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### 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]

this does not work…

Mobile Ambient [Hausmann, Mossakowski, Schröder – TCS 2006]

Formalisms with asynchronous message passing

Petri Nets

…

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

Petri Nets

P is a set of places

T is a set of transitions

Pre:TP

Post:TP

l:TL 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}…}

Open Petri Nets

Petri net + interface

Interface=(Input Places, Output Places)

Output Place

Input Places

Input Places

$

b

interface

a

a

b

Closed

Place

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

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

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’

l

l

l

l

THM: it is always the largest bisimulation congruence

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

I have 5$ and

I need 6 b

Running ExamplesI 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

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$.

Plan of the Talk

- Compositionality
- Saturated Semantics
- Symbolic Semantics
- Saturated Coalgebras
- Normalized Coalgebras

As running example, we will use Petri nets

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

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

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

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)

SetCAlgГ(C)

for our nets

One object: {$}

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

Saturated Transition System as a coalgebra

C[-]

Ordinary LTS having as labels ||C|| and Λ

F:SetSetF(X)=(||C||ΛX)

We lift F to F: AlgГ(C) AlgГ(C)

(saturated transition system as a bialgebra)

p

q

l

Adding the Inference Relation

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

The set of deduction rules induces an ordering

on||C||ΛX

$$$

a

n

$

b$$$

a

X

b$

n

a

b

a

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 Coalgebras

THM: Saturated Coalgebras are not bialgebras

THM:

CoalgS

is a covariety of

CoalgF

CoalgF

1F

1S

CoalgS

Redundant Transitions

Saturated Set

…

…

…

…

…

…

X

partial order

||C||ΛX,

Given a set A in(||C||ΛX),

a transition is redundant

if it is not minimal

Normalized Set

Saturated Set

Normalization

Saturation

…

…

…

…

…

…

X

Normalized Set

partial order

||C||ΛX,

A set in(||C||ΛX) is normalized

if it contains only NOT redundant

transitions

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

…

…

…

…

…

…

X

y

…

…

…

…

This is redundant

||C||ΛX,

||C||ΛY,

Isomorphism Theorem

CoalgF

Proof: Saturation and Normalization are two natural isomorphisms between

S and N

CoalgS

Saturation

Normalization

CoalgN

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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