Ctegoricl pproches to bisimilrity PPS seminr, IRIF, Pris 7 Jérémy Dubut Ntionl Institute of Informtics Jpnese-French Lbortory for Informtics April 2nd Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 1 / 40
Bisimilrity of Trnsition Systems Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 2 / 40
Trnsition Systems Trnsition system : A TS T = (Q, i, ) on the lphbet Σ is the following dt: set Q (of sttes); n initil stte i Q; set of trnsitions Q Σ Q. Σ = {, b, c}, Q = {0, 1, 2, 3}, b 1 i = 0, 0 b c 3 = {(0,, 0), (0, b, 1), (0,, 2), (1, c, 2), (2, b, 0), (2,, 3)}. 2 Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 3 / 40
Bisimultions of Trnsition Systems Bisimultions [Prk81] : A bisimultion between T 1 = (Q 1, i 1, 1 ) nd T 2 = (Q 2, i 2, 2 ) is reltion R Q 1 Q 2 such tht: (i) (i 1, i 2 ) R; (ii) if (q 1, q 2 ) R nd (q 1,, q 1 ) 1 then there is q 2 Q 2 such tht (q 2,, q 2 ) 2 nd (q 1, q 2 ) R; (iii) if (q 1, q 2 ) R nd (q 2,, q 2 ) 2 then there is q 1 Q 1 such tht (q 1,, q 1 ) 1 nd (q 1, q 2 ) R. b b c c Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 4 / 40
Severl Chrcteristions of Bisimilrity Bisimilrity: Given two TS T nd T, the following re equivlent: [Prk81] There is bisimultion between T nd T. [Stirling96] Defender hs strtegy to never loose in 2-plyer gme on T nd T. [Hennessy80] T nd T stisfy the sme formule of the Hennessy-Milner logic. In this cse, we sy tht T nd T re bisimilr. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 5 / 40
Morphisms of Trnsition Systems Morphism of TS: A morphism of TS f : T 1 = (Q 1, i 1, 1 ) T 2 = (Q 2, i 2, 2 ) is function f : Q 1 Q 2 such tht: preserving the initil stte: f (i 1 ) = i 2, preserving the trnsitions: for every (p,, q) 1, (f (p),, f (q)) 2. TS(Σ) = ctegory of trnsition systems nd morphisms b c b c Morphisms re functionl simultions: Morphisms re precisely functions f between sttes whose grph {(q, f (q)) q Q 1 } is simultion. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 6 / 40
Ctegoricl Chrcteristions Bisimilrity, using morphisms: Two TS T nd T re bisimilr iff there is spn of functionl bisimultions between them. T f g T T Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 7 / 40
Bisimilrity from Colgebr J. Rutten. Universl colgebr: theory of systems. Theoreticl Computer Science 249(1), 3 80 (2000) Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 8 / 40
Trnsition systems, s pointed colgebrs Set of trnsitions, s functions: There is bijection between sets of trnsitions Q Σ Q nd functions of type: δ : Q P(Σ Q) where P(X ) is the powerset {U U X }. Initil sttes, s functions: There is bijection between initil sttes i Q nd functions of type: ι : Q where is singleton. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 9 / 40
Exmple Σ = {, b, c}, Q = {0, 1, 2, 3}, b 1 i = 0, 0 b c 3 = {(0,, 0), (0, b, 1), (0,, 2), (1, c, 2), (2, b, 0), (2,, 3)}. 2 ι : Q 0 δ : Q P(Σ Q) 0 {(, 0), (b, 1), (, 2)} 1 {(c, 2)} 2 {(b, 0)} 3 Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 10 / 40
Pointed colgebrs Pointed colgebrs: Given n endofunctor G : C C nd n object I C, pointed colgebr is the following dt: n object Q C, morphism ι : I Q of C, morphism σ : Q G(Q) of C. G is often decomposed s T F, where: T : brnching type, e.g, non-deterministic, probbilistic, weighted. For TS: T = P. F : trnsition type. For TS: F = Σ _. I is often the finl object, but we will see other exmples. For TS: I =, the finl object. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 11 / 40
Morphisms of TS, using Pointed Colgebrs Morphisms of TS re lx morphisms of pointed colgebrs A morphism of TS, seen s pointed colgebrs T = (Q 1, ι 1, δ 1 ) nd T = (Q 2, ι 2, δ 2 ) is the sme s function stisfying I Q 1 f : Q 1 Q 2 ι 1 σ 1 ι 2 P(Σ Q 1 ) f P(Σ f ) Q 2 σ 2 P(Σ Q 2 ) Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 12 / 40
Lx Morphisms of Pointed Colgebrs Lx Morphisms: Assume there is n order on every Hom-set of the form C(X, G(Y )). A lx morphism from (Q 1, ι 1, δ 1 ) to (Q 2, ι 2, δ 2 ) is morphism of C stisfying I Q 1 f : Q 1 Q 2 ι 1 σ 1 ι 2 G(Q 1 ) f G(f ) Q 2 σ 2 G(Q 2 ) Col lx (G, I ) = ctegory of pointed colgebrs nd lx morphisms. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 13 / 40
