CS780 Discrete-State Models Instructor: Peter Kemper R 006, phone - - PowerPoint PPT Presentation

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CS780 Discrete-State Models Instructor: Peter Kemper R 006, phone - - PowerPoint PPT Presentation

Sep 11, 2022 119 likes •729 views

CS780 Discrete-State Models Instructor: Peter Kemper R 006, phone 221-3462, email:kemper@cs.wm.edu Office hours: Mon,Wed 3-5 pm Today: Milners Calculus of Communicating Systems Strong & Weak Bisimulation Observational Congruence Quick

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CS780 Discrete-State Models

Today: Milner‘s Calculus of Communicating Systems Strong & Weak Bisimulation Observational Congruence Quick Reference:

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

Instructor: Peter Kemper

R 006, phone 221-3462, email:kemper@cs.wm.edu Office hours: Mon,Wed 3-5 pm

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Outline Origin of Process Algebras: Calculus of Communicating Systems (CCS) Trace Equivalence Bisimulation

 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 abstract specification 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 on partition refinement