Semantics of Non-terminating Rewrite Systems Using Minimal Coverings

We propose a new semantics for rewrite systems ba.~ed on interpreting rewrite rules as in­ equatioIlB between terms in an ordered algebra. In part.icular, we show thai the algebra. of normal forms in a terminating system is a uniqnely minimal covering of the term algebra. In the non-terminating ca..~e, the existence of this minimal covering is established in the comple­ tion of an ordered algebra formed by rewrit.ing sequences. We thus generalize the properties of normal forms far: non-terminating systelil~ to this minimal covering. ThesE' include the exi~tence of normal forms for arbitrary rewrite ~ystems, and their uniqueness for conBue-nt ~ystems, in which Ca the benefits of alge­ braic semantics to systems with non-determinist.ic and non-t.erminating computations. V·le first study properties of abstract. order~, and then instantiat.e the~e to term rewriting sy~tems.