Bisimulation for Neighbourhood Structures

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) March 23 & 30, 2007 / TCS seminar

Outline 1 Classical modal logic, neighbourhood structures. 2 Examples (on blackboard). 3 Neighbourhood structures as coalgebras. 4 Coalgebraic equivalence notions. 5 Modal saturation, Hennessy-Milner Thm. 6 Characterisation theorem (cf. Van Benthem).

Syntax and Semantics Basic modal language L(At) (At countable set of atomic propositions) φ ::= p φ φ φ φ (p At). Neighbourhood model M = (W, ν, V ): W is a set of worlds, ν : W P(P(W )) is a neighbourhood function. V : W P(At) is a valuation. (W, ν) is neighbourhood frame,

Classical Modal Logic Inductive def. of [[φ]] M = {w W M, w φ} M, w p iff p V (w) M, w φ iff M, w φ M, w φ ψ iff M, w φ and M, w ψ M, w φ iff [[φ]] M ν(w) Modal equivalence: s t iff s and t satisfy the same formulas. Classical modal logic Λ: propositional tautologies, closed under substitution, modus ponens and φ ψ φ ψ (RE) Λ is sound and complete w.r.t the class of nbhd. models.

Generalisation of Kripke semantics Def. augmented neighbourhood model A neighbourhood model (W, ν, V ) is augmented if for all w W : ν(w) ν(w); Thm U V W : if U ν(w) then V ν(w). Kripke models are in 1-1 correspondence with augmented neighbourhood models. Proof sketch: Let (W, R, V ), be a Kripke model. Let R[w] = {w wrw }. Define ν(w) = {U W R[w] U}. Then (W, ν, V ) is augmented. Now let (W, ν, V ) be augmented. Define (W, R, V ) by R[w] = ν(w). Corresponding models are pointwise modally equivalent (formula induction).

Examples Are the following frames bisimilar? Should they be?

Bisimulation for Neighbourhood Structures? Logic criteria for equivalence notion E: (truth) Modal truth is invariant under E (i.e. E ). (rel) E has relational characterisation (forward-backward). (hm) The class of finite neighbourhod models is a Hennessy-Milner class w.r.t E, i.e. modal equivalence implies E-equivalence ( E over finite models). (chr) Classical modal logic is the E-invariant fragment of first-order logic (Van Benthem characterisation thm.)

Functors for Neighbourhood Structures Let X, Y be sets, f : X Y a function, then Contravariant powerset functor 2 ( ) : Set op Set 2 X = P(X) 2 f = f 1 [ ] : P(Y ) P(X) Neighbourhood functor 2 2( ) = 2 2 2 2 (X) = P(P((X)) 2 2 (f ) = (f 1 [ ]) 1 [ ] : P(P(X) P(P(Y )) U P(X), D Y : D 2 2 (f )(U) f 1 [D] U. 2 2 P(At)(X) = 2 2 (X) P(At), 2 2 P(At)(f ) = 2 2 (f ) id.

Coalgebraic modelling Def. T-coalgebra A T-coalgebra for a functor T : Set Set is a pair (X, ν) where ν : X TX). Neighbourhood structures are coalgebras Neighbourhood frame (W, ν) is 2 2 -coalgebra: ν : W 2 2 W = P(P(W ). Neighbourhod model (W, ν, V ) is 2 2 P(At)-coalgebra: ν, V : W P(P(W ) P(At) = (2 2 P(At))(W )

Morphisms of Coalgebras T-coalgebra morphism A function f : W W is a T-coalgebra morphism from (W, ν) to (W, ν ) if Tf ν = ν f. As commuting diagram: W f ν W ν TW Tf TW

Bounded Morphisms (frames) 2 2 -coalgebra morphism A function f : W W is a 2 2 -coalgebra morphism from (W, ν) to (W, ν ) if 2 2 f ν = ν f. Morphisms of neighbourhood frames Def. A function f : W W is a bounded morphism from F = (W, ν) to F = (W, ν ) if w W : 2 2 f (ν(w)) = ν (f (w)), i.e. X W : f 1 [X ] ν(w) X ν (f (w)).

