Perpetuality and Uniform Normalization

We define a perpetual one-step reduction strategy which enables one to construct minimal (w.r.t. Levy's ordering ⊴ on reductions) infinite reductions in Conditional Orthogonal Expression Reduction Systems. We use this strategy to derive two characterizations of perpetual redexes, i.e., redexes whose contractions retain the existence of infinite reductions. These characterizations generalize existing related criteria for perpetuality of redexes. We give a number of applications of our results, demonstrating their usefulness. In particular, we prove equivalence of weak and strong normalization (the uniform normalization property) for various restricted λ-calculi, which cannot be derived from previously known perpetuality criteria.