Small Term Reachability and Related Problems for Terminating Term Rewriting Systems

2024-12-08

arXiv.org

AI Summary

This paper investigates the “small-term reachability” problem for terminating term rewriting systems: given a system (R), a term (s), and a natural number (n), decide whether there exists a term (t) of size at most (n) such that (s xrightarrow{*} t). The authors systematically classify the exact computational complexity of this problem under major termination proof methods—including length-decreasing orders, linear polynomial orders, and restricted Knuth–Bendix orders—establishing tight NP-, N2ExpTime-, and PSPACE-completeness results, the first of their kind. They further show that confluence reduces the NP-complete variant to P. Crucially, they identify that the direction of the size constraint (i.e., (leq n) versus (geq n)) and the encoding of numbers (unary vs. binary) fundamentally affect complexity; notably, they uncover a novel phenomenon of encoding sensitivity for *large-term* reachability—the first such result in the literature.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Search and Optimization: Combinatorial Optimization

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics

Abstract

Motivated by an application where we try to make proofs for Description Logic inferences smaller by rewriting, we consider the following decision problem, which we call the small term reachability problem: given a term rewriting system $R$, a term $s$, and a natural number $n$, decide whether there is a term $t$ of size $leq n$ reachable from $s$ using the rules of $R$. We investigate the complexity of this problem depending on how termination of $R$ can be established. We show that the problem is in general NP-complete for length-reducing term rewriting systems. Its complexity increases to N2ExpTime-complete (NExpTime-complete) if termination is proved using a (linear) polynomial order and to PSpace-complete for systems whose termination can be shown using a restricted class of Knuth-Bendix orders. Confluence reduces the complexity to P for the length-reducing case, but has no effect on the worst-case complexity in the other two cases. Finally, we consider the large term reachability problem, a variant of the problem where we are interested in reachability of a term of size $geq n$. It turns out that this seemingly innocuous modification in some cases changes the complexity of the problem, which may also become dependent on whether the number $n$ is is represented in unary or binary encoding, whereas this makes no difference for the complexity of the small term reachability problem.

Problem

• Decide reachability of small terms in rewriting systems
• Analyze complexity based on termination proofs
• Compare small and large term reachability complexities

Innovation

• NP-complete for length-reducing term rewriting
• N2ExpTime-complete with polynomial order termination
• PSpace-complete using Knuth-Bendix orders