Uncurrying for Innermost Termination and Derivational Complexity

TL;DR

This paper studies how an uncurrying transformation affects innermost termination and derivational complexity in first-order applicative term rewriting systems, showing it preserves and reflects certain properties but not all.

Contribution

It extends previous work by analyzing the uncurrying transformation's behavior for innermost termination and complexity, providing theoretical insights and counterexamples.

Findings

Uncurrying reflects innermost termination and derivational complexity.

It preserves and reflects polynomial derivational complexity.

Counterexamples show it does not always preserve innermost termination.

Abstract

First-order applicative term rewriting systems provide a natural framework for modeling higher-order aspects. In earlier work we introduced an uncurrying transformation which is termination preserving and reflecting. In this paper we investigate how this transformation behaves for innermost termination and (innermost) derivational complexity. We prove that it reflects innermost termination and innermost derivational complexity and that it preserves and reflects polynomial derivational complexity. For the preservation of innermost termination and innermost derivational complexity we give counterexamples. Hence uncurrying may be used as a preprocessing transformation for innermost termination proofs and establishing polynomial upper and lower bounds on the derivational complexity. Additionally it may be used to establish upper bounds on the innermost derivational complexity while it neither is sound for proving innermost non-termination nor for obtaining lower bounds on the innermost derivational complexity.