Termination of rewriting under strategies

Abstract

A termination proof method for rewriting under strategies, based on an explicit induction on the termination property, is presented and instantiated for the innermost, outermost, and local strategies. Rewriting trees are simulated by proof trees generated with an abstraction mechanism, narrowing and constraints representing sets of ground terms. Abstraction introduces variables to represent normal forms without computing them and to control the narrowing mechanism, well known to easily diverge. The induction ordering is not given a priori, but defined with ordering constraints, incrementally set during the proof. It is established that termination under strategy is equivalent to the construction of finite proof trees schematizing terminating rewriting trees. Sufficient effective conditions to ensure finiteness are studied and the method is illustrated on several examples for each specific strategy.

Termination of Priority Rewriting

“…This technique was inspired from the one we proposed for proving the innermost termination of classical rewrite systems in [16]. We now give the concepts needed to formalize and automate it.…”

“…Otherwise, automatic constraint solvers can provide adequate polynomial orderings. See [16] for experiments.…”

“…To schematize innermost rewriting on ground terms, in [16], we introduced a specific narrowing definition involving constrained substitutions. The problem here is to see how to express priorities in the narrowing mechanism and how to integrate them in the previous constraint based definition.…”

“…The problem here is to see how to express priorities in the narrowing mechanism and how to integrate them in the previous constraint based definition. In [16], the usual notion of narrowing was refined as follows. With the usual innermost narrowing relation, if a position p in a term t is a narrowing position, no suffix position of p can be a narrowing position as well.…”

“…We use an inductive approach, whose principle has already been applied for termination of rewriting under strategies [16]. The idea is to prove, in developing proof trees simulating the rewriting trees, that every derivation starting from any term terminates, supposing that it is true for terms smaller than the starting terms.…”