Incremental checking of well-founded recursive specifications modulo axioms

Abstract

Abstract. We introduce the notion of well-founded recursive order-sorted equational logic (OS) theories modulo axioms. Such theories define functions by well-founded recursion and are inherently terminating. Moreover, for well-founded recursive theories important properties such as confluence and sufficient completeness are modular for so-called fair extensions. This enables us to incrementally check these properties for hierarchies of such theories that occur naturally in modular rule-based functional programs. Well-founded recursive OS theories modulo axioms contain only commutativity and associativity-commutativity axioms. In order to support arbitrary combinations of associativity, commutativity and identity axioms, we show how to eliminate identity and (under certain conditions) associativity (without commutativity) axioms by theory transformations in the last part of the paper.

Twenty years of rewriting logic

“…At the vanilla-flavored level of untyped rewrite theories of the form R = ( , ∅, R), there is already a substantial body of such techniques available (see, e.g., [358,456]), and even some very useful work for untyped rewrite theories of the form R = ( , AC, R), with AC associative-commutative axioms [296]. Schernhammer and I have initiated the study of modularity techniques for the termination of unconditional order-sorted specifications modulo combinations of associativity and/or commutativity and/or identity axioms in [415].…”