Bounded Morphisms (models) 2 2 P(At)-coalgebra morphism A function f : W W is a 2 2 P(At)-coalgebra morphism from (W, ν, V ) to (W, ν, V ) if 2 2 f ν = ν f, and V = V f Morphisms of neighbourhood models Def. A function f : W W is a bounded morphism from M = (W, ν, V ) to M = (W, ν, V ) if w W : X W : f 1 [X ] ν(w) X ν (f (w)), and p At : p V (w) p V (f (w)) that is: f 1 [[[p]] ] = [[p]]

Truth Invariance (bounded morphisms) Lemma Let f : M M be a bounded morphism and φ L. We have w W : M, w φ M, f (w) φ. That is, [[φ]] M = f 1 [[[φ]] M ]. Proof (structural induction on φ) Base case: f is bounded morphism (valuation) [[p]] M = f 1 [[[p]] M ] p At. Step: Boolean connectives: standard. Modal case: IH M, w φ [[φ]] M ν(w) f 1 [[[φ]] M ] ν(w) b.m. [[φ]] M ν (f (w))

Basic Idea: Equivalence and Morphisms Equivalence as identification via morphisms: same image f 1 X 1 f Y 2 X 2 x 1 f 1 (x 1 ) = f 2 (x 2 ) x 2 same source X 1 f 1 Y f 2 X 2 x 1 = f 1 (y) y f 2 (y) = x 2

Category Theory Tool Kit CT Tool kit: Pullbacks Def. Pullback Y, p 1, p 2 is a (weak) pullback of X, g 1, g 2 if: g 1 p 1 = g 2 p 2, and for all Y, p 1, p 2 such that g 1 p 1 = g 2 p 2 there is a (not necessarily) unique h : Y Y such that p 1 h = p 1 and p 2 h = p 2. Y p 1 h p 2 Y p 1 p 2 X1 g 1 g 2 X 2 X

Category Theory Tool Kit CT Tool kit: Pushouts Def. Pushout Y, p 1, p 2 is a pushout of X, g 1, g 2 if: p 1 g 1 = p 2 g 2, and for all Y, p 1, p 2 such that p 1 g 1 = p 2 g 2 there is a unique h : Y Y such that h p 1 = p 1 and h p 2 = p 2. X g 1 X 1 p 1 p 1 Y!h Y g 2 p 2 p 2 X 2

Category Theory Tool Kit CT Tool kit: Pushouts and Pullbacks in Set Def. Canonical pullback in Set: For functions f 1 : W 1 Y and f 2 : W 2 Y, the pullback of Y, f 1, f 2 is the relation Z = pb(f 1, f 2 ) = { w 1, w 2 W 1 W 2 f 1 (w 1 ) = f 2 (w 2 )}. together with the projections π 1 : Z W 1 and π 2 : Z W 2. Def. Canonical pushout in Set: For a relation Z W 1 W 2, define Y Z = (W 1 + W 2 )/Ẑ, where Ẑ is the smallest equiv.rel. on W 1 + W 2 s.t. Z Ẑ. Let p i : W i Y Z, i = 1, 2, denote the natural quotient maps. Y Z, p 1, p 2 is called the canonical pushout of Z, π 1, π 2.

2 2 -bisimulation 2 2 -bisimilarity (same source) Def. 2 2 -bisimilarity Given F 1 = (W 1, ν 1 ), F 2 = (W 2, ν 2 ), two states s 1 W 1 and s 2 W 2 are 2 2 -bisimilar if there exists a 2 2 -coalgebra (Y, γ), and bounded morphisms f i : (Y, γ) (W i, ν i ), i = 1, 2, such that s 1 = f 1 (y) and s 2 = f 2 (y) for some y Y. W 1 ν 1 f 1 2 2 f 1 γ Y 2 2 W 1 2 2 Y f 2 W 2 ν 2 2 2 f 2 2 2 W 2 (Y, γ) (or just Y ) is called a 2 2 -bisimulation. In fact, w.l.o.g. we can assume Y is a relation Y W 1 W 2 (next slide).

