Coalgebraic Symbolic Semantics

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Coalgebraic Symbolic Semantics
Filippo Bonchi Ugo Montanari
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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
  05Miculan MFPS 08

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• in many interesting cases,
• this does not work
• Mobile Ambient Hausmann, Mossakowski, Schröder
  TCS 2006
• Formalisms with asynchronous message passing
• Petri Nets

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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
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Petri Nets
P is a set of places T is a set of
transitions PreT?P? PostT?P ? lT?L is a
labelling
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p
a marking is a multiset over P
p
The semantics is quite intuitive
pc
qcB
Given a set A, A? is the set of all multisets
over A, e.g., for Aa,b ,then
A??,a,b,aa,bb,ab ,aab
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Open Petri Nets

• Petri net interface

Interface(Input Places, Output Places)
Output Place
Input Places
Input Places
interface
Closed Place
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Petri Nets Contexts

• Petri nets Inner interfaces Outer Interface

Outer Interface
Inner Interface
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Bisimilarity is not a congruence
c
c
a
They are bisimilar
c
e
c
c
a
They are not
cx
ex
C
e
f
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Plan of the Talk

• Compositionality
• Saturated Semantics
• Symbolic Semantics
• Saturated Coalgebras
• Normalized Coalgebras

As running example, we will use Petri nets
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Saturated Bisimilarity

• A relation R is a saturated bisimulation
• iff whenever pRq, then ?C-
• If Cp?p then ? q s.t. Cq?q and pRq
• If Cq?q then ? p s.t. Cp?p and pRq

l
l
l
l
THM it is always the largest bisimulation
congruence
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Saturated Transition System
C- is a context
l is a label
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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
?



a
a
a
a
b
b
b
aa
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Running Examples
The activation a is free. The service b costs 1.
The activation a costs 5. The service b is free.
The activation a costs 3. The service b is free
for 3 times and then it costs 1.
THEY ARE ALL DIFFERENT
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Running Examples
IS IT DIFFERENT FROM ALL THE PREVIOUS???
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.
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Plan of the Talk

• Compositionality
• Saturated Semantics
• Symbolic Semantics
• Saturated Coalgebras
• Normalized Coalgebras

As running example, we will use Petri nets
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Symbolic Transition System
C- is a context
l is a label
intuitively C- is the smallest context that
allows such transition
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Symbolic Transition System
a
b
b
c
d
a
b
a
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Symbolic Semantics

• a symbolic LTS a set of deduction rules

In our running example
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Inference relation

• Given a symbolic transition system and a set of
  deduction rules, we can infer other transitions

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Inference relation
a
b
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Bisimilarity over the Symbolic TS is too strict
?
?
l
m
n
o
b
b
a
?
a
b
q
p
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Plan of the Talk

• Compositionality
• Saturated Semantics
• Symbolic Semantics
• Saturated Coalgebras
• Normalized Coalgebras

As running example, we will use Petri nets
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Category of interfaces and contexts

• Objects are interfaces
• Arrows are contexts
• Functors from C to Set are algebras for ?(C)
• SetC ? Alg?(C)

for our nets
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Saturated Transition System as a coalgebra

• Ordinary LTS having as labels C and ?
• FSet?Set F(X)?(C???X)
• We lift F to F Alg?(C)? Alg?(C)
• (saturated transition system as a bialgebra)

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Adding the Inference Relation

• An F-Coalgebra is a pair (X, ?X?F(X))
• The set of deduction rules induces an ordering
• onC???X

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Saturated Coalgebras

• A set in?(C???X) is saturated in X if it is
  closed wrt
• S Alg?(C)? Alg?(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

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Saturated Coalgebras
CoalgF
CoalgS
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Redundant Transitions
Saturated Set
partial order C???X,
Given a set A in?(C???X), a transition is
redundant if it is not minimal
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Normalized Set
Saturated Set
Normalization
Saturation
Normalized Set
partial order C???X,
A set in?(C???X) is normalized if it contains
only NOT redundant transitions
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Normalized Coalgebras

• N Alg?(C)? Alg?(C)
• the carrier set of N(X)
• is the set of all normalized sets of transitions
• For hX?Y, the definition of N(h) is peculiar

This is redundant
C???X,
C???Y,
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Running Example
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Isomorphism Theorem
CoalgF

• Proof Saturation and Normalization are two
  natural isomorphisms between
• S and N

CoalgS
Saturation
Normalization
CoalgN
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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

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Questions ?