A graphical Fusion Calculus

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A graphical Fusion Calculus
Ivan Lanese Dipartimento di Informatica Università
di Pisa
Joint work with
Ugo Montanari
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Roadmap

• Aims
• Fusion Calculus
• Synchronized Hyperedge Replacement
• Mapping Fusion Calculus into SHR
• Beyond SHR logic programming
• Conclusion

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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)

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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
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Fusion Calculus

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

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Syntax for Fusion Calculus

• Agents
• S?i ?i.Pi
• P0 S P1P2 (x)P rec X. P X

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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

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SOS semantics
PREF
SUM
PAR
COM
SCOPE
PASS
OPEN
STRUCT
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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
• ...

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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

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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
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Edge Replacement Systems

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

Many concurrent rewritings are allowed
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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, ...)

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Synchronized Hyperedge Replacement

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

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SHR with mobility
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Example
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Algebraic notation for graphs

• Example ring

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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)
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Rewritings as syntactic judgements

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

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Fusion Calculus vs SHR

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

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Translation
Using structural congruence we can avoid
recursion at top-level
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Productions

• One for each possible action of a standard
  process

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Correspondence theorem

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

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

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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
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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

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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

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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

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Example
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Recursion example
Production
Corresponding rewriting