Abstract. Given a set of processes and a set of tests on these processes we show how to define in a natural way three different eyuitalences on processes. ThesP equivalences are applied to a particular language CCS. We give associated complete proof systems and fully abstract models. These models have a simple representation in terms of trees.

This paper is a contribution to the algebraic theory of recognizable languages. The main topic of this paper is the polynomial closure, an operation that mixes together the operations of union and concatenation. Formally, the polynomial closure of a class of languages L of A ∗ is the set of languages

Fokkink and Zantema ((1994) Computer Journal 37:259–267) have shown that bisimulation equivalence has a finite equational axiomatization over the language of Basic Process Algebra with the binary Kleene star operation (BPA ∗). In the light of this positive result on the mathematical tractability of bisimulation equivalence over BPA ∗ , a natural question to ask is whether any other (pre)congruence relation in van Glabbeek’s linear time/branching time spectrum is finitely (in)equationally axiomatizable over it. In this paper, we prove that, unlike bisimulation equivalence, none of the preorders and equivalences in van Glabbeek’s linear time/branching time spectrum, whose discriminating power lies in between that of ready simulation and that of completed traces, has a finite equational axiomatization. This we achieve by exhibiting a family of (in)equivalences that holds in ready simulation semantics, the finest semantics that we consider, whose instances cannot all be proven by means of any finite set of (in)equations

this paper was published as Walicki, M.A. and Meldal, S., 1995, Nondeterministic Operators in Algebraic Frameworks, Tehnical Report No. CSL--TR--95--664, Stanford University

. Rational terms (possibly infinite terms with finitely many subterms) can be represented in a finite way via -terms, that is, terms over a signature extended with self-instantiation operators. For example, f ! = f(f(f(: : :))) can be represented as x :f(x) (or also as x :f(f(x)), f(x :f(x)), . . . ). Now, if we reduce a -term t to s via a rewriting rule using standard notions of the theory of Term Rewriting Systems, how are the rational terms corresponding to t and to s related? We answer to this question in a satisfactory way, resorting to the definition of infinite parallel rewriting proposed in [7]. We also provide a simple, algebraic description of -term rewriting through a variation of Meseguer's Rewriting Logic formalism. 1 Introduction Rational terms are possibly infinite terms with a finite set of subterms. They show up in a natural way in Theoretical Computer Science whenever some finite cyclic structures are of concern (for example data flow diagrams, cyclic te...

We extend Reiterman's theorem to first-order structures: a class of finite first-order structures is a pseudovariety if and only if it is defined by a set of identities in a certain relatively free profinite structure (pseudoidentities) .

Salomaa ((1969) Theory of Automata, page 143) asked whether the equational theory of regular expressions over a singleton alphabet has a finite equational base. In this paper, we provide a negative answer to this long standing question. The proof of our main result rests upon a modeltheoretic argument. For every finite collection of equations, that are sound in the algebra of regular expressions over a singleton alphabet, we build a model in which some valid regular equation fails. The construction of the model mimics the one used by Conway ((1971) Regular Algebra and Finite Machines, page 105) in his proof of a result, originally due to Redko, to the effect that infinitely many equations are needed to axiomatize equality of regular expressions. Our analysis of the model, however, needs to be more refined than the one provided by Conway ibidem.

We propose a method to axiomatize by equations the least pre xed point of an order preserving function. We discuss its domain of application and show that the Boolean Modal -Calculus has a complete equational axiomatization. The method relies on the existence of a \closed structure" and its relationship to the equational axiomatization of Action Logic is made explicit. The implication operation of a closed strucure is not monotonic in one of its variables; we show that the existence of such a term that does not preserve the order is an essential condition for de ning by equations the least pre xed point. We stress the interplay between closed structures and xed point operators by showing that the theory of Boolean modal -algebras is not a conservative extension of the theory of modal -algebras. The latter is shown to lack the nite model property.

been a major theme of Joseph Goguen’s research, perhaps even the major theme. One strand of this work concerns algebraic datatypes. Recently there has been some interest in what one may call algebraic computation types. As we will show, these are also given by equational theories, if one only understands the notion of equational logic in somewhat broader senses than usual. One moral of our work is that, suitably considered, equational logic is not tied to the usual first-order syntax of terms and equations. Standard equational logic has proved a useful tool in several branches of computer science, see, for example, the RTA conference series [9] and textbooks, such as [1]. Perhaps the possibilities for richer varieties of equational logic discussed here will lead to further applications. We begin with an explanation of computation types. Starting around 1989, Eugenio Moggi introduced the idea of monadic notions of computation [11, 12]

this article, I would like to emphasize the role of three of these tools: topology, partial orders and categories. The three parts are relatively independent