Conservation and uniform normalization in Lambda calculi with erasing reductions

• Authors: Peter Møller Neergaard;Morten Heine Sørensen
• Venue: Information and Computation
• Year: 2002

For a notion of reduction in a λ-calculus one can ask whether a term satisfies conservation and uniform normalization. Conservation means that single-step reductions of the term preserve infinite reduction paths from the term. Uniform normalization means that either the term will have no reduction paths leading to a normal form or all reduction paths will lead to a normal form. In the classical conservation theorem for ΛI the distinction between the two notions is not clear: uniform normalization implies conservation, and conservation also implies uniform normalization. The reason for this is that ΛI is closed under reduction, due to the fact that reductions never erase terms in ΛI. More generally for nonerasing reductions, the two notions are equivalent on a set closed under reduction. However, when turning to erasing reductions the distinction becomes important as conservation no longer implies uniform normalization. This paper presents a new technique for finding uniformly normalizing subsets of a λ-calculus. This is done by combining a syntactic and a semantic criterion. The technique is demonstrated by several applications. The technique is used to present a new uniformly normalizing subset of the pure λ-calculus; this subset is a superset of ΛI and thus contains erasing K-redexes. The technique is also used to prove strong normalization from weak normalization of the simply typed λ-calculus extended with pairs; this is an extension of techniques developed recently by Sørensen and Xi. Before presenting the technique the paper presents a simple proof of a slightly weaker form of the characterization of perpetual redexes by Bergstra and Klop; this is a step for the later applications of the technique.