1 CoMeta, final workshop, 15-16-17/12/2003 Ivan Lanese Dipartimento di Informatica Università di...

1CoMeta, final workshop, 15-16-17/12/2003

Ivan LaneseDipartimento di Informatica

Università di Pisa

Ugo Montanari

A graphical Fusion Calculus

Joint work with:


Roadmap

Aims

Fusion Calculus

Synchronized Hyperedge Replacement

Mapping Fusion Calculus into SHR

Beyond SHR: logic programming

Conclusion


Aims of the work

Give a more intuitive presentation for Fusion Calculus

Applying process calculi to distributed systems is not intuitive because of:

– Interleaving semantics– The same operators describes topology and

allowed behaviours Comparing two apparently quite different

formalisms: Fusion Calculus and Synchronized Hyperedge Replacement

– Extending a similar work on π-calculus (Hirsch)


Fusion Calculus vs SHR: an overview

Fusion Calculus SHR

Process calculus Graph transformation

Algebraic model Graphical representation (but also algebraic notation...)

Interleaving Concurrent

Milner synchronization Different allowed synchronization models


Fusion Calculus

It is an evolution of -calculus Simpler and more symmetric but also more

expressive Introduce fusion of names


Syntax for Fusion Calculus

Agents:

S::=i i.Pi

P::=0 | S | P1|P2 | (x)P | rec X. P | X

nameson relation eequivalenc:

||::

:actions Free

xuxu


Structural congruence

Process: agent up-to the following laws:– | and + are associative, commutative and with 0 as

unit -conversion– (x)0 = 0, (x)(y)P=(y)(x) P– P|(x)Q=(x)(P|Q) if x not free in P– rec X.P=P[rec X.P/X]


SOS semantics

''',,'

')(,,\,'

')()()n(,'

/')(,,'

'|'|

,','

|'|

'

'

'

.

)(

)(

\

QQQPQPPP

PPzzzuyxzPP

PzPzzPP

zxPPzxzxzPP

QPQP

yxQQPP

QPQP

PP

PQP

PP

PP

xuyz

xuy

z

yx

yuxu

PREF

PAR

COM

SUM

PASS

SCOPE

STRUCT

OPEN


Graph transformation

Graphs naturally represent the topology of the system

Synchronized Hyperedge Replacement for modeling computation, synchronization, reconfiguration

Powerful metamodel:

– Different process calculi: Ambient, π, ...

– Software architecture

– ...


SHR: a 2 step approach

Productions to describe the behaviour of single hyperedges:

– Local effects (easyer to implement)– Hyperedges rewritten into generic graphs– Constraints on surrounding nodes

Global constraint-solving algorithm:– Allows to define complex transformations


Edge Replacement Systems

A production describes how the hyperedge L is rewritten into the graph R

R

1

2 3 4

L

1

2 3 4 H


Edge Replacement Systems

A production describes how the hyperedge L is transformed into the graph R

R

R’

1

2 3 4

1

2

3

Many concurrent rewritings are allowed

L

L’

1

2 3 4

1

2

3

H



Synchronized rewritings: we associate actions

to surrounding nodes. A rewriting is allowed iff the

synchronization constraints associated to nodes

are satisfied

Many synchronization models are possible

(Hoare, Milner, ...)



Milner synchronization: pair of edges can synchronize by doing complementary actions

aa a

3 3

B1 A1

B2 A2


SHR with mobility

a<x> a<y>

(x) (y)

B1 A1

a<x> ~ a<y>

B2 A2

a<x> a<y>

x= y

– Actions carry tuple of references to nodes (new or already

existant)

– References associated to synchronized actions are matched and

corresponding nodes are merged

We introduce name mobility


Example

b)

x CBrother

C

C

C

C

C

C

CC CBrother Brother

(4)(3)(2)(1)

Star Rec.S

S

SS

(5)

x

Initial Graph

C

Brother:

