Coalgebraic Symbolic Semantics

Filippo Bonchi Ugo Montanari

Many formalisms modelling Interactive Systems

Algebras - SyntaxCoalgebras - 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, Schröder – TCS 2006]

Formalisms with asynchronous message passingPetri 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

q

B

c

d

P is a set of placesT is a set of transitionsPre:TP

Post:TP

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 qcB

Open Petri NetsPetri net + interface

a b

$

interface

Input PlacesInput Places

Output Place

ClosedPlace

Interface=(Input Places, Output Places)

Petri Nets Contexts

Petri nets + Inner interfaces + Outer Interface

a

$

c

c

c c

c c

InnerInterface

OuterInterface

a b

$

a b

$

a b

$

x3

$

Bisimilarity is not a congruence

c d

$

5

c e

x3

$

cx exC$$$

e$$$

f

They are bisimilar

They are not

x3

$

e f

$

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 bisimulationiff 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

p qC[-]

C[p] q

C[-] is a context is a label

Saturated Semantics for Open NetsAt 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

aa

+a

+a

-$ -$ -$

b

b$ +$ b$$

+$

a$ a$$

a$$$

+$ +$ +$ +$

+a +a +a$

a

Running Examples

a b

$

e f

$

3

g

i

h

c d

$

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

q

m

$

3

n

p

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 b

$

The activation is free.The service costs 1$.

$

$

a b

$

e f

$

3

g

i

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

p qC[-]

C[p] q

C[-] is a context is a label

intuitively C[-] is “the smallest context” that allows such transition

Symbolic Transition System

a b

$

c d

$

5

e f

$

3

g

i

h

a b $

c d5$

e

f

g

h

i

3$

$

Symbolic Semanticsa symbolic LTS + a set of deduction rules

In our running example

m nm$ n$

p qD[p] ’ E[q]

p,q

Inference relation

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

p qC[-] p ’ q’C’[-]

Inference relation

a b

b$$$

$$$

b$n

$n

m n

m$ n$

a b

$

Bisimilarity over the Symbolic TS is too strict

l

q

m

$

3

n

p

o

l m n o

p

3$

$

q $

a b

$

a b $

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)SetC AlgГ(C)

One object: {$}

Arrows: -$n: {$}{$}

for our nets

Saturated Transition System as a coalgebra

Ordinary LTS having as labels ||C|| and ΛF:SetSet F(X)=(||C||ΛX)

We lift F to F: AlgГ(C) AlgГ(C)

(saturated transition system as a bialgebra)

p qC[-]

Adding the Inference Relation

An F-Coalgebra is a pair (X, :XF(X))

The set of deduction rules induces an ordering on||C||ΛX

X

a b

b$$$

$$$

b$n

$n

Saturated Coalgebras

• A set in(||C||ΛX) 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

Saturated CoalgebrasCoalgF

CoalgS

THM: CoalgS

is a covariety of CoalgF

THM: Saturated Coalgebras are not bialgebras

1F

1S

Redundant Transitions

… … … … … …

partial order ||C||ΛX,

X

Saturated Set

Given a set A in(||C||ΛX), a transition is redundant

if it is not minimal

Normalized Set

… … … … … …

partial order ||C||ΛX,

X

Saturated Set

A set in(||C||ΛX) is normalizedif it contains only NOT redundant

transitions

Normalized Set

SaturationNormalization

Normalized Coalgebras

N: AlgГ(C) AlgГ(C)

the carrier set of N(X) is the set of all normalized sets of transitions For h:XY, the definition of N(h) is peculiar

… … … … … … … …… …

||C||ΛX, X

||C||ΛY, y

This is redundant

Running Example

l m n o

p

3$

$

q $

a b $ b$

$$b$$b$

3$

lq m

$

3

n

p

o

a b

$

Isomorphism Theorem

Proof: Saturation

and Normalization

are two natural isomorphisms

between S and N

CoalgF

CoalgS

CoalgN

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?