Bisimulation for Neighbourhood Structures

Transcription

1 Bisimulation for Neighbourhood Structures Helle Hvid Hansen 1,2 Clemens Kupke 2 Eric Pacuit 3 1 Vrije Universiteit Amsterdam (VUA) 2 Centrum voor Wiskunde en Informatica (CWI) 3 Universiteit van Amsterdam (UvA) CALCO 2007/ Bergen, Norway

2 Outline 1 Classical modal logic, neighbourhood structures 2 Bisimulation for neighbourhood structures 3 Hennessy-Milner theorem (finite models) 4 Main result: analogue of Van Benthem characterisation theorem

3 Classical Modal Logic Basic modal language L(At) φ ::= p φ φ φ φ (p At). Classical modal logic is minimal only requirement for the -operator: ϕ ψ ϕ ψ congruence rule

4 Neighbourhood semantics Neighbourhood model M = (S, ν, V ): S is a set of states, ν : S P(P(S)) is a neighbourhood function. V : S P(At) is a valuation. (S, ν) is neighbourhood frame. Truth: [[φ]] M = {s S M, s φ} M, s φ [[φ]] M ν(s)

5 Instances of neighbourhood semantics Kripke semantics Kripke frame (S, R S S) corresponds to (S, ν R ) with ν R (s) := {U S R[s] U}. Similarly for topological semantics.

6 Applications and Motivation Neighbourhood semantics: Non-normal ML: strategic ability, knowledge assessment Coalgebraic ML: semantics of (unary) modalities is obtained by transforming T -coalgebras into neighbourhood structures

7 Bisimulation for Neighbourhood Structures? Logic criteria for equivalence notion : (TRUTH) Modal truth is invariant under (s t s t) (REL) has relational characterisation. (à la forward-backward) (HM) The class of finite neighbourhood models is a Hennessy-Milner class w.r.t. (s t s t over finite models) (CHR) Classical modal logic is the -invariant fragment of first-order logic (cf. Van Benthem char. thm.)

8 Neighbourhood Structures as Coalgebras Basic observations Neighbourhood frames are 2 2 -coalgebras Neighbourhood models are 2 2 P(At)-coalgebras Bounded morphisms between neighbourhood frames/models are coalgebra morphisms

9 Morphisms of Neighbourhood Frames 2 2 -coalgebra morphism S f S ν ν 2 2 (S) 22 (f ) 2 2 (S ) Bounded morphism For all s S and U S : f 1 [U] ν(s) U ν (f (s)).

10 Morphisms of Neighbourhood Models 2 2 P(At)-coalgebra morphism S f S ν,v 2 2 (S) P(At) 2 2 (f ) Id ν,v 2 2 (S ) P(At) Bounded morphism For all s S and U S : f 1 [U] ν(s) U ν (f (s)), p V (s) p V (f (s)).

11 Two candidates Note: 2 2 does not preserve weak pullbacks bisimulation S 1 π 1 π 2 Z S 2 ν 2 µ ν (S 1 ) 2 π (Z ) 2 2 π (S 2 )

12 Two candidates Note: 2 2 does not preserve weak pullbacks behavioural equivalence S 1 ν 1 Z π 1 π 2 f 1 f 2 Y γ 2 2 f 2 S (S 1 ) 22 f (Y ) 2 2 (S 2 ) ν 2

13 2 2 -bisimulation Features S 1 π 1 π 2 Z S 2 ν 2 µ ν (S 1 ) 2 π (Z ) preserves truth of modal formulas has a relational characterisation Problematic The following example π (S 2 )

14 Example 1 x y

15 Example 1 x x and y have same neighbourhoods x and y are logically equivalent y y has no "definable" neighbourhoods

16 Example 1 x y F 1 F logically 2 equivalent

17 Example 1 x y F 1 F 2 not - bisimilar!

18 2 2 -Behavioural equivalence Preserves truth of modal formulas. Relational characterisation? Given (S 1, ν 1 ), (S 2, ν 2 ) and Z S 1 S 2, how to check whether there exist (S 1, ν 1 ) f 1 f (Y, γ) 2 (S 2, ν 2 ) such that Z = pb(f 1, f 2 )? Candidate for Y : Try the pushout (in Set) of Z, π 1, π 2.

