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

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: 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. 1

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 2

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Meaning of CCS Operators 5

Notational Conventions 6

Labelled Transition System 7

Semantics of CCS 8

Semantics of CCS 9

Semantics of CCS 10

Semantics of CCS 11

Recursion 12

Equivalence 13

Equivalence of CCS Processes 14

Desired Properties of Equivalence 15

Congruence 16

Trace Equivalence 17

Trace Equivalence 18

Congruence 19

Trace Equivalence 20

Trace Equivalence 21

Deadlock 22

Deadlock 23

Desired Properties of Equivalence 24

Strong Bisimulation 25

Strong Bisimulation 26

Strong Bisimulation 27

Strong Bisimulation How to prove this? 28

Strong Bisimulation How to prove this? 29

Properties of Strong Bisimulation 30

Strong Bisimulation => Trace Equivalence 31

Congruence Property Makes use of following Lemma 32

Congruence 33

Deadlock 34

Summary 35

Traces and Deadlocks 36

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 37

Partition Refinement Algorithm 38

Strong Simulation 39

Strong Simulation and Bisimulation 40

Strong Bisimulation is not an ideal solution! 41

Weak Bisimulation 42

Weak Bisimulation 43

Weak Bisimulation 44

Weak Bisimulation 45

Weak Bisimulation 46

Weak Bisimulation 47

Weak Bisimulation 48

Properties of Weak Bisimulation 49

Properties of Weak Bisimulation 50

Observation Congruence 51

Observation Congruence 52

Observation Congruence 53

Observation Congruence 54

Observation Congruence 55

Observation Congruence 56

Observation Congruence 57

Observation Congruence 58

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 59