Unification and Matching on Compressed Terms

TL;DR

This paper investigates unification and matching of compressed terms, introducing polynomial algorithms for dags and first-order cases, and analyzing complexity for Singleton Tree Grammars.

Contribution

It presents new polynomial algorithms for unification and matching of compressed terms and analyzes their computational complexity.

Findings

Polynomial-time algorithm for context matching with dags when context variables are fixed

NP-completeness of context matching with Singleton Tree Grammars

Improved polynomial algorithms for first-order unification and matching

Abstract

Term unification plays an important role in many areas of computer science, especially in those related to logic. The universal mechanism of grammar-based compression for terms, in particular the so-called Singleton Tree Grammars (STG), have recently drawn considerable attention. Using STGs, terms of exponential size and height can be represented in linear space. Furthermore, the term representation by directed acyclic graphs (dags) can be efficiently simulated. The present paper is the result of an investigation on term unification and matching when the terms given as input are represented using different compression mechanisms for terms such as dags and Singleton Tree Grammars. We describe a polynomial time algorithm for context matching with dags, when the number of different context variables is fixed for the problem. For the same problem, NP-completeness is obtained when the terms are represented using the more general formalism of Singleton Tree Grammars. For first-order unification and matching polynomial time algorithms are presented, each of them improving previous results for those problems.