Frankowski BISIMULATIONS AND p - MORPHISMS

One of the fundamental notions of model theory for modal logics is a bisimulation. Indeed it not only preserves logical value of formulas but it plays the important role for the modal definable classes of models ([2]). The main scope of the paper is to present several types of bisimulations that are proper for the different types of model structures – ordinary Kripke models, their many-valued counterparts, or those based on topological spaces. Moreover, the important part of our work is to show connections between bisimulation and the so-called p-morphism. However, we will not limit ourselves to the well known statement that every p-morphism is a bisimulation. Some reduction of the notion of bisimulation to the notion of p-morphism will be shown. 1. Frames and Kripke models In this section we define some known notions. Let F = (W,R), where W 6= ∅, R ⊆W 2 be a relational structure, that will be called frame. By Kripke model based on F we understand a pair M = (F, V ), where V : V ar −→ P(W ) is a valuation. The formula value at the point is defined recursively: F, V, w ‖– p iff w ∈ V (p) F, V, w ‖– ¬φ iff F, V, w ‖6– φ F, V, w ‖– φ ∧ ψ iff F, V, w ‖– φ & F, V, w ‖– ψ F, V, w ‖– φ iff F, V, v ‖– φ for every v ∈W, Rwv 230 Szymon Frankowski Assume that the frames F = (W,R) and G = (W ′, S) are given. Bisimulation between F and G, is a relation ∅ 6 = Z ⊆W ×W ′ fulfilling (for wZw′): • if Rwv then there exists v′ ∈W ′ such that Sw′v′ and vZv′ • if Sw′v′, then for some v ∈W the following relations Rwv and vZv′ hold. The following notations will be used: • Z : F↔G when Z is a bisimulation between F and G • Z : F, w↔G, w′ when Z : F↔G and wZw′ • Z : (F, V )↔(G, V ′), if Z : F↔G and w ∈ V (p) iff w′ ∈ V ′(p) for wZw′ • Z : (F, V ), w↔(G, V ′), w′ etc. Relevance of bisimulation is expressed in: Theorem 1.1. (see [2]) If Z : (F, V ), w↔(G, V ′), w′, for any formula φ : F, V, w ‖– φ iff G, V ′, w′ ‖– φ. Any function f : W −→W ′ such that: • Rwv ⇒ Sfwfv • Sfwv′ ⇒ ∃v∈W (Rwv & fv = v′) will be called p-morphism between F and G (symbolically, f : F G). Naturally, a relation f is p-morphism iff f is a bisimulation that is a function. Let Z : F↔G and Dom(Z) = W . Put ≡Z= (Z−1 ◦ Z)∗. It is easy to see that ≡Z is an equivalence relation on the set W ′. Moreover, let G/ ≡Z= (W ′ ≡Z , S/ ≡Z) where S/ ≡Z is defined in the following manner: S/ ≡Z [w][v] iff ∃w1∈[w],v1∈[v]Sw1v1. Let Z• : W −→ W ′ be any function that validates Z•(w) ∈ −→ Z (w). (By the way we can notice that the usage of the axiom of choice is not essential and just simplifies the construction). Finally, put fZ : W −→W/ ≡Z where: fZ(w) = [Z•(w)]. Bisimulations and p-morphisms 231 Reassuming – the set W ′ ≡Z contains two types of classes of abstraction. The classes of the first type are just one element sets, for example those classes that contain w′ ∈W ′ not linked by Z with any element of W . Any class of abstraction of the second type has the following property: for its two different elements, say w′ and v′, the latter can be reached from w′ by the finite sequence wZw1Zw 1Z w2 . . . wkZv ′ back and forth steps through the relation Z. According to Theorem 1.1 any ≡Z-connected points w′ and