C

C

C

C S

Star Reconfiguration:

(w)

r(w)

r(w)


x,y z, w. C(x,w) | C(w,y) | C (y,z) | C(z,x)

Algebraic notation for graphs

Example: ring

w z


: (A x N* ) (x, a , y) if (x) = (a , y)

Associate to each external node its action and its tuple of names: is an idempotent substitution (forces some merges on nodes)

Rewritings as syntactic judgements

Rewriting:

G1 G2

,


Rewritings as syntactic judgements

x1,…,xn L(x1,…,xn) G

Productions

,

Rewritings generated from productions by applying a suitable set of inference rules (determined by the synchronization model)

Derivations

0 G0 1 G1 … n Gn

1,1 2,2 n,n


Fusion Calculus vs SHR

Fusion SHR

Processes Graphs

Sequential processes Hyperedges

Names Nodes

Parallel comp. Parallel comp.

Scope Restriction

Transitions Rewritings


Translation

PxPx

PPPP

SLSS

)(

||

))(fnarray(

nil0

2121

ˆ

with agreeing on substituti

xoutuxu

xinuxu

n

n

names) canonical(with for form standard : ˆ SS

Using structural congruence we can avoid recursion at top-level


Productions

One for each possible action of a standard process

'ˆ PP '|ˆ| PP

'ˆ PP )'(|)(ˆ| PP


Correspondence theorem

We use special rules to force an interleaving behaviour in SHR

Bijective correspondence between transitions and rewritings

'PP '|,| PP E

'PP )'(|)(| PP


Example

).|.|.)(( RzyQyxPuxxy

)||)((

).|.|)(().|.|.)(( )(

RQPy

RzyQyxPyRzyQyxPuxxyzx

uxx

With normal SHR we can execute both the steps at the same time


Beyond SHR: logic programming

Logic programming is a well developed programming paradigm– Useful for implementation purposes

Which logic programming?– Not only refutations but general partial computation

– Limit logic programming to have a correspondence between it and SHR systems: Synchronized Logic Programming» Can be meta-interpreted into standard logic programming


The correspondence

Correspondence between SHR with Hoare synchronization and Synchronized Logic Programming– Do you remember last year presentation?

SHR SLP

Graphs Goals

Nodes Variables

Productions Clauses

Transitions Big-steps

Synchronization Unification


Synchronized Logic Programming

Graphs are goals without functional symbols and constants

Big-steps: sequence of SLD steps between “graphs”

Functional symbols for modeling actions (unification for synchronization)

Logic programming has no restriction operator: we can introduce it


From fusion to logic programming

One further step is needed:– Implementing Milner synchronization using Hoare

synchronization– Not so easy in a mobile environment– Auxiliary structures for implementing Milner nodes


Conclusion

Fusion Calculus a “subcalculus” of interleaving Milner SHR

– Essentially two kinds of actions– One action at the time

Graphical representation for Fusion Calculus– Separation between topology of the system and behaviour

Cross-fertilization between the two models:– Concurrent semantics for Fusion Calculus– Extension of hyperequivalence to general SHR rewritings– Changing the synchronization model of Fusion Calculus


Example

PPuxy uxy.

QPQzwuPuxy wyzx |.|. ,

xzQPQzwuPuxyx wy /)|().|.)((


Recursion example

).)(.().)(.( )( XuxxXrecXuxxXrec uyy

),,(|,

),,(|

).)(.(.

),,().)(.(.

321

1

321

uzuLzyu

uyuLyu

XxxxXrecxx

yinuXxxxXrecxx

),,()(|,,

),,(|,,

33).)(.(.321

),,(321).)(.(.321

321

211

321

xxxLxxxx

xxxLxxx

XxxxXrecxx

xinxXxxxXrecxx

Production:

Corresponding rewriting:

1 CoMeta, final workshop, 15-16-17/12/2003 Ivan Lanese Dipartimento di Informatica Università di...

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