Clocks, Trees and Stars in Process Theory
Wan Fokkink
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,