Uncurrying for Termination

Abstract

Abstract. First-order applicative term rewrite systems provide a natural framework for modeling higher-order aspects. In this paper we present a transformation from untyped applicative term rewrite systems to functional term rewrite systems that preserves and reflects termination. Our transformation is less restrictive than other approaches. In particular, head variables in right-hand sides of rewrite rules can be handled. To further increase the applicability of our transformation, we present a version for dependency pairs.

Uncurrying for Termination and Complexity

“…So is the material in Section 6.2. Moreover, we close a non-trivial gap in the proof of [18,Theorem 33]. Some of the new contributions go beyond the scope of uncurrying, e.g., Theorem 11 gives a condition when the length of reductions is the same for innermost and full rewriting, which has an immediate impact on (automated) complexity analysis.…”

“…The first line of research is based on types [1-3, 7, 21, 28, 41] and allows to study properties like termination or strong computability directly. The second approach [13,18] aims for transformations that recover the structure of applicative rewrite rules to enable methods that do not rely on types. The benefit of the first approach is that the type information may make proving termination properties easier.…”

“…A preliminary version of this article appeared in [18]. Several of the results on innermost rewriting and derivational complexity have been published in [44].…”

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 [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. This transformation is sound and complete with respect to termination.…”