v′ fulfill G, V ′, w′ ‖– φ iff G, V ′, v′ ‖– φ for every valuation V ′. However, bisimilarity is stronger property than the above (see [2]). Moreover, it can be easily proved that (fZ(W ), S/ ≡Z |fZ(W )) is a generated subframe of G/ ≡Z , that is for every [w] ∈ fZ(W ): if [v] ∈W ′/ ≡Z and S/ ≡Z [w][v], then [v] ∈ fZ(W ), in other words fZ(W ) is closed under S/ ≡Z . Theorem 1.2. fZ is a function (it does not depend on Z•) and fZ : F G/ ≡Z . Proof. Functionality of fZ is obvious. Assume thatRwv. SinceDom(Z) = W and Z is a bisimulation. There can be found w′, v′ ∈ W ′ such that wZw′, w′Sv′ and vZv′. Naturally, [w′] ≡Z fZ(w), [v′] ≡Z fZ(v) and, consequently, S/ ≡Z [w′][v′]. Thus, S/ ≡Z fZ(w)fZ(v). On the other hand, let S/ ≡Z fZ(w)[v]. Thus there are: (i) v′ 1 ≡Z v′ and w′ 1 ∈ fZ(w) which validate Sw′ 1v 1. According to the definition of Z•, we can write w′ 1Z w1Zw ′ 2 . . . wn−1Z wn−1Zw ′ nZ wn = w for some n. We are showing by induction that: for any i ≤ n there are ui such that Sw ′ iu ′ i and ui such that Rwiui (1) and {u1, . . . , un} ⊆ [v′ 1], uiZ◦ ≡Z v′ 1. If i = 1 we have Sw′ 1v ′ 1 and due to the definition of bisimulation, there exists u that forms the square with w1, w′ 1 and v ′ 1, that is Rw1u and uZv ′ 1. Assume that our hypothesis holds for i < i0 ≤ n. So far, we have shown that Rwi0−1ui0−1 and (i) ui0−1Z◦ ≡Z v′ 1. From wi0−1Zw i0 we obtain the existence of ui0 for which (ii) Sw ′ i0 ui0 and ui0−1Zu ′ i0 hold true. By (i) we 232 Szymon Frankowski obtain ui0 ∈ [v ′ 1]. On the other hand, by (ii) and wi0Zw ′ i0 there is ui0 such that Rwi0ui0 and ui0Zu ′ i0 , that is ui0Z◦ ≡Z v′ 1. Since (1) there is un ∈ W which fulfills Rwnun and unZ◦ ≡Z v′ 1, that is fZ(un) = [v′ 1]. For V ′ : V ar −→ P(W ′) we put V ′/ ≡Z (p) = π(V ′(p)) where π is the canonical function. Theorem 1.3. If Z : (F, V )↔(G, V ′), then fZ : (F, V ) (G/≡Z , V/≡Z) 2. Many valuedness in frames and Kripke models Definition 2.1. [3] Assume that a propositional language L = (L, f1, . . . , fn) is given. Let A = (A,≤A) be such that A = (A,F1, . . . , Fn) is an algebra similar to L and ≤A is a partial order on A that determines complete Heyting algebra. By A-frame we understand F = (W, r) where W 6= ∅ and r : W ×W −→ A. By A-Kripke model we shall understand a pair M = (F, V ) where V : V ar ×W −→ A. Every valuation V can be extended on the set of all formulas: V (w, f(φ1, . . . , φσ(f))) = F (V (w,φ1), . . . , V (w,φσ(f))) V (w, φ) = ∧ v∈W (Rwv →A V (v, φ)) V (w,♦φ) = ∨ v∈W (Rwv ∧A V (v, φ)). Let A = (A,≤A) and B = (B,≤B). Put HOM(A,B) for the family of homomorphisms of algebras A and B that preserve arbitrary infima and suprema. Definition 2.2. For any A-frame F = (W, r), B-frame G = (W ′, s) and h ∈ HOM(A,B) by bisimulation w.r.t. h we shall understand any relation Z ⊆W ×W ′ fulfilling for wZw′ the following conditions: 1. ∀v∈W∃v′∈W ′ [vZv′ & h(r(w, v)) ≤B s(w′, v′)] 2. ∀v′∈W ′∃v∈W [vZv′ & s(w, v) ≤B h(r(w′, v′))] Bisimulation between many valued Kripke models is additionally assumed to validate hV (w, p) = V ′(w′, p) for any propositional variable p. Bisimulations and p-morphisms 233 Definition 2.3. By p-morphism w.r.t. h we shall understand a function, graph of which is a bisimulation w.r.t. h. In a