2 2 -bisimulation 2 2 -bisimulation (concrete representation) Suppose Y is a 2 2 -bisimulation. Then Z = f 1, f 2 [Y ] W 1 W 2 (image under f 1, f 2 ) is a 2 2 -bisimulation. Since f 1, f 2 : Y Z surjective, it has a right inverse g : Z Y, i.e. f 1, f 2 g = id Z. Define µ: Z 2 2 Z by µ = 2 2 f 1, f 2 γ g. Then D1 and D2 commute: D2: Z D1: Y f 1,f 2 Z g π 1 π 2 γ µ 2 2 Y 22 f 1,f 2 f 1,f 2 g 2 2 f 1 f 2 Z W 1 Y W 2 It follows: π 1, π 2 are bounded morphisms (case i = 1): 2 2 π 1 µ = 2 2 π 1 2 2 f 1, f 2 γ g (2 2 is functor) = 2 2 (π 1 f 1, f 2 ) γ g (π 1 f 1, f 2 = f 1 ) = 2 2 (f 1 ) γ g (f 1 is b.m.) = ν 1 f 1 g (D2) = ν 1 π 1

2 2 -bisimulation 2 2 -bisimulation (relation) Def. 2 2 -bisimulation Given F 1 = (W 1, ν 1 ), F 2 = (W 2, ν 2 ), and Z W 1 W 2, Z is a 2 2 -bisimulation between F 1 and F 2 if there is a function µ : Z 2 2 Z such that the projections π 1, π 2 are bounded morphisms: i.e. π 1 π 2 W 1 Z W 2 ν 1 µ ν 2 2 2 2 2 π 1 W 1 2 2 2 2 π 2 Z 2 2 W 2 2 2 π 1 µ = ν 1 π 1 2 2 π 2 µ = ν 2 π 2

2 2 -bisimulation 2 2 -bisimulation (set-theoretic) Morphism conditions on π 1, π 2 π i : (Z, µ) (W i, ν i ), i = 1, 2, are bounded morphisms iff for all (s 1, s 2 ) Z : for all U 1 W 1 : π 1 1 [U 1] µ(s 1, s 2 ) U 1 ν 1 (s 1 ). for all U 2 W 2 : π 1 2 [U 2] µ(s 1, s 2 ) U 2 ν 2 (s 2 ).... necessarily (factoring out µ): for i = 1, 2, U i, V i W i : if π 1 i [U i ] = π 1 i [V i ] then for all U 1 W 1 and U 2 W 2 : U i ν i (s i ) V i ν i (s i ). if π 1 1 [U 1] = π 1 2 [U 2] then U 1 ν 1 (s 1 ) U 2 ν 2 (s 2 ).

2 2 -bisimulation 2 2 -bisimulation (set-theoretic) reformulate guard (factoring out π 1, π 2 ) Let span(z ) = dom(z ) rng(z ), and U V = (U \ V ) (V \ U). for i = 1, 2, U i, V i W i : π 1 i [U i ] = π 1 i [V i ] (U i V i ) span(z ) = ( U i V i is Z -unrelated ). for all U 1 W 1 and U 2 W 2 : π 1 1 [U 1] = π 1 2 [U 2] Z [U 1 ] U 2 and Z 1 [U 2 ] U 1 ( U 1, U 2 are Z -coherent )

2 2 -bisimulation 2 2 -bisimulation (relational characterisation) Theorem Z is a 2 2 -bisimulation between (W 1, ν 1 ) and (W 2, ν 2 ) iff (s 1, s 2 ) Z : (i) for i = 1, 2, U i, V i W i : if U i V i Z -unrelated then U i ν i (s i ) V i ν i (s i ). (ii) for all U 1 W 1 and U 2 W 2 : if U 1, U 2 Z -coherent then U 1 ν 1 (s 1 ) U 2 ν 2 (s 2 ). Now, check examples again.

