Linear Compressed Pattern Matching for Polynomial Rewriting (Extended Abstract)

TL;DR

This paper introduces polynomial-time algorithms for term rewriting with singleton tree grammar compression, enabling efficient detection of rewrite applicability and execution of rewriting sequences under this compression method.

Contribution

It develops the first efficient algorithms for submatching and rewriting sequences in STG-compressed terms, extending the applicability of compression in term rewriting systems.

Findings

Submatching for left-linear rules is polynomial-time under STG compression.

Rewriting sequences of length n can be performed in nondeterministic polynomial time.

Polynomial-time rewriting is possible for certain compressed rules and target terms.

Abstract

This paper is an extended abstract of an analysis of term rewriting where the terms in the rewrite rules as well as the term to be rewritten are compressed by a singleton tree grammar (STG). This form of compression is more general than node sharing or representing terms as dags since also partial trees (contexts) can be shared in the compression. In the first part efficient but complex algorithms for detecting applicability of a rewrite rule under STG-compression are constructed and analyzed. The second part applies these results to term rewriting sequences. The main result for submatching is that finding a redex of a left-linear rule can be performed in polynomial time under STG-compression. The main implications for rewriting and (single-position or parallel) rewriting steps are: (i) under STG-compression, n rewriting steps can be performed in nondeterministic polynomial time. (ii) under STG-compression and for left-linear rewrite rules a sequence of n rewriting steps can be performed in polynomial time, and (iii) for compressed rewrite rules where the left hand sides are either DAG-compressed or ground and STG-compressed, and an STG-compressed target term, n rewriting steps can be performed in polynomial time.