many-valued case the notation Z : M↔hN, Z : M, w↔hN, w′ etc. is used. p-morphism w.r.t. will be noted as f : F h G and f : M h N . Now Theorem 1.1 takes the form (see [3]): Theorem 2.4. If Z : M, w↔hN, w′, then for every formula: φ V (w,φ) = hV ′(w′, φ). In the case of p-morphism one has: Theorem 2.5. If f : M h N, then for every formula: φ V (w,φ) = hV ′(fw, φ). For Z : M↔hN (Dom(Z) = W ) define ≡Z in the same way like in the classical case. Z is an ordinary binary relation, so there is no risk of involving any many-valued components. Next we construct a quotient model N/≡Z= (W/≡Z , S/≡Z ,B, V/≡Z) where S/ ≡Z [w′][v′] = ∨ w′ 1∈[w],v 1∈[v] Sw′ 1v ′ 1 and a valuation is defined as V/ ≡Z ([w], p) = V ′(w, p). Similarly like in the classical case put fZ(w) = [Z•(w)] where Z• preserves the previous meaning, that is wZ•(w). Theorem 2.6. (see [3]) fZ : M h N/ ≡Z . Theorem 1.2 is, in fact, a special case of the above but we have decided to give the full proof of the classical case. There are two reasons. Theorem 2.4 from [3] is not the same as Theorem 2.6 because there was assumed in [3] that the codomain of Z equals the whole set W ′. However, the generalization contained in Theorem 2.6 requires insignificant modifications only. On the other hand, the proof of Theorem 1.2 explains, in our opinion, the construction of the frame F/ ≡Z . 3. Topological Kripke models Let M = (W,O, V ) be a topological Kripke model (see [1]), i.e. (W,O) is a topological space and V ∈ P(W ) ar is a valuation. Truth conditions for modalities are 234 Szymon Frankowski M, w ‖– φ iff exists o ∈ O : w ∈ o & for every v ∈ o : M, v ‖– φ M, w ‖– ♦φ iff for every o ∈ O : w ∈ o ⇒ there exists v ∈ o : M, v ‖– φ Let F = (W,O),G = (W ′,O′) be topological spaces. Definition 3.1. !⊆W ×W ′ is topobisimulation iff for w! w′: (i) w ∈ o ∈ O ⇒ ∃o′∈O′ [w′ ∈ o′ & ∀v′∈o′∃v∈ov! v′] (ii) w′ ∈ o′ ∈ O′ ⇒ ∃o∈O[w ∈ o & ∀v∈o∃v′∈o′v! v′]. f : W −→W ′ is topo-p–morphism iff f is open and continuous. Fact 3.2. Any function f is topobisimulation iff f is topo-p–morphism. Proof. (⇒) Let f fulfills the conditions contained in 3.1. Then for o ∈ O: −→ f (o) = ⋃ {o′ ∈ O′ : o′ ⊆ −→ f (o)}, (2) Let w′ ∈ −→ f (o), thus f(w) = w′ for some w ∈ o. According to 3.1 (i) there exists o′ ∈ O′ such that w′ ∈ o′ and ∀v′∈o′∃v∈of(v) = v′; in other words o′ ⊆ −→ f (o). We have proved (⊆). The opposite inclusion is obvious. Moreover: ←− f (o′) = ⋃ {o ∈ O : o ⊆ ←− f (o′)} (3) Assume that w ∈ ←− f (o′), i.e. f(w) ∈ o′. Similarly, as above, there exists o ∈ O such that w ∈ o and ∀v∈o∃v′∈o′f(v) = v′ that is: o ⊆ ←− f (o′). (⇐) Assume that f is topo–p–morphism. Thus it validates (2) and (3). It is clear that it is the other formulation of the fact that f is topo-p– morphism. Definition 3.3. Let (W,O) be a topological space and let R ⊆W 2 be an equivalence relation. Then the quotient space (W/R,O/R) can be defined, where O/R = {o′ ⊆W/R : ⋃ o′ ∈ O}. For any topobisimulation! put R(!) = (! ◦!−1)∗ and R∼(!) = (!−1 ◦!)∗. Further, we shall write just R and R∼. Moreover, let • : W −→W ′ be any function for such that •w ∈ −→ R∼(w). Bisimulations and p-morphisms 235 Theorem 3.4. For a total bisimulation !⊆ W ×W ′ a function f! : W −→W /R∼ defined as follows: w 7→ [•w]R∼ (4)