Well-structured transition systems (WSTS's) are a general class of infinite state systems for which decidability results rely on the existence of a well-quasi-ordering between states that is compatible with the transitions. In this article, we provide an extensive treatment of the WSTS idea and show several new results. Our improved definitions allow many examples of classical systems to be seen as instances of WSTS's.

Introduction Over the past decade much attention has been devoted to the study of process calculi such as CCS, ACP and CSP [13]. Of particular interest has been the study of the behavioural semantics of these calculi as given by labelled transition graphs. One important question is when processes can be said to exhibit the same behaviour, and a plethora of behavioural equivalences exists today. Their main rationale has been to capture behavioural aspects that language or trace equivalences do not take into account. The theory of finite-state systems and their equivalences can now be said to be well-established. There are many automatic verification tools for their analysis which incorporate equivalence checking. Sound and complete equational theories exist for the various known equivalences, an elegant example is [18]. One may be led to wonder what the results will look like for infinite-state systems. Although language equivalence is decidable

We develop a model-checking algorithm that decides for a given context-free process whether it satisfies a property written in the alternation-free modal mu-calculus. The central idea behind this algorithm is to raise the standard iterative model-checking techniques to higher order: in contrast to the usual approaches, in which the set of formulas that are satisfied by a certain state are iteratively computed, our algorithm iteratively computes a property transformer for each state class of the finite process representation. These property transformers can then simply be applied to solve the model-checking problem. The complexity of our algorithm is linear in the size of the system's representation and exponential in the size of the property being investigated.

this paper have been presented at the 4th European Summer School in Logic, Language and Information, University of Essex, 1992; at the Tempus Summer School for Algebraic and Categorical Methods in Computer Science, Masaryk University, Brno, 1993; and the Summer School in Logic Methods in Concurrency, Aarhus University, 1993. I would like to thank the organisers and the participants of these summer schools, and of the Banff higher order workshop. I would also like to thank Julian Bradfield for use of his Tex tree constructor for building derivation trees and Carron Kirkwood, Faron Moller, Perdita Stevens and David Walker for comments on earlier drafts.

In this chapter, we present a hierarchy of infinite-state systems based on the primitive operations of sequential and parallel composition; the hierarchy includes a variety of commonly-studied classes of systems such as context-free and pushdown automata, and Petri net processes. We then examine the equivalence and regularity checking problems for these classes, with special emphasis on bisimulation equivalence, stressing the structural techniques which have been devised for solving these problems. Finally, we explore the model checking problem over these classes with respect to various linear- and branching-time temporal logics.

. We show that recent progress in extending the automatatheoretic approach to model-checking beyond the class of finite-state processes finds a natural application in the area of interprocedural dataflow analysis. Keywords: Interprocedural data-flow analysis, model-checking, automata theory, program optimisation. 1 Introduction Recent work [15, 24] has shown that model-checking algorithms for abstract classes of infinite-state systems, like context-free processes [1, 5] and pushdown processes [6], find a natural application in the area of data-flow analysis (DFA) for programming languages with procedures [16], usually called interprocedural DFA. A large variety of DFA problems, whose solution is required by optimising compilers in order to apply performance improving transformations, can be solved by means of a unique model-checking technique. The techniques of [5, 6] are based on what could be called the fixpoint approach to model-checking [24], in which the set of states satisfying...

Baeten, Bergstra, and Klop (and later Caucal) have proved the remarkable result that bisimulation equivalence is decidable for irredundant context-free grammars. In this paper we provide a much simpler and much more direct proof of this result using a tableau decision method involving goal-directed rules. The decision procedure also provides the essential part of the bisimulation relation between two processes which underlies their equivalence. We also show how to obtain a sound and complete sequent-based equational theory for such processes from the tableau system and how one can extract what Caucal calls a fundamental relation from a successful tableau.

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. We consider the factorization (collapse) of infinite transition graphs wrt. bisimulation equivalence. It turns out that almost none of the more complex classes of the process taxonomy, which has been established in the last years, are preserved by this operation. However, for the class of BPA graphs (i.e. prefix transition graphs of context-free grammars) we can show that the factorization is effectively a regular graph, i.e. finitely representable by means of a deterministic hypergraph grammar. Since finiteness of regular graphs is decidable, this yields, as a corollary, a decision procedure for the finiteness problem of context-free processes wrt. bisimulation equivalence. 1 Introduction In concurrency theory, process calculi are widely accepted as algebraic description languages for concurrent systems. Their semantics are usually formulated in terms of labelled transition graphs which model the dynamic behaviour together with some notion of behavioural equivalence. Since there is...

PA is the process algebra allowing non-determinism, sequential and parallel compositions, and recursion. We suggest viewing PA-processes as trees, and using tree-automata techniques for verification problems on PA. Our main result is that the set of iterated predecessors of a regular set of PA-processes is a regular tree language, and similarly for iterated successors. Furthermore, the corresponding tree-automata can be built effectively in polynomial-time. This has many immediate applications to verification problems for PA-processes, among which a simple and general model-checking algorithm.