Decidability Results in Automata and Process Theory

Preface The study of Process Algebra has received a great deal of attention since the pioneering work in the 1970s of the likes of R. Milner and C.A.R. Hoare. This attention has been merited as the formalism provides a natural framework for describing and analysing systems: concurrent systems are described naturally using constructs which have intuitive interpretations, such as notions of abstractions and sequential and parallel composition. The goal of such a formalism is to provide techniques for verifying the cor-rectness of a system. Typically this verification takes the form of demonstrating the equivalence of two systems expressed within the formalism, respectively representing an abstract specification of the system in question and its implementation. However, any reasonable process algebra allows the description of any computable function, and the equivalence problem-regardless of what reasonable notion of equivalence you consider-is readily seen to be undecidable in general. Much can be accomplished by restricting attention to (communicating) finite-state systems where the equivalence problem is just as quickly seen to be decidable. However, realistic applications, which typically involve infinite entities such as counters or timing aspects, can only be approximated by finite-state systems. Much interest therefore lies in the problem of identifying classes of infinite-state systems in which the equivalence problem is decidable. Such questions are not new in the field of theoretical computer science. Since the proof by Moore [50] in 1956 of the decidability of language equivalence for finite-state automata, language theorists have been studying the decidability problem over classes of automata which express languages which are more expressive than the class of regular languages generated by finite-state automata. Bar-Hillel, Perles and Shamir [3] were the first to demonstrate in 1961 that the class of languages defined by context-free grammars was too wide to permit a 103 decidable theory for language equivalence. The search for a more precise dividing line is still active, with the most outstanding open problem concerning the decidability of language equivalence between deterministic push-down automata. When exploring the decidability of the equivalence checking problem, the first point to settle is the notion of equivalence which you wish to consider. In these notes we shall be particularly interested not in language equivalence but in bisimulation equivalence as defined by Park and used to great effect by Milner. Apart from being the fundamental notion of equivalence for several process algebraic formalisms, this behavioural equivalence has several pleasing mathematical properties, not …