Behavioural equivalence 2 2 -Behaviourally equivalent (same image) Def. Behaviourally equivalent Given F 1 = (W 1, ν 1 ), F 2 = (W 2, ν 2 ), two states s 1 W 1 and s 2 W 2 are 2 2 -behaviourally equivalent if there exist (Y, γ) and bounded morphisms f i : (W i, ν i ) (Y, γ), i = 1, 2, s.t. f 1 (s 1 ) = f 2 (s 2 ). W 1 f 1 Y f 2 W 2 ν 1 γ 2 2 f 2 ν 2 2 2 2 2 f 1 W 1 2 2 Y 2 2 W 2

Behavioural equivalence 2 2 -Behaviourally equivalent (same image) Def. Behavioural equivalence Given F 1 = (W 1, ν 1 ), F 2 = (W 2, ν 2 ), two states s 1 W 1 and s 2 W 2 are 2 2 -behaviourally equivalent if there exist (Y, γ) and bounded morphisms f i : (W i, ν i ) (Y, γ), i = 1, 2, s.t. f 1 (s 1 ) = f 2 (s 2 ). Z π 1 π 2 f 1 f 2 Y W 1 ν 1 γ 2 2 f 2 W 2 2 2 2 2 f W 1 1 2 2 Y 2 2 W 2 Z = pb(f 1, f 2 ) = { s 1, s 2 f 1 (s 1 ) = f 2 (s 2 )} is called a behavioural equivalence. ν 2

Behavioural equivalence Weak pullback preservation If functor T weakly preserves pullbacks then... If Z = pb(y, f 1, f 2 ) for some Y, f 1, f 2, then TZ is a (weak) pullback of Tf 1, Tf 2. We get µ : Z TZ from universal property of TZ : Z ν 1 π 1 µ ν 2 π 2 TZ TW 1 T π 1 Tf 1 T π 2 Tf 2 TW 2 TY Hence Z is T -bisimulation. But: 2 2 does not weakly preserve pullbacks!

Behavioural equivalence 2 2 -Behavioural equivalence How to obtain (Y, γ)? Given Z W 1 W 2, how to check whether there exist (Y, γ) and bounded morphisms f i : (W i, ν i ) (Y, γ), i = 1, 2, s.t. for all s 1, s 2 Z, s 1 and s 2 are behaviourally equivalent? Candidate for Y : Try Y Z, p 1, p 2 the canonical pushout of Z (in Set). This does not always work, only for special relations Z...

Neighbourhood bisimulation (2 2 -Relational equivalence) Neighbourhood bisimulation Def. Neighbourhood bisimulation Z W 1 W 2 is a neighbourhood bisimulation if its canonical pushout Y Z, p 1, p 2 gives rise to a behavioural equivalence:, i.e., γ : Y Z 2 2 Y Z s.t. Z π 1 π 2 p 1 W 1 p Y Z 2 W 2 Note: Z pb(p 1, p 2 ). ν 1 γ ν 2 2 2 2 2 p 1 W 1 2 2 2 2 p Y 2 Z 2 2 W 2

Neighbourhood bisimulation (2 2 -Relational equivalence) Neighbourhood bisimulation (relational char.) Theorem: Z W 1 W 2 is a neighbourhood bisimulation between (W 1, ν 1 ) and (W 2, ν 2 ) iff for all (s 1, s 2 ) Z, and for all U 1 W 1, U 2 W 2 : if U 1 and U 2 are Z -coherent then U 1 ν 1 (s) U 2 ν 2 (s 2 ). Corollary Z is 2 2 -bisimulation implies Z is a neighbourhood bisimulation implies Z is contained in a behavioural equivalence. Now, check examples...

Neighbourhood bisimulation (2 2 -Relational equivalence) So far Criteria 2 2 -bis. Nbhd-bis. Behav.Eq. (truth) Y Y Y (rel) Y Y (N) (hm) N N? (chr)???

Hennessy-Milner theorem (wrt Kripke bisim.) Theorem (finite Kripke models): Over the class of finite (finitely branching) Kripke models, modal equivalence implies bisimilarity. Theorem (modally saturated Kripke models) Over a class of modally saturated Kripke models, modal equivalence implies bisimilarity. (Finitely branching Kripke models are modally saturated.) Modal saturation (Kripke) A Kripke model (W, R, V ) is modally saturated if Φ L, w W : if all finite Φ 0 Φ are satisfiable in R[w], then Φ is satisfiable in R[w].