Wht bout functionl bisimultions? Functionl bisimultions re homomorphisms of pointed colgebrs For two TS, seen s pointed colgebrs T = (Q 1, ι 1, δ 1 ) nd T = (Q 2, ι 2, δ 2 ), nd for function of the form f : Q 1 Q 2, the following re equivlent: The grph {(q, f (q)) q Q 1 } of f is bisimultion. f is homomorphism of pointed colgebrs, tht is, the following digrm commutes: ι 1 σ 1 I Q 1 ι 2 P(Σ Q 1 ) f P(Σ f ) Q 2 σ 2 P(Σ Q 2 ) Bisimilrity, using homomorphisms of pointed colgebrs For two TS T nd T, the following re equivlent: T nd T re bisimilr. There is spn of homomorphisms of pointed colgebrs between T nd T. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 14 / 40
Homomorphisms of Pointed Colgebrs Morphisms: A homomorphism from (Q 1, ι 1, δ 1 ) to (Q 2, ι 2, δ 2 ) is morphism of C stisfying I Q 1 f : Q 1 Q 2 ι 1 σ 1 ι 2 G(Q 1 ) f G(f ) Q 2 σ 2 G(Q 2 ) Col(G, I ) = ctegory of pointed colgebrs nd homomorphisms. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 15 / 40
Summry [Wißmn, D., Ktsumt, Hsuo FoSSCS 19] Non-deterministic brnching colgebr open mps dt type G : C C, I C on C(X, G(Y )) J : P M systems pointed colgebrs objects of M functionl simultions lx morphisms morphisms of M functionl bisimultions homomorphisms open mps bisimilrity existence of spn of functionl bisimultions [Lsot 02] Smll ctegory of pths Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 16 / 40
Bisimilrity from Open Mps A. Joyl, M. Nielsen, G. Winskel. Bisimultion from Open Mps. Informtion nd Computtion 127, 164 185 (1996) Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 17 / 40
Runs in Trnsition System Run A run in trnsition system (Q, i, ) is sequence written s: with: q j Q nd j Σ q 0 = i for every j, (q j, j+1, q j+1 ) q 1 0 2 q1... n qn q 0 q 0 b q 1 c q 2 q 1 b q 3 q 0 b c q 3 q 2 Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 18 / 40
Runs, Ctegoriclly Finite Liner Systems: A finite liner system is TS of the form 1,..., n = ([n], 0, ) where: [n] is the set {0,..., n}; is of the form {(i, i+1, i + 1) i [n 1]} for some 1,..., n in Σ. Runs re morphisms 1 2 0 1 2... n n 1 n There is bijection between runs of T nd morphisms of TS between finite liner system to T. 0 b c 1 2 3 4 q 0 b b q 1 c q 3 q 2 Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 19 / 40
Functionl Bisimultions, from Lifting Properties of Pths Functionl bisimultions re open mps: For morphism f of TS from T to T, the following re equivlent: The rechble grph of f, tht is, {(q, f (q)) q rechble} is bisimultion. f hs the right lifting property w.r.t. pth extensions, tht for every commuttive squre (in plin): 1,..., n T inj 1,..., n, n+1,..., n+p T there is lifting (in dot), mking the two tringles commute. Bisimilrity, using open mps For two TS T nd T, the following re equivlent: T nd T re bisimilr. There is spn of open mps between T nd T. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 20 / 40 θ ρ ρ f
Open mps Open mp sitution: An open mp sitution is ctegory M (of systems) together with subctegory J : P M (of pths). M = ctegory of systems (Ex : TS(Σ)), P = sub-ctegory of finite liner systems. Open mps: A morphism f : T T of M is sid to be open if for every commuttive squre (in plin): J(p) J(P) T J(Q) T where p : P Q is morphism of P, there is lifting (in dot) mking the two tringles commute. ρ θ ρ f Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 21 / 40
Summry [Wißmn, D., Ktsumt, Hsuo FoSSCS 19] Non-deterministic brnching colgebr open mps dt type G : C C, I C on C(X, G(Y )) J : P M systems pointed colgebrs objects of M functionl simultions lx morphisms morphisms of M functionl bisimultions homomorphisms open mps bisimilrity existence of spn of functionl bisimultions [Lsot 02] Smll ctegory of pths Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 22 / 40
From Open Mps to Colgebr S. Lsot. Colgebr morphisms subsume open mps. Theor. Comput. Sci. 280(1 2): 123 135 (2002) Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 23 / 40
Problem Setting Input: An open mp sitution P M such tht: P is smll, P nd M hs common initil object 0 Problem: Construct colgebr sitution: G : C C, I C, on C(X, G(Y )). such tht functor Beh : M Col lx (G, I ) f is n open mp iff Beh(f ) is homomorphism. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 24 / 40
Solution C = Set ob(p), G((X P ) P ob(p) ) = ( Q P ( P(XQ )) P(P,Q)) P P I 0 =, I P = otherwise, point-wise inclusion, Beh(X ): XP = set of runs lbelled by P, i.e., the set M(P, X ), ι : (IP ) (X P ) mps to unique morphism from 0 to X, σp = (σ P,Q ) Q : X P Q P ( P(XQ )) P(P,Q), where σ P,Q mps run ρ lbelled by P to the set of runs lbelled by Q tht extend ρ. Theorem [Lsot02]: f is n open mp iff Beh(f ) is homomorphism. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 25 / 40
