cs780 discrete-state models - william & marykemper/cs780_08/slides/v_ccs.pdf · 37 computing...

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CS780 Discrete-State Models

Today:Milner‘s Calculus of Communicating SystemsStrong & Weak BisimulationObservational Congruence

Quick Reference:Robin Milner, A Calculus of Communicating Systems,Springer, LNCS 92, 1980.Robin Miner, Communication and Concurrency, Prentice Hall,1989.

Instructor: Peter KemperR 006, phone 221-3462, email:kemper@cs.wm.eduOffice hours: Mon,Wed 3-5 pm

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Outline

Origin of Process Algebras:Calculus of Communicating Systems (CCS)Trace EquivalenceBisimulation Strong Weak

Observational Congruence

Credits: Slides from Noll, Katoen, RWTH Aachen, Germany, 2007/08

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Meaning of CCS Operators

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Notational Conventions

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Labelled Transition System

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Semantics of CCS

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Semantics of CCS

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Semantics of CCS

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Semantics of CCS

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Recursion

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Equivalence

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Equivalence of CCS Processes

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Desired Properties of Equivalence

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Congruence

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Trace Equivalence

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Trace Equivalence

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Congruence

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Trace Equivalence

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Trace Equivalence

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Deadlock

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Deadlock

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Desired Properties of Equivalence

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Strong Bisimulation

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Strong Bisimulation

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Strong Bisimulation

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Strong Bisimulation

How to prove this?

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Strong Bisimulation

How to prove this?

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Properties of Strong Bisimulation

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Strong Bisimulation => Trace Equivalence

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Congruence Property

Makes use of following Lemma

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Congruence

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Deadlock

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Summary

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Traces and Deadlocks

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Computing Equivalences

Basic Algorithm: Paige, Tarjan: Three partition refinement algoriths, SIAM J. Computing,

16, 1987.Multiple variants and refinements, in particular wrt stochastic models P. Buchholz. Exact and ordinary lumpability in finite Markov chains.

Journal of Applied Probability, 31:59–75, 1994. S. Derisavi, H. Hermanns, and W. H. Sanders. Optimal State-Space

Lumping in Markov Chains, Information Proc. Letters, 87, 6, 2003

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Partition Refinement Algorithm

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Strong Simulation

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Strong Simulation and Bisimulation

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Strong Bisimulation is not an ideal solution!

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Weak Bisimulation

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Weak Bisimulation

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Weak Bisimulation

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Weak Bisimulation

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Weak Bisimulation

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Weak Bisimulation

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Weak Bisimulation

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Properties of Weak Bisimulation

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Properties of Weak Bisimulation

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Observation Congruence

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Observation Congruence

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Observation Congruence

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Observation Congruence

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Observation Congruence

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Observation Congruence

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Observation Congruence

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Observation Congruence

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Summary

Origin of Process Algebras:Calculus of Communicating Systems (CCS)Trace Equivalence Insensitive to deadlocks!

Bisimulation Strong Bisimulation:

too restrictive to be used for an equivalence between an abstractspecification and a detailed implementation model,we need to abstract from internal operations

Weak Bisimulation:no congruence wrt to choice, problem is an initial Tau step

Observational Congruence Compromise between strong and week bisimulation Yields congruence wrt CCS operations

Equivalence classes can be determined with algorithms based onpartition refinement