Memoization for Unary Logic Programming: Characterizing PTIME

TL;DR

This paper characterizes PTIME computation using an algebraic structure called the resolution semiring, focusing on unary logic programs, and introduces a memoization technique to demonstrate polynomial time soundness and completeness.

Contribution

It provides a novel algebraic framework for unary logic programming that characterizes PTIME and introduces a memoization method for complexity analysis.

Findings

Resolution semiring characterizes PTIME for unary logic programs.

Memoization technique proves PTIME soundness of the framework.

Unary logic programming queries are PTIME-complete.

Abstract

We give a characterization of deterministic polynomial time computation based on an algebraic structure called the resolution semiring, whose elements can be understood as logic programs or sets of rewriting rules over first-order terms. More precisely, we study the restriction of this framework to terms (and logic programs, rewriting rules) using only unary symbols. We prove it is complete for polynomial time computation, using an encoding of pushdown automata. We then introduce an algebraic counterpart of the memoization technique in order to show its PTIME soundness. We finally relate our approach and complexity results to complexity of logic programming. As an application of our techniques, we show a PTIME-completeness result for a class of logic programming queries which use only unary function symbols.