From Colgebr to Open Mps T. Wißmn, J. Dubut, S. Ktsumt, I. Hsuo. Pth Ctegory For Free Open Morphisms From Colgebrs With Non-Deterministic Brnching. FoSSCS 19 Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 26 / 40
Problem Setting Input: A colgebr sitution: G = T F : C C, I C, on C(X, G(Y )). stisfying some xioms. Problem: Construct n open mp sitution J : P Col lx (G, I ) such tht lx homomorphism f : c 1 c 2 : if f is homomorphism then f is open if f is open nd c 2 is rechble, then f is homomorphism Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 27 / 40
THE key notion: F -precise morphisms F -precise morphisms: A morphism s : S FR of C is F -precise if for ll f, g, h: s S FR f Fg FC FD Fh d = s S FR f Fd FC & R d g C D h Intuition: morphism s : S FR is precise iff every element of R is used exctly once in the definition of s. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 28 / 40
Exmples FX = X X + X f Y X f Y not precise precise Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 29 / 40
The Pth Ctegory P = Pth(I, F ) A pth consists in: finite sequence P 0,..., P n of objects of C with P 0 = I, finite sequence of F + 1-precise mps: f k : P k FP k+1 + 1 A morphism between pths, from (P k, p k ) k n to (Q j, q j ) j m consists in sequence of isomorphisms φ k : P k Q k such tht: φ k p k P k FP k+1 + 1 q k F φ k+1 +1 Q k FQ k+1 + 1 Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 30 / 40
Exmples FX = {} X + X X, I =, p k : P k FP k+1 + { } p 1 P p 1 0 P 0 P 2 p 2 P 3 p 3 P 4 P 1 p 1 P 2 p 2 P 3 p 3 P p 4 0 P 0 Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 31 / 40
The Functor J Assumptions on C nd T : 1 C hs finite coproducts, 2 η : Id C T, 3 : 1 T such tht X C(1, T (X )) is the lest element for, 4 some others. Typicl exmple: the powerset functor P Set hs disjoint unions nd empty set, η is the unit η X (x) = {x}, is given by the empty subset X ( ) =,... Theorem: There is functor J : Pth(I, F ) Col lx (TF, I ) given by J(P k, p k ) := I in 0 k n [ ] inl [F in k+1 p k ] k<n,inr! P k F k n P k + 1 [η, ] TF k n P k Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 32 / 40
Wrpping up Theorem: A homomorphism of pointed colgebrs in Col lx (TF, I ) is open. Proposition-Definition: For pointed colgebr c = (Q, ι, σ), the following re equivlent: c hs no proper subcolgebr, the set of ll morphisms of the form J(P k, p k ) c is jointly epic. In this cse, we sy tht c is rechble. Theorem: An open mp h : c c in Col lx (TF, I ) where c is rechble is homomorphism. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 33 / 40
Instnces Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 34 / 40
Lbelled Trnsition Systems C = Set, F (X ) = Σ X, T = P, I =. Pths re given by words. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 35 / 40
Vrious Tree-like Automt C = Set, F nlytic, i.e., F (X ) = X n /G σ, T = P, I =. σ/n Σ Pths re given by prtil trees. P 2 p 2 P p 3 1 p 3 P 1 P p 4 0 P 0 Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 36 / 40
Multi-Sorted Trnsition Systems [Lsot 02] C = Set ob(p), F ((X P ) P P ) = ( P(P, Q) X Q )P P, Q P T ((X P ) P P ) = (P(X P )) P P, I 0 =, I P =. Pths re given by sequences of pth extensions from the initil pth ctegory: 0 m1 m P 2 1 P 2 mn P n Consequence: We cnnot expect more generl trnsltion from colgebr to open mps. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 37 / 40
Regulr Nondeterministic Nominl Automt [Schröder et l. 17] C = Nom, F (X ) = 1 + A X + [A]X, where [A]_ is binding opertor, T = P ufs, the set of uniformly finitely supported, I = A #n, the set of n-tuples of distincts toms. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 38 / 40
Generl Kripke Frmes [Kupke et l. 04] C = Stone, the ctegory of Stone spces F = Id, T = V, the Vietoris topology on the set of compct subsets, I =. Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 39 / 40
Conclusion [Wißmn, D., Ktsumt, Hsuo FoSSCS 19] Non-deterministic brnching colgebr open mps dt type G : C C, I C on C(X, G(Y )) J : P M systems pointed colgebrs objects of M functionl simultions lx morphisms morphisms of M functionl bisimultions homomorphisms open mps bisimilrity existence of spn of functionl bisimultions [Lsot 02] Smll ctegory of pths Jérémy Dubut (NII & JFLI) Ctegoricl pproches to bisimilrity April 2nd 40 / 40