19 Neighbourhood Bisimulation Definition Z S 1 S 2 is a neighbourhood bisimulation if its pushout Y Z, p 1, p 2 gives rise to a behavioural equivalence, i.e., there exists γ : Y Z 2 2 (Y Z ) such that Z π 1 π 2 p 1 S 1 p 2 Y Z S 2 ν 1 γ ν (S 1 ) 22 p (Y Z ) 2 p (S 2 )

20 Neighbourhood Bisimulation Theorem (relational characterisation): Z S 1 S 2 is a neighbourhood bisimulation between (S 1, ν 1 ) and (S 2, ν 2 ) iff for all (s 1, s 2 ) Z, and all U 1 S 1, U 2 S 2 : 1 If Z [U 1 ] U 2 & Z 1 [U 2 ] U 1 ( U 1, U 2 are Z -coherent ) then U 1 ν 1 (s 1 ) U 2 ν 2 (s 2 ). Comparing bisimilarity = neighbourhood bisimilarity = behavioural equivalence. On single frame: all three coincide.

21 Example 2 s 3 s 2 logically equivalent! s 1 neighbourhood bisimilar? t F 1 F 2

22 Example 2 s 3 s 2 Z s 1 t F 1 F 2

23 Example 2 and are Z-coherent s 3 s 2 Z s 1 t F 1 F 2

24 Example 2 s 3 s 2 Z s 1 t F 1 F 2

25 Example 2 {s 2 } and are Z-coherent s 3 s 2 Z s 1 t F 1 F 2

26 Example 2 Behaviourally equivalent! s 3 s 2 f g s 1 t F 1 G F 2

27 So far Criteria 2 2 -bis. Nbhd-bis. Behav.Eq. (TRUTH) Y Y Y (REL) Y Y (N) (HM)??? (CHR)???

28 Hennessy-Milner Theorem for Finite Models From examples: Finite neighbourhood models are not a H-M class with respect to 2 2 -bisimilarity and neighbourhood bisimilarity. How about behavioural equivalence? Quotienting with Suppose M 1, s 1 M 2, s 2 (modally equivalent). Idea: Identify s 1 and s 2 in modal quotient: (M 1 + M 2 )/ in 2 in 1 M 1 M 1 + M 2 ε ε in 1 ε in 2 (M 1 + M 2 )/ M 2

29 Hennessy-Milner Theorem for Finite Models From examples: Finite neighbourhood models are not a H-M class with respect to 2 2 -bisimilarity and neighbourhood bisimilarity. How about behavioural equivalence? Quotienting with Suppose M 1, s 1 M 2, s 2 (modally equivalent). Idea: Identify s 1 and s 2 in modal quotient: (M 1 + M 2 )/ in 2 in 1 M 1 M 1 + M 2 ε ε in 1 ε in 2 (M 1 + M 2 )/ M 2

30 Hennessy-Milner Theorem for Finite Models From examples: Finite neighbourhood models are not a H-M class with respect to 2 2 -bisimilarity and neighbourhood bisimilarity. How about behavioural equivalence? Quotienting with Suppose M 1, s 1 M 2, s 2 (modally equivalent). Idea: Identify s 1 and s 2 in modal quotient: (M 1 + M 2 )/ in 2 in 1 M 1 M 1 + M 2 ε ε in 1 ε in 2 (M 1 + M 2 )/ M 2

31 Modal Saturation Want: ε : S S/ is bounded morphism. Need: s t U S/ : ε 1 [U] ν(s) iff ε 1 [U] ν(t). Suffices: ε 1 [U] = [[δ]] M (definability). Def. Modal coherence A set of states X in a neighbourhood model is modally coherent if s t implies s X iff t X (i.e. X = x X ε(x)). Def. Modal saturation A neighbourhood model (S, ν, V ) is modally saturated if for all Φ L, and all modally coherent neighbourhoods X: if Φ is finitely satisfiable in X, then Φ is satisfiable in X. if Φ is finitely satisfiable in X c, then Φ is satisfiable in X c.

