Coalgebraic Symbolic Semantics

1. Coalgebraic Symbolic Semantics Filippo Bonchi Ugo Montanari

2. Many formalisms modelling Interactive Systems Algebras - Syntax Coalgebras - Semantics Bialgebras – Semantics of the composite system in terms of the semantics of the components (compositionality of final semantics) CCS [Turi, Plotkin – LICS 97] Pi-calculus [Fiore, Turi – LICS 01] [Ferrari, Montanari, Tuosto – TCS 05] Fusion Calculus [Ferrari et al. – CALCO 05][Miculan – MFPS 08]

3. … in many interesting cases, this does not work… Mobile Ambient [Hausmann, Mossakowski, Schröder – TCS 2006] Formalisms with asynchronous message passing Petri Nets …

4. Plan of the Talk • Compositionality • Saturated Semantics • Symbolic Semantics • Saturated Coalgebras • Normalized Coalgebras Bonchi, Montanari – FOSSACS 08 As running example, we will use Petri nets

5. Petri Nets P is a set of places T is a set of transitions Pre:TP Post:TP  l:TL is a labelling p c 2 p p B c q d a marking is a multiset over P i i p The semantics is quite intuitive pc qcB Given a set A, A is the set of all multisets over A, e.g., for A={a,b} ,then A={,{a},{b},{aa},{bb},{ab} ,{aab}…}

6. Open Petri Nets Petri net + interface Interface=(Input Places, Output Places) Output Place Input Places Input Places $ b interface a a b Closed Place

7. Petri Nets Contexts Petri nets + Inner interfaces + Outer Interface Outer Interface a c a c c $ $ $ $ c c b b b Inner Interface c a c a a a a a a a a b b b

8. 3 3 3 $ $ $ Bisimilarity is not a congruence $ $ 3 5 c e a a d f c c x x x c c c a They are bisimilar c e c They are not c a cx C$$$ ex e$$$ f

9. Plan of the Talk • Compositionality • Saturated Semantics • Symbolic Semantics • Saturated Coalgebras • Normalized Coalgebras As running example, we will use Petri nets

10. Saturated Bisimilarity A relation R is a saturated bisimulation iff whenever pRq, then C[-] • If C[p]→p’ then q’ s.t. C[q]→q’ and p’Rq’ • If C[q]→q’ then p’ s.t. C[p]→p’ and p’Rq’ l l l l THM: it is always the largest bisimulation congruence

11. Saturated Transition System C[-] C[-] is a context lis a label p q l C[p] q l

12. Saturated Semantics for Open Nets At any moment of their execution a token can be inserted into an input place and one can be removed from an output place -$ -$ -$ $ +$ +$ +$ +$ +$ +$ +$ +$  $ $$ $$$ $ b +a +a +a +a b a a b a a$ a$$ a$$$ a +$ +$ a a +a b b$ b$$ b aa

13. $ I have 5$ and I need 6 b Running Examples I have 1$ and I need 1 b 5 b $ The activation a is free. The service b costs 1$. c a d b $ The activation a costs 5$. The service b is free. a a b b i b h 3 The activation a costs 3$. The service b is free for 3 times and then it costs 1$. e a f b g b THEY ARE ALL DIFFERENT

14. Running Examples $ $ $ $ IS IT DIFFERENT FROM ALL THE PREVIOUS??? b $ b b $ a b a b p b o a a a a b b 3 b i b h 3 m l a b n b e a f b g b a q b This behaves as a or e: either the activation a is free and the service b costs 1$. Or the activation costs 3$ and then for 3 times the service is free and then it costs 1$. The activation a is free. The service b costs 1$.

15. Plan of the Talk • Compositionality • Saturated Semantics • Symbolic Semantics • Saturated Coalgebras • Normalized Coalgebras As running example, we will use Petri nets

16. Symbolic Transition System C[-] C[-] is a context lis a label p q l C[p] q l intuitivelyC[-] is “the smallest context” that allows such transition

17. $ Symbolic Transition System e 5 b b $   a a b 3$ $ b c a d b f b $  5$ b b  c d  a b a a b b i b h 3 g e a f b g b a a h b $ a i b

18. Symbolic Semantics a symbolic LTS + a set of deduction rules l’  p,q l l D[p] E[q] p q m n l m$ n$ In our running example

19. Inference relation Given a symbolic transition system and a set of deduction rules, we can infer other transitions C’[-] C[-] l’ l p q’ p q

20. Inference relation  $ a b b l m n $$$ a a a b b$$$ l m$ n$ n $ a a b$ n

21. Bisimilarity over the Symbolic TS is too strict b $ $   $ b b $ b b p b o a a b 3 m l a b n b   3$ l m n o b b a q b a a a  a b $ q p b

22. Plan of the Talk • Compositionality • Saturated Semantics • Symbolic Semantics • Saturated Coalgebras • Normalized Coalgebras As running example, we will use Petri nets

23. Category of interfaces and contexts • Objects are interfaces • Arrows are contexts Functors from C to Set are algebras for Г(C) SetCAlgГ(C) for our nets One object: {$} Arrows: -$n: {$}{$}

24. Saturated Transition System as a coalgebra C[-] Ordinary LTS having as labels ||C|| and Λ F:SetSetF(X)=(||C||ΛX) We lift F to F: AlgГ(C) AlgГ(C) (saturated transition system as a bialgebra) p q l

25. Adding the Inference Relation  An F-Coalgebra is a pair (X, :XF(X)) The set of deduction rules induces an ordering on||C||ΛX $$$ a n $ b$$$ a X b$ n a b a

26. Saturated Coalgebras • A set in(||C||ΛX) is saturated in X if it is closed wrt S: AlgГ(C) AlgГ(C) the carrier set of S(X) is the set of all saturated sets of transitions • E.g: the saturated transition system is always an S-coalgebra X

27. Saturated Coalgebras THM: Saturated Coalgebras are not bialgebras THM: CoalgS is a covariety of CoalgF CoalgF 1F 1S CoalgS

28. Redundant Transitions Saturated Set … … … … … … X partial order ||C||ΛX, Given a set A in(||C||ΛX), a transition is redundant if it is not minimal

29. Normalized Set Saturated Set Normalization Saturation … … … … … … X Normalized Set partial order ||C||ΛX, A set in(||C||ΛX) is normalized if it contains only NOT redundant transitions

30. Normalized Coalgebras N: AlgГ(C) AlgГ(C) the carrier set of N(X) is the set of all normalized sets of transitions For h:XY, the definition of N(h) is peculiar … … … … … … X y … … … … This is redundant ||C||ΛX, ||C||ΛY,

31. Running Example b $ $   $ b b   $ l m n o a b b b b a b p b o  a a b 3 q p m l a b n b 3$ a q b a a 3$ a    $ b$ b$$ b$$$ b b b b

32. Isomorphism Theorem CoalgF Proof: Saturation and Normalization are two natural isomorphisms between S and N CoalgS Saturation Normalization CoalgN

33. Conclusions • Bisimilarity of Normalized Colagebras coincides with Saturated Bisimilarity • Minimal Symbolic Automata • Symbolic Minimization Algorithm [Bonchi, Montanari - ESOP 09] • Coalgebraic Semantics for several formalisms (asynchronous PC, Ambients, Open nets …) • Normalized Coalgebras are not Bialgebras

34. Questions ?