Abstract not found

Monads have been recognized by Moggi as an elegant device for dealing with stateful computation in functional programming languages. It is thus natural

The question of how to combine monads arises naturally in many areas with much recent interest focusing on the coproduct of two monads. In general, the coproduct of arbitrary monads does not always exist. Although a rather general construction was given by

Modular rewriting seeks criteria under which rewrite systems inherit properties from their smaller subsystems. This divide and conquer methodology is particularly useful for reasoning about large systems where other techniques fail to scale adequately. Research has typically focused on reasoning about the modularity of specific properties for specific ways of combining specific forms of rewriting. This paper is, we believe, the first to ask a much more general question. Namely, what can be said about modularity independently of the specific form of rewriting, combination and property at hand. A priori there is no reason to believe that anything can actually be said about modularity without reference to the specifics of the particular systems etc. However, this paper shows that, quite surprisingly, much can indeed be said.

Abstract. In [17], Toyama proved that the union of two confluent term-rewriting systems that share absolutely no function symbols or constants is likewise confluent, a property called modularity. The proof of this beautiful modularity result, technically based on slicing terms into an homogeneous cap and a so called alien, possibly heterogeneous substitution, was later substantially simplified in [7, 9]. In this paper 3 we present a further simplification of the proof of Toyama’s result for confluence, which shows that the crux of the problem lies in two different properties: a cleaning lemma, whose goal is to anticipate the application of collapsing reductions; a modularity property of ordered completion, that allows to pairwise match the caps and alien substitutions of two equivalent terms obtained from the cleaning lemma. The approach allows for rules with extra variables on the right and scales up to rewriting modulo arbitrary sets of equations. 1

Recently there has been a great deal of interest in higherorder syntax which seeks to extend standard initial algebra semantics to cover languages with variable binding by using functor categories. The canonical example studied in the literature is that of the untyped λ-calculus which is handled as an instance of the general theory of binding algebras, cf. Fiore, Plotkin, Turi [8]. Another important syntactic construction is that of explicit substitutions. The syntax of a language with explicit substitutions does not form a binding algebra as an explicit substitution may bind an arbitrary number of variables. Nevertheless we show that the language given by a standard signature Σ and explicit substitutions is naturally modelled as the initial algebra of the endofunctor Id + FΣ ◦ + ◦ on a functor category. We also comment on the apparent lack of modularity in syntax with variable binding as compared to first-order languages. Categories and Subject Descriptors