Efficient Implementation of Rewriting Revisited Technical Report

TL;DR

This paper extends previous results on polynomial runtime complexity of confluent term rewrite systems to full rewriting, providing a new proof and an implementation that relates rewrite system complexity to Turing machine complexity.

Contribution

The paper generalizes complexity results to full rewriting, introduces a new proof for graph rewriting adequacy, and describes an implementation linking rewrite complexity to Turing machine complexity.

Findings

Runtime complexity of TRS and Turing machine are polynomially related

Graph rewriting can precisely control resource copying

Classification of non-deterministic polytime computations is possible

Abstract

Recently, many techniques have been introduced that allow the (automated) classification of the runtime complexity of term rewrite systems (TRSs for short). In earlier work, the authors have shown that for confluent TRSs, innermost polynomial runtime complexity induces polytime computability of the functions defined. In this paper, we generalise the above result to full rewriting. Following our previous work, we exploit graph rewriting. We give a new proof of the adequacy of graph rewriting for full rewriting that allows for a precise control of the resources copied. In sum we completely describe an implementation of rewriting on a Turing machine (TM for short). We show that the runtime complexity of the TRS and the runtime complexity of the TM is polynomially related. Our result strengthens the evidence that the complexity of a rewrite system is truthfully represented through the length of derivations. Moreover our result allows the classification of non-deterministic polytime-computation based on runtime complexity analysis of rewrite systems.