Coalgebraic Symbolic Semantics Filippo Bonchi Ugo Montanari

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]

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

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

Petri Nets p p   q q B B   c c   d d   P is a set of places T is a set of transitions Pre:T  P  Post:T  P  l:T   is a labelling 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}…} 2 a marking is a multiset over P The semantics is quite intuitive pc qc B   

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

Petri Nets Contexts Petri nets + Inner interfaces + Outer Interface a a $ $     c c     c c c c c c c c c c   Inner Interface Outer Interface a a b b $ $     a a b b $ $     a a b b $ $    

  x x 3 3 $ $ Bisimilarity is not a congruence c c d d $ $   5 5 ce   x x 3 3 $ $ cxex C$ $$  e$ $$  f They are bisimilar They are not   x x 3 3 $ $ e e f f $ $   3 3      

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

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’ THM: it is always the largest bisimulation congruence

Saturated Transition System pq C[-] C[ p ] q C[-] is a context is a label

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 b     $ $ a a  $$$ $$ $ +$ a aa +a -$ b  b$ +$ b$ $ +$  a$ a$ $ a$ $$ +$ +a $ $ a a    

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

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

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

Symbolic Transition System pq C[-] C[ p ] q C[-] is a context is a label intuitively C[-] is “the smallest context” that allows such transition

Symbolic Transition System a a b b $ $     c c d d $ $     5 5 e e f f $ $     3 3 g g   i i   h h   ab  $   cd  5$   e f g h i 3$        $         

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

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

Inference relation ab   b$ $$ $$$  b$ n $ n  m n m$m$n$n$ a a b b $ $    

Bisimilarity over the Symbolic TS is too strict l l q q     m m $ $     3 3 n n   p p   o o   lmno p 3$     $    q  $    a a b b $ $     ab  $  

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

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

Saturated Transition System as a coalgebra Ordinary LTS having as labels ||C|| and Λ F:Set  Set F(X)=  (||C||  Λ  X) We lift F to F: Alg Г(C)  Alg Г(C) (saturated transition system as a bialgebra) pq C[-]

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 X ab   b$ $$ $$$  b$ n $ n 

Saturated Coalgebras A set in  (||C||  Λ  X) is saturated in X if it is closed wrt S: Alg Г(C)  Alg Г(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

Saturated Coalgebras Coalg F Coalg S THM: Coalg S is a covariety of Coalg F THM: Saturated Coalgebras are not bialgebras 1F1F 1S1S

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

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

Normalized Coalgebras N: Alg Г(C)  Alg Г(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 ……… … …… ………… ||C||  Λ  X, X ||C||  Λ  Y, y This is redundant

Running Example lmno p 3$     $    q  $    ab  $   b$ $$ b$ $ b$       3$  l l q q     m m $ $     3 3 n n   p p   o o   a a b b $ $    

Isomorphism Theorem Proof: Saturation and Normalization are two natural isomorphisms between S and N Coalg F Coalg S Coalg N Saturation Normalization

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

Questions ?