Minimal Relative Normalization in Orthogonal Expression Reduction Systems

? Abstract. In previous papers, the authors studied normalization relative to desirable sets S of`partial results', where it is shown that such sets must be stable. For example, the sets of normal forms, head-normal-forms, and weak head-normal-forms in the-calculus, are all stable. They showed that, for any stable S, S-needed reductions are S-normalizing. This paper continues the investigation into the theory of relative normalization. In particular, we prove existence of minimal normalizing reductions for regular stable sets of results. All the above mentioned sets are regular. We give a suucient and necessary criterion for a normalizing reduction (w.r.t. a regular stable S) to be minimal. Finally, we establish a relationship between relative minimal and optimal reductions, revealing a connict between minimality and optimality: for regular stable sets of results, a term need not possess a reduction that is minimal and optimal at the same time.