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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
• 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
• 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