Programming in equational logic: Beyond strong sequentiality

The authors consider whether it is possible to devise a complete normalization algorithm that minimizes (rather than eliminates) the wasteful reductions for the entire class of regular systems. A solution is proposed to this problem using the concept of a necessary set of redexes. In such a set, at least one of the redexes must be reduced to normalize a term. An algorithm is devised to compute a necessary set for any term not in normal form, and it is shown that a strategy that repeatedly reduces all redexes in such a set is complete for regular programs. It is also shown that the algorithm is optimal among all normalization algorithms that are based on left-hand sides alone. This means that the algorithm is lazy (like Huet-Levy's) on strongly sequential parts of a program, relaxes laziness minimally to handle the other parts, and thus does not sacrifice generality for the sake of efficiency.