Some Varieties of Equational Logic

The application of ideas from universal algebra to computer science has long 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] with the idea that, for appropriately chosen monads T on, e.g., Set, the category of sets, one thinks of T (X) as the type of computations of an element of X. For example, for side-effects one takes the monad TS(X) =def (S ×X) where S is the set of states. Below, we take S =def V Loc where V is a countably infinite set of values such as the natural numbers, and Loc is a finite set of locations. See [2] for a recent exposition of Moggi’s ideas, particularly emphasising the connections with functional programming, where the monadic approach has been very influential. As is well known, equational theories give rise to free algebra monads. For example the free semilattice monad arises from the theory of a binary operation ∪ subject to the axioms of associativity, commutativity and idempotence, where the last is the equation x ∪ x = x. The induced monad TN (X) is the collection of all non-empty finite subsets of X. In general, the equational theories with operations of finite arity induce exactly those monads which have finite rank, see, e.g., [19]. In denotational semantics one typically employs a category of ordered structures, such as ω-Cpo, the category of ω-cpos, which are partial orders with lubs of increasing ω-chains, and with morphisms those monotonic functions preserving the ω-lubs. An ω-Cpo-semilattice is a semilattice in ω-Cpo, that is an ω-cpo together with a continuous binary function satisfying the semilattice axioms; the free ω-Cpo-semilattice monad is (a generalisation of) the convex powerdo-