID02/ 32/2009 of Bulgarian Science Fund. Thanks are due to the anonymous referee and to Yde Venema for pointing me out some typos and for discussing some problems of the paper. Let me note that the referee presented a sketch of a new proof of decidability of LDM yielding PSPACE complexity for the satisfiability problem. If everything is correct this will be a nice result, because I expected a NEXPTIME complexity. So, I propose the referee to present or publish somewhere the result under his own name. References [1] Blackburn, P., M. de Rijke, and Y. Venema, Modal Logic, Cambridge University Press, 2001. [2] Chagrov, A., and M. Zakharyaschev, Modal Logic, Oxford, 1997.
Dynamic Modalities
397
[3] Demri, S., and E. Orlowska, Incomplete Information: Structure, Inference, Complexity, Springer 2002. [4] Rybakov, V. V., ‘Temporal logic with interacting agents’, Journal of Applied NonClassical Logics 18(2-3):293–308, 2008. [5] Vakarelov, D., ‘Dynamic Mereotopology: A Point-free Theory of Changing Regions. I. Stable and unstable mereotopological relations’, Fundam. Inform. 100(1-4): 159–180, 2010.
Dimiter Vakarelov Faculty of Mathematics and Informatics Blvd James Bourchier 5 Sofia, Bulgaria
[email protected]
