Uncurrying for Termination and Complexity

Abstract

First-order applicative rewrite systems provide a natural framework for modeling higher-order aspects. In this article 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 study the method for innermost rewriting and derivational complexity, and present a version for dependency pairs.

A Transfinite Knuth–Bendix Order for Lambda-Free Higher-Order Terms

“…There are, however, a considerable number of higher-order variants of RPO [17,18,28,31,32,35] and many encodings of higher-order term rewriting systems into first-order systems [2,22,24,24,46]. The encoding approaches are more suitable to term rewriting systems than to superposition and similar proof calculi.…”

“…This is at odds with the need for complete higher-order proof calculi to synthesize arbitrary terms during proof search [10], in which a symbol f may be applied to fewer arguments than anywhere in the problem. A scheme by Hirokawa et al [24] circumvents this issue but requires additional symbols and rewrite rules.…”

“…The system is also beyond the scope of the uncurrying approach of Hirokawa et al [24], because of the applied variable f on the left-hand side of rule 1.…”

“…Moreover, the above examples are easy for modern first-order termination provers, which use uncurrying techniques [24,43] to transform applicative rewrite systems into functional systems that can be analyzed by standard techniques. This is somewhat to be expected: Even with transfinite weights and argument coefficients, KBO tends to consider syntactically smaller terms smaller.…”

Analysing the complexity of functional programs: higher-order meets first-order

“…All these methods focus on the analysis of the given defined symbols (like for instance the application symbol in the example above) and fail if their recursive definition is too complicated. Naturally this calls for a special treatment of the applicative structure of the system [31].…”