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Ivan LaneseComputer Science Department

University of BolognaItaly

Concurrent and located synchronizationsin π-calculus

Roadmap

Process calculi

The standard semantics of π-calculus

A new located semantics

A new concurrent semantics

Conclusions

Roadmap

Process calculi

The standard semantics of π-calculus

A new located semantics

A new concurrent semantics

Conclusions

Process calculi

Behavioral models of concurrent interacting systems Useful to analyze the properties of systems before

building them– System satisfies some properties

– Equivalence of different implementations

– Correctness of optimizations

Systems modeled as terms in a suitable algebra– Constants for basic behaviours

– Operators of composition (parallel composition, declaration of a resource, …)

Operational semantics

Allows to describe system behaviour Two styles: reduction semantics and labelled semantics Reduction semantics

– Describes the evolution of a closed system

– Easy to use and understand

– Rules of the form

Labelled semantics– Describes the interactions between the system and the

environment

– Useful to describe open systems and analyze properties

– Rules of the form P1®¡! P2

P1 ! P2

Bisimilarity ≈

Equivalence relation– Abstracts from internal details

– Equates systems indistinguishable from the outside

– Built on top of a reduction or a labelled semantics

Two processes are bisimilar if one can simulate the other (do the same actions going to bisimilar processes) and vice versa– Using reductions requires also context closure

– More distinguishing than trace equivalence

Bisimilarity is compositional if preserved by contexts– Known as “bisimilarity is a congruence” property

– Allows to substitute bisimilar processes without changing the behaviour (e.g., optimization)

≈/a ab c b c

a

Which calculus?

Different process calculi have been proposed, focused on different aspects (locations, cryptography, wireless communication, …)

We choose the π-calculus Apt to model distributed mobile systems in an easy way Used both in academia and industry

– Basis for BPEL

Easy to extend to deal with different features (spi, dPi, stochastic pi, …)

Roadmap

Process calculi

The standard semantics of π-calculus

A new located semantics

A new concurrent semantics

Conclusions

π-calculus syntax

Names a,b,x,… represent communication channels Channel names are the only data Enough to encode booleans, integers, …P ::= ab:P1 Output pre x

a(x):P1 Input pre xP1jP2 Parallel compositionP1+P2 Nondeterministic choiceºa P1 New namedeclaration!P1 Replication0 Null process

π-calculus reductions

react-S (a(x):P +M )j(ab:Q +N ) ! S P fb=xgjQ

par-SP ! S P 0

P jQ ! S P 0jQres-S

P ! S P 0

ºa P ! S ºa P 0

congr-SP1 ´ P2 ! S P 02 ´ P

01

P1 ! S P 01

≡ is an equivalence relation stating basic properties (e.g., parallel composition is associative and commutative)

Example

ºn an:P [n] a(x):bx:Q

b(x):R[x]

a

b

Example

b(x):R[x]

b

bn:QP [n]n

Example

P [n]

n

Q

R[n]

π-calculus labelled semantics

out-s ab:P ab¡! s P

sum-s¤P ®¡! s P 0

P +Q ®¡! s P 0

com-s¤P ab¡! s P 0 Q ab¡! s Q0

P jQ ¿¡! s P 0jQ0

res-sP ®¡! s P 0 a =2 n(®)

ºaP ®¡! s ºaP 0

rep-sP j!P ®¡! s P 0

!P ®¡! s P 0

inp-s a(x):P ab¡! s P fb=xg

par-s¤P ®¡! s P 0 bn(®) \ fn(Q) = ;

P jQ ®¡! s P 0jQ

close-s¤P ab¡! s P 0 Q

a(b)¡ ¡! s Q0 b=2 fn(P )

P jQ ¿¡! s ºb(P 0jQ0)

open-sP ab¡! s P 0 a 6= b

ºbPa(b)¡ ¡ ! s P 0

Observations on the standard semantics

The semantics shows inputs, outputs, and synchronizations (τ)

All the synchronizations are equal– On a restricted or on a free channel

– All free channels are equal

At each step exactly one action is performed– Concurrency indistinguishable from interleaving

