An Abstract Concurrent Machine for Rewriting

1 I n t r o d u c t i o n Term rewriting is a powerful computat ional paradigm. On the one hand, it has the full power of Turing machines, because they can be formulated as rewrite systems, and on the other hand, i t lends itself to parallel execution, since different subterms of a term can be rewrit ten concurrently. A rewrite rule over a set of (first-order) terms is an ordered pair of terms, which we write as l -* r. A rewr/te system is a finite set of rules. Given a rewrite system R, we say that a term s rewrites to a term t at some position p, denoted by s ~ R t, if there is a rule I ~ r in R and a subst i tut ion a , such that the subterm of s at position p is la and t is s with that subterm replaced by r a . The position at which a rewrite applies is called a redez. A term is said to be in normal form if no rewrite rule can be applied to it. It should be stressed that any variables in s remain uninstant ia ted by rewriting. We also consider a more general operation, known as "narrowing", in which a rule is applied if i ts left-hand side unifies with any subterm of s. More precisely, we say tha t a term s narrows to a term t (at position p), denoted s ~'*R t, if there is a rule l ~ r in R such that a is the most general unifier of the subterm of s (at p) with l, and t is the result of applying a to s and then replacing that subterm la of sa with ra . If s a is different from s, we will call it a proper narrowing. We assume tha t unification always t reats variables in rules as dist inct from those in the term. In this paper , we show how different rewriting and narrowing strategies can be implemented on an abst ract concurrent machine. The way our abstract machine works for rewriting is tha t