Comparing Curried and Uncurried Rewriting

Abstract

Currying is a transformation of term rewrite systems which may contain symbols of arbitrary arity into systems which contain only nullary symbols, together with a single binary symbol called application. We show that for all term rewrite systems (whether orthogonal or not) the following properties are preserved by this transformation: strong normalization, weak normalization, weak Church-Rosser, completeness, semi-completeness, and the non-convertibility of distinct normal forms. Under the condition of leftlinearity we show preservation of the properties NF (if a term is reducible to a normal form, then its reducts are all reducible to the same normal form) and UN → (a term is reducible to at most one normal form). We exhibit counterexamples to the preservation of NF and UN → for non-left-linear systems. The results extend to partial currying (where some subset of the symbols are curried), and imply some modularity properties for unions of applicative systems.

Uncurrying for Termination and Complexity

“…We remark that Corollary 1 preserves derivational complexity. This is straightforward from [23 Table 3 our transformation admits significant gains in power. We only tested the direct transformations, because proofs with dependency pairs give upper bounds on the derivational complexity much beyond exponential [32].…”

Simplifying Algebraic Functional Systems

“…In first-order rewriting, the question whether properties such as confluence and termination are preserved under currying or uncurrying is studied in [5,6,2]. In [6] a currying transformation from (functional) term rewriting systems (TRSs) into applicative term rewriting systems (ATRSs) is defined; a TRS is terminating if and only if its curried form is. In [2], an uncurrying transformation from ATRSs to TRSs is defined that can deal with partial application and leading variables, as long as they do not occur in the left-hand side of rewrite rules.…”

On the Modularity of Deciding Call-by-Need

“…Most properties of rewrite systems are preserved under signature extension. Two notable exceptions are the normal form property and the unique normal form property (with respect to reduction), see Kennaway et al [11]. Also some properties dealing with ground terms are not preserved under signature extension.…”