We want to change these assumptions

The ideas of our approach

Many works considered locations and concurrency in π-calculus

Locations explicitly added: l::P Concurrency using mappings to other formalisms (Petri

nets, graphs, …) or complex algebraic structures (event structures, …)

We want to analyze what can be done without changing the framework– Standard π-calculus syntax– Direct semantics using standard labelled and reduction style– Trying to preserve the good properties of the standard

framework

Roadmap

Process calculi

The standard semantics of π-calculus

A new located semantics

A new concurrent semantics

Conclusions

Located synchronizations

We want to see where a synchronization is performed Different channels can have different properties

– Accounting

– Security policies

– …

π-calculus located reductions

We need labels also for reductions Labels show which (free) channel is used

react-L (a(x):P +M )j(ab:Q +N )f ag¡¡! L P fb=xgjQ

par-LP S¡! L P 0

P jQ S¡! L P 0jQres-L

P S¡! L P 0

ºa PSnf ag¡¡ ¡ ¡! L ºa P 0

congr-LP1 ´ P2

S¡! L P 02 ´ P01

P1S¡! L P 01

Properties of the located semantics

Can be expressed also using the labelled semantics– Label aτ for a synchronization on free channel a

– Label τ for a synchronization on a hidden channel

Full correspondance between reduction and labelled semantics– Reductions correspond to labelled synchronizations

– They induce the same bisimilarity» Closure under substitutions needed for labelled semantics

Located bisimilarity

Located bisimilarity refines standard one

aja¼/ L ºb(a+b)j(a+b)

aja¼S ºb(a+b)j(a+b) but

This allows to observe which channels are used For instance we can distinguish between communication

on a local network (free, safe) from communication via Internet (under accounting, unsafe)

Roadmap

Process calculi

The standard semantics of π-calculus

A new located semantics

A new concurrent semantics

Conclusions

Concurrent synchronizations

We want to see which actions can be performed concurrently

Actions can be executed concurrently provided that:– they are performed by different sequential processes

– they are executed on different channels

One concurrent transition corresponds to one or more located transitions

π-calculus concurrent reductions

Labels contain the set of used channels

react-C (a(x):P +M )j(ab:Q +N )f ag¡¡! C P fb=xgjQ

par-CP S¡! C P 0

P jQ S¡! C P 0jQres-C

P S¡! C P 0

ºa PSnf ag¡¡ ¡ ¡! C ºa P 0

comp-CP S1¡ ! C P 0 Q S2¡ ! C Q S1 \ S2 = ;

P jQ S1[ S2¡¡ ¡ ¡! C P jQ

congr-CP1 ´ P2

S¡! C P 02 ´ P01

P1S¡! C P 01

Properties of the concurrent semantics

Can be expressed also using the labelled semantics– Labels are (essentially) multisets of located labels

Full correspondance between reduction and labelled semantics– Reductions correspond to labelled synchronizations

– They induce the same bisimilarity

Concurrent bisimilarity

Concurrent bisimilarity refines the located one

but

This allows to distinguish concurrency from nondeterminism

Actions on the same channel are sequentialized

ajb¼L a:b+b:a

ajb¼/ C a:b+b:a

aja¼C a:a

Compositionality

Concurrent bisimilarity is a congruence– This allows compositional reasoning about system behaviour

– Bisimilar subcomponents can be substituted one for the other

This property does not hold for standard or located semantics– Standard and located bisimilarity not preserved by contexts

that perform substitutions

ajb¼L a:b+b:a but

bjb¼/ L b:b+b:b

Roadmap

Process calculi

The standard semantics of π-calculus

A new located semantics

A new concurrent semantics

Conclusions

Conclusions

Two new semantics for π-calculus highlighting– where synchronizations are performed

– which synchronizations can be performed concurrently

The semantics capture these behaviours– More expressive power

Many good properties of standard semantics are preserved by the extensions– Correspondance between reduction and labelled semantics

– Bisimilarities refine standard one

Additional property: concurrent bisimilarity is compositional

Future work

Further analysis on the new semantics– Weak semantics (first results in the paper)

– Analysis techniques

– Applications

Looking for semantics in the same style for other calculi Analysing the effects of more concurrency

– What happens if many actions are allowed on the same channel?

End of talk