Fokkink and Zantema (Fokkink and Zantema 1994) 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 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 (which is the finest semantics that we consider) and whose instances cannot all be proved by means of any finite set of (in)equations that is sound in completed trace semantics (which is the coarsest semantics that is appropriate for the language BPA*). To this end, for every finite collection of (in)equations that are sound in completed trace semantics, we build a model in which some of the (in)equivalences of the family under consideration fail. The construction of the model mimics the one used by Conway (Conway 1971, p. 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 non-finite axiomatizability results apply to the language BPA* over an arbitrary non-empty set of actions. In particular, we show that completed trace equivalence is not finitely based over BPA* even when the set of actions is a singleton. Our proof of this result may be adapted to the standard language of regular expressions to yield a solution to an open problem posed by Salomaa (Salomaa 1969, p. 143).

Another semantics that is usually considered in process theory is trace semantics. Trace semantics is, in general, not preserved by sequential composition, and is therefore inappropriate for the language BPA*. We show that, if the set of actions is a singleton, trace equivalence and preorder are preserved by all the operators in the signature of BPA*, and coincide with simulation equivalence and preorder, respectively. In that case, unlike all the other semantics considered in this paper, trace semantics have finite, complete equational axiomatizations over closed terms.

In object-oriented programming, there are many notions of ‘collection with members in X’. This paper offers an axiomatic theory of collections based on monads in the category of sets and total functions. Heuristically, the axioms defining a collection monad state that each collection has a finite set of members of X, that pure 1-element collections exist and that a collection of collections flattens to a single collection whose members are the union of the members of the constituent collections. The relationship between monads and universal algebra leads to a formal definition of collection implementation in terms of tree-processing. Ideas from elementary category theory underly the classification of collections. For example, collections can be zipped if and only if the monad's endofunctor preserves pullbacks. Or, a collection can be uniquely specified by its shape and list of data if the morphisms of the Kleisli category of the monad are all deterministic, and the converse holds for commutative monads. Again, a collection monad is ordered if the monad's functor preserves equalizers of monomorphisms (so, in particular, if collections can be zipped the monad is ordered). Every implementable monad is ordered. It is shown using the well-ordering principle that a collection monad is ordered if and only if its functor admits an appropriated list-valued natural transformation that lists the members of each collection.

We develop temporal logic from the theory of complete lattices, Galois connections and fixed points. In particular, we prove that all seventeen axioms of Manna and Pnueli's sound and complete proof system for linear temporal logic can be derived from just two postulates, namely that ([oplus ], &[ominus ]tilde;) is a Galois connection and that ([ominus ], [oplus ]) is a perfect Galois connection. We also obtain a similar result for the branching time logic CTL.

A surprising insight is that most of the theory can be developed without the use of negation. In effect, we are studying intuitionistic temporal logic. Several examples of such structures occurring in computer science are given. Finally, we show temporal algebra at work in the derivation of a simple graph-theoretic algorithm.

This paper is tutorial in style and there are no difficult technical results. To the experts in temporal logics, we hope to convey the simplicity and beauty of algebraic reasoning as opposed to the machine-orientedness of logical deduction. To those familiar with the calculational approach to programming, we want to show that their methods extend easily and smoothly to temporal reasoning. For anybody else, this text may serve as a gentle introduction to both areas.