32 Modal Saturation Want: ε : S S/ is bounded morphism. Need: s t U S/ : ε 1 [U] ν(s) iff ε 1 [U] ν(t). Suffices: ε 1 [U] = [[δ]] M (definability). Def. Modal coherence A set of states X in a neighbourhood model is modally coherent if s t implies s X iff t X (i.e. X = x X ε(x)). Def. Modal saturation A neighbourhood model (S, ν, V ) is modally saturated if for all Φ L, and all modally coherent neighbourhoods X: if Φ is finitely satisfiable in X, then Φ is satisfiable in X. if Φ is finitely satisfiable in X c, then Φ is satisfiable in X c.

33 Modal Saturation Want: ε : S S/ is bounded morphism. Need: s t U S/ : ε 1 [U] ν(s) iff ε 1 [U] ν(t). Suffices: ε 1 [U] = [[δ]] M (definability). Def. Modal coherence A set of states X in a neighbourhood model is modally coherent if s t implies s X iff t X (i.e. X = x X ε(x)). Def. Modal saturation A neighbourhood model (S, ν, V ) is modally saturated if for all Φ L, and all modally coherent neighbourhoods X: if Φ is finitely satisfiable in X, then Φ is satisfiable in X. if Φ is finitely satisfiable in X c, then Φ is satisfiable in X c.

34 Hennessy-Milner Theorem for Finite Models Theorem Over the class of finite neighbourhood models modal equivalence implies behavioural equivalence. Proof: Finite neighbourhood models are closed under coproducts (disjoint unions), and are modally saturated (ε is morphism). in 2 in 1 M 1 M 1 + M 2 ε ε in 1 ε in 2 (M 1 + M 2 )/ M 2

35 Hennessy-Milner Theorem for Finite Models Theorem Over the class of finite neighbourhood models modal equivalence implies behavioural equivalence. Proof: Finite neighbourhood models are closed under coproducts (disjoint unions), and are modally saturated (ε is morphism). in 2 in 1 M 1 M 1 + M 2 ε ε in 1 ε in 2 (M 1 + M 2 )/ M 2

36 So far Criteria 2 2 -bis. Nbhd-bis. Behav.Eq. (TRUTH) Y Y Y (REL) Y Y (N) (HM) N N Y (CHR)???

37 Characterisation Theorem Van Benthem Characterisation Thm. (Normal ML): Normal modal logic is the Kripke-bisimulation invariant fragment of first-order logic. Translation into first-order logic translate modal formula φ into first-order language L 1, translate nbhd-model M into first-order L 1 -model M.

38 Characterisation Theorem Van Benthem Characterisation Thm. (Normal ML): Normal modal logic is the Kripke-bisimulation invariant fragment of first-order logic. Translation into first-order logic translate modal formula φ into first-order language L 1, translate nbhd-model M into first-order L 1 -model M.

39 Classical Modal Fragment of First-Order Logic Idea: View neighbourhoods as first-class citizens. Neighbourhood model M = (S, ν, V ) as FO-model M : two-sorted domain: S + ν[s]. constants: {P i p i At}. neighbourhood relation: R ν = {(s, U) U ν(s)}. element-of relation: R = {(U, s) s U}.

40 Classical Modal Fragment of First-Order Logic Standard translation {st x : L L 1 x VAR} Two-sorted first-order language L 1 : state variables (x, y,...). neighbourhood variables (u, v,...). unary predicates: {P i p i At}. binary predicates: N and E. st x (p i ) = P i x, st x ( φ) = st x (φ), st x (φ ψ) = st x (φ) st x (ψ), st x ( φ) = u (xnu y (uey st y (φ)).

