DS-1994-09:
Fokkink, Wan
(1994)
*Clocks, Trees and Stars in Process Theory.*
Doctoral thesis, University of Amsterdam.

## Abstract

Process algebra, or process theory, constitutes an attempt to reason about

`behaviours of systems' in a mathematical framework. Starting from a syntax,

each syntactic object is supplied with some kind of behaviour, and a semantic

equivalence says which behaviours are to be identified. (Bisimulation is such

a semantic equivalence.) Process algebra expresses semantic equivalences in

axioms, or equational laws. It requires that a set of axioms is sound (i.e. if

two behaviours can be equated, then they are semantically equivalent), and it

desires that it is complete (i.e. if two behaviours are semantically

equivalent, then they can be equated). Process algebra can be applied to prove

correctness of system behaviour. It enables to express (un)desirable properties

of the behaviour of a system in an abstract way, and to deduce by mathematical

manipulations whether or not the behaviour satisfies such a property.

Forty years ago, Kleene introduced a binary operator x*y, called iteration or

Kleene star. The process x*y can choose to execute either x, after which it

evolves into x*y again, or y, after which it terminates. In Chapter 3 it is

proved that BPA extended with iteration, modulo bisimulation, is axiomatized

completely by the five standard axioms for BPA together with three extra

axioms for iteration. In Chapter 2 it is proved that basic CCS extended with

prefix iteration a*x, where the left argument is restricted to atomic actions,

is axiomatized completely, modulo bisimulation, by the four standard axioms

for basic CCS together with two extra axioms for prefix iteration.

In order to describe a semantic equivalence by means of axioms, it is essential

that such an equivalence is a congruence. Groote and Vaandrager proved that

behaviours generated by well-founded Plotkin rules in the tyft/tyxt format are

always a congruence for strong bisimulation. In Chapter 4 it is shown that the

well-foundedness restriction can be omitted in this congruence result. This

follows from a stronger result, which says that for each collection of

transitions rules in tyft/tyxt format, there is an equivalent collection of

transition rules in the more restrictive tree format.

In Chapter 5, a classic result from unification theory, which says that if a

finite set of equations allows a unifier then it allows an idempotent most

general unifier, is generalized to infinite sets.

In Chapter 6, a subalgebra of the regular processes in ACPrI with recursion is

determined, such that for each pair of processes p and q in this algebra, their

merge p||q is in this algebra too. The subalgebra is very specific, because if

it is enlarged or restricted in any obvious way, then this elimination result

is lost. The discovered algebra is equal to the class of timed automata of

Alur and Dill.

In Chapter 7 it is proved that strong bisimulation equivalence for the timed

process algebra ACPrI is decidable, i.e. for each pair of terms in ACPrI it

can be decided whether or not they are bisimilar.

Finally, Chapter 8 presents a complete axiomatization for Basic Process Algebra

extended with the constants silent step and deadlock and recursion and time,

Item Type: | Thesis (Doctoral) |
---|---|

Report Nr: | DS-1994-09 |

Series Name: | ILLC Dissertation (DS) Series |

Year: | 1994 |

Depositing User: | Dr Marco Vervoort |

Date Deposited: | 14 Jun 2022 15:16 |

Last Modified: | 14 Jun 2022 15:16 |

URI: | https://eprints.illc.uva.nl/id/eprint/1972 |

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