Refining weakly outermost-needed rewriting and narrowing

Outermost-needed rewriting/narrowing is a sound and complete optimal demand-driven strategy for the class of inductively sequential constructor systems. Its parallel extension (known as weakly) deals with non-inductively sequential constructor systems. In this paper, we present natural rewriting, a suitable extension of (weakly) outermost-needed rewriting which is based on a refinement of the demandness notion associated to the latter, and we extend it to narrowing. Intuitively, natural rewriting (narrowing) always reduces (narrows) the most often demanded position in a term. We formalize the strategy for left-linear constructor systems though, for the class of inductively sequential constructor systems, natural rewriting (narrowing) behaves even better than outermost-needed rewriting (narrowing) in the avoidance of failing computations. With regard to inductively sequential constructor systems, we introduce a larger class of systems called inductively sequential preserving where natural rewriting and narrowing preserve optimality for sequential parts of the program. We also provide a prototype interpreter of natural rewriting and narrowing.

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