41 Classical Modal Fragment of First-Order Logic Lemma: M, s φ iff M = st x (φ)[s]. But: Not all L 1 -models are neighbourhood models. Kripke case: All FO-models (of appropriate signature) are Kripke models.

42 Classical Modal Fragment of First-Order Logic Lemma: M, s φ iff M = st x (φ)[s]. But: Not all L 1 -models are neighbourhood models. Kripke case: All FO-models (of appropriate signature) are Kripke models.

43 Classical Modal Fragment of First-Order Logic Lemma: M, s φ iff M = st x (φ)[s]. But: Not all L 1 -models are neighbourhood models. Kripke case: All FO-models (of appropriate signature) are Kripke models. Characterisation w.r.t.? all L 1 -models: define beh.eq. for arbitrary L 1 -models, or neighbourhood models: ensure compactness etc. still hold.

44 Classical Modal Fragment of First-Order Logic Lemma: M, s φ iff M = st x (φ)[s]. But: Not all L 1 -models are neighbourhood models. Kripke case: All FO-models (of appropriate signature) are Kripke models. Characterisation w.r.t. neighbourhood models: ensure compactness etc. still hold.

45 Axiomatising Neighbourhood Models NAX (neighbourhood axioms): (Sts) x (x = x) (Nbh) u x (xnu) (Ext) u, v ( (u = v) x ((uex vex) ( uex vex))) Proposition: Let N = {M M = NAX}. If M N, then there is a neighbourhood model M such that M = (M ). Definition (invariance over N): α(x) L 1 is invariant for over N if for all M, M in N: if M, s M, s then M = α[s] M = α[s ].

46 Axiomatising Neighbourhood Models NAX (neighbourhood axioms): (Sts) x (x = x) (Nbh) u x (xnu) (Ext) u, v ( (u = v) x ((uex vex) ( uex vex))) Proposition: Let N = {M M = NAX}. If M N, then there is a neighbourhood model M such that M = (M ). Definition (invariance over N): α(x) L 1 is invariant for over N if for all M, M in N: if M, s M, s then M = α[s] M = α[s ].

47 Characterisation Theorem We have: 1 = N is compact. 2 N closed under elementary extensions. 3 ω-saturated N-models are modally saturated. Theorem Let α(x) be an L 1 -formula. T.F.A.E. 1 = N α(x) st x (φ) for some modal formula φ L. 2 α(x) is invariant for behavioural equivalence over N. 3 α(x) is invariant for neighbourhood bisimilarity over N. 4 α(x) is invariant for 2 2 -bisimilarity over N.

48 Characterisation Theorem We have: 1 = N is compact. 2 N closed under elementary extensions. 3 ω-saturated N-models are modally saturated. Theorem Let α(x) be an L 1 -formula. T.F.A.E. 1 = N α(x) st x (φ) for some modal formula φ L. 2 α(x) is invariant for behavioural equivalence over N. 3 α(x) is invariant for neighbourhood bisimilarity over N. 4 α(x) is invariant for 2 2 -bisimilarity over N.

49 Summary Criteria 2 2 -bis. Nbhd-bis. Behav.Eq. (TRUTH) Y Y Y (REL) Y Y (N) (HM) N N Y (CHR) Y Y Y Summary Beh.eq: Good logical properties, optimal equivalence bis: Good computational properties, too strict. Nbhd-bis: Good compromise of logical and computational properties.

50 Summary Criteria 2 2 -bis. Nbhd-bis. Behav.Eq. (TRUTH) Y Y Y (REL) Y Y (N) (HM) N N Y (CHR) Y Y Y Summary Beh.eq: Good logical properties, optimal equivalence bis: Good computational properties, too strict. Nbhd-bis: Good compromise of logical and computational properties.

51 Further Questions Other properties of classical ML: Interpolation: Yes. Łoś-Tarski: Probably not (Which formulas are preserved under taking submodels?) Goldblatt-Thomason (model-theoretic proof): Probably not (Which frame classes are modally definable?) Neighbourhood frames are Chu spaces: (S, R ν, R, N). General characterisation theorem for coalgebraic ML.