Process Bisimulation via a Graphical Encoding

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Process Bisimulation via a Graphical Encoding
Filippo Bonchi Fabio Gadducci Barbara König
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Leifer and Milners Theory of Reactive System

• Given a reduction systems,
• derive a labeled transition system
• where
• bisimilarity is a congruence

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Borrowed Context Rewriting
Rewriting rules L?I?R DPO Rewriting Reduction
System BC Rewriting Labeled Transition System
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From Process to Graph
Gadducci Montanari propose several encodings
of Process in Graph Correct and Complete w.r.t.
Structural Congruence Processes Reductions are
simulated by Graphs Rewrites
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Our Approach
Processes
Graphs
Reduction Rules
Rewriting Rules
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Our Approach
Processes
Graphs
Reduction Rules
Rewriting Rules
LTS Semantics
BC Rewriting
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Plan of the Talk

• Two operational semantics for CCS
• From Processes to Graphs with Interfaces
• From Process Reductions
• to DPO Graph Rewrites
• Borrowed Contexts Rewritings
• Conclusions

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Plan of the Talk

• Two operational semantics for CCS
• From Processes to Graphs with Interfaces
• From Process Reductions
• to DPO Graph Rewrites
• Borrowed Contexts Rewritings
• Conclusions

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CCS Syntax
A process is a term where each occurence of a
variable x is in the scope of a recx
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Reduction Semantics
The reduction is closed under the Structural
Congruence
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Interactive Semantics
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Two Operational Semantics

• Reduction Semantics
• Reduction System Structural Congruence
• Specify how the whole system behaves
• More natural and easy
• Trivial behavioral equivalence

• Interactive Semantics
• Labeled Transition System
• labels are interactions and observations
• Specify how the system interact with the
  envoirment
• More expressive
• Good behavioral equivalence

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Plan of the Talk

• Two operational semantics for CCS
• From Processes to Graphs with Interfaces
• From Process Reductions
• to DPO Graph Rewrites
• Borrowed Contexts Rewritings
• Conclusions

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Graphs with Interfaces

• J?G?K

A process is a graph Ø?G?K, where K contains all
the free name of the process Graphs K?G?K are
contexts where processes can be inserted
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The encoding
summations
processes
names
(for label op either snd or rcv)
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Sequential Process
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Parallel Process
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Non Deterministic Process
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Restriced Process
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A Context
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Interaction with The Environment
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Interaction with The Environment
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Interaction with The Environment
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Interaction with The Environment
The process cannot interact on a since it is not
in the interface
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Plan of the Talk

• Two operational semantics for CCS
• From Processes to Graphs with Interfaces
• From Process Reductions
• to DPO Graph Rewrites
• Borrowed Contexts Rewritings
• Conclusions

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DPO rewriting on Graph with Interface

• Productions are span

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DPO rewriting on Graph with Interface

• Productions are span

• Find a match m of L inside G

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DPO rewriting on Graph with Interface

• Productions are span

• Find a match m of L inside G
• Construct C such that (1) is a pushout and k
  exists

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DPO rewriting on Graph with Interface

• Productions are span

• Find a match m of L inside G
• Construct C such that (1) is a pushout and k
  exists
• Construct H such that (2) is a pushout

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The synchronization rule
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The internal action rule
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Example
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Example
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Example
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Example
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Plan of the Talk

• Two operational semantics for CCS
• From Processes to Graphs with Interfaces
• From Process Reductions
• to DPO Graph Rewrites
• Borrowed Contexts Rewritings
• Conclusions

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Borrowed Context Rewriting
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Borrowed Context Rewriting
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Borrowed Context Rewriting
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Borrowed Context Rewriting
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Example
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Plan of the Talk

• Two operational semantics for CCS
• From Processes to Graphs with Interfaces
• From Process Reductions
• to DPO Graph Rewrites
• Borrowed Contexts Rewritings
• Conclusions

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The synthetized LTS

• It is finite branching
• It is really close to the standard
• Bisimilarity coincides with the standard

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CCS encoding in Bigraphs

• Infinite rules
• Infinite branching LTS
• Only finite processes
• Bisimilarity is finer than the standard

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However

• Reasoning on a such LTS is really hard
• This approach works badly for calculi with
  scoping

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Future Works

• Define a rule format for describing the labeled
  transitions of a graph
• Find adhesive structure to encode calculi with
  scoping

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Gadducci Montanari Approach
Process
Graphs
Reduction Rules
Rewriting Rules
Reduction Semantics
DPO Rewriting