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A graphical Fusion Calculus

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Title: A graphical Fusion Calculus Subject: Presentation at Udine, CoMeta final workshop Author: Ivan Lanese Last modified by: Ivan Created Date: 2/22/1999 11:07:13 AM – PowerPoint PPT presentation

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Title: A graphical Fusion Calculus


1
A graphical Fusion Calculus
Ivan Lanese Dipartimento di Informatica Università
di Pisa
Joint work with
Ugo Montanari
2
Roadmap
  • Aims
  • Fusion Calculus
  • Synchronized Hyperedge Replacement
  • Mapping Fusion Calculus into SHR
  • Beyond SHR logic programming
  • Conclusion

3
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 p-calculus (Hirsch)

4
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
5
Fusion Calculus
  • It is an evolution of ?-calculus
  • Simpler and more symmetric but also more
    expressive
  • Introduce fusion of names

6
Syntax for Fusion Calculus
  • Agents
  • S?i ?i.Pi
  • P0 S P1P2 (x)P rec X. P X

7
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)(PQ) if x not free in P
  • rec X.PPrec X.P/X

8
SOS semantics
PREF
SUM
PAR
COM
SCOPE
PASS
OPEN
STRUCT
9
Graph transformation
  • Graphs naturally represent the topology of the
    system
  • Synchronized Hyperedge Replacement for modeling
    computation, synchronization, reconfiguration
  • Powerful metamodel
  • Different process calculi Ambient, p, ...
  • Software architecture
  • ...

10
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

11
Edge Replacement Systems
  • A production describes how the hyperedge L is
    rewritten into the graph R

L
R
H
3
3
4
4
2
2
1
1
12
Edge Replacement Systems
  • A production describes how the hyperedge L is
    transformed into the graph R

Many concurrent rewritings are allowed
13
Synchronized Hyperedge Replacement
  • 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, ...)

14
Synchronized Hyperedge Replacement
  • Milner synchronization pair of edges can
    synchronize by doing complementary actions

15
SHR with mobility
16
Example
17
Algebraic notation for graphs
  • Example ring

18
Rewritings as syntactic judgements
  • Rewriting

?,?
? G1 ?? ? G2
?
?
? ? ? (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)
19
Rewritings as syntactic judgements
  • Rewritings
  • generated from productions by applying a
    suitable set of inference rules (determined by
    the synchronization model)

20
Fusion Calculus vs SHR
  • Fusion SHR
  • Processes Graphs
  • Sequential processes Hyperedges
  • Names Nodes
  • Parallel comp. Parallel comp.
  • Scope Restriction
  • Transitions Rewritings

21
Translation
Using structural congruence we can avoid
recursion at top-level
22
Productions
  • One for each possible action of a standard
    process

23
Correspondence theorem
  • We use special rules to force an interleaving
    behaviour in SHR
  • Bijective correspondence between transitions and
    rewritings

24
Example
With normal SHR we can execute both the steps at
the same time
25
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

26
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
27
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

28
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

29
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

30
Example
31
Recursion example
Production
Corresponding rewriting
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