Modal saturation (nbhd-models) Modal coherence A set of states X in a neighbourhood model is modally coherent if for all states s, t: s t s X iff t X. (X is -coherent with itself). Def. Modal saturation (nbhd) A neighbourhood model (W, ν, V ) is modally saturated if 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. Lemma (easy) Finite neighbourhood models are modally saturated.

Hennessy-Milner theorem (wrt Beh.eq.) Theorem: Over a class of modally saturated neighbourhood models, modal equivalence implies behavioural equivalence.

Hennessy-Milner theorem (wrt Beh.eq.) Theorem: Over a class of modally saturated neighbourhood models, modal equivalence implies behavioural equivalence. Proof: Can show: If M modally saturated, and X W is modally coherent, then X = [[δ]] M for some δ L (definability). It follows: given modally saturated M 1 and M 2, we can construct (M 1 + M 2 )/ s.t. quotient map ε is a bounded morphism. Now if s 1 s 2, then ε ι 1 (s 1 ) = ε ι 2 (s 2 ).

Introduction Characterisation Theorem Char. Thm. (informally): Over the class of neighbourhood models, classical modal logic is the 2 2 -bis./nbhd-bis./beh.eq.-invariant fragment of first-order logic. Classical modal fragment of first-order logic translate modal formula φ into first-order language L 1, translate nbhd-model M into first-order L 1 -model M. Invariance for equivalence notion : An L 1 -formula α(x) is invariant for if for all nbhd-models M 1 and M 2 s.t. M 1, s 1 M 2, s 2 we have: M 1 = α(x)[x/s 1] M 2 = α(x)[x/s 2].

Translation into FOL Modal formulas as first-order formulas first-order language L 1 Two sorts: s (state sort), n (neighbourhood sort). Notational convention: x, y, z,... are state variables; and u, v,... are neighbourhood variables. α ::= x = y u = v P i x xnu uex α α α xα uα, i ω; standard translation st x : L L 1 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))).

Translation into FOL Neighbourhood models as first-order models first-order translation of M = (W, ν, V ) is the L 1 -model M = D, R ν, R, {P i i ω} where D s = S, D n = ν[s] = s S ν(s) P i = V (p i ) for each p i At, R ν = {(s, U) s D s, U ν(s)}, R = {(U, s) s D s, s U}. Lemma: For all nbhd-models M, and all modal φ L: M, s φ iff M = st x (φ)[x/s]. But: Not all L 1 -models are neighbourhood models.

Translation into FOL Axiomatising Neighbourhood L 1 -models NAX-axioms: (Sts) x(x = x) (D s ) (Nbh) u x(xnu) (D n = R N [D s ]) (Ext) u, v( (u = v) x((uex vex) ( uex vex))) (extensionality) Proposition: If M = (D, R N, R E, {P i i ω}) is an L 1 -model s.t. M = NAX, then there is a nbhd-model M s.t. M = (M ). Prf: Let W = D s, define η : D n P(W ) by η(n) = {s D s (n, s) R E }. Now define: ν : W P(P(W )) by ν(w) = {η(n) wr N n}, and V (w) = {p i w P i }. Take M = (D s, ν, V ). Then id + η : M (M ) is an isomorphism.

Characterisation Theorem Characterisation Theorem Let N = {M M = NAX}. Theorem: Let α(x) be an L 1 -formula. TFAE: 1 = N α(x) st x (φ) for some 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. Proof sketch Clear: 1 2 3 4. Also 4 2 (b.m. are functional 2 2 -bis). Proof of 2 1: Similar to Kripke case...

Characterisation Theorem 1 2 Show that MOC N (α) = N α(x). Recall MOC N (α) = {st x (φ) α(x) = N st x (φ)}. Let M N s.t. M = MOC N (α)[x/s]. Then T (x) = {st x (φ) M, s = φ} {α(x)} is N-consistent, i.e. has a model: N = T (x)[x/t]. Now: N = α[x/t] and s t. Construct M + N, take ω-saturated extension U. which is again a NAX-model (which is also modally saturated). Identify s and t in quotient U/.

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. 2 2 -bis: Good computational properties, too strict. Nbhd-bis: Good compromise of logical and computational properties.

CALCO 2007 Paper accepted for: Conference on Algebra and Coalgebra in Computer Science Bergen, Norway, August 20-24, 2007.