On Problems Dual to Unification: The String-Rewriting Case

TL;DR

This paper explores the dual problems to unification in string rewriting systems, analyzing their decidability and complexity, with implications for computing fixed points like loop invariants in programming languages.

Contribution

It establishes new decidability and complexity results for fixed point, common term, and common equation problems in various classes of string rewriting systems.

Findings

Fixed point problem undecidable in finite convergent systems

Fixed point problem decidable in polynomial time for dwindling systems

Common term problem undecidable for dwindling systems

Abstract

In this paper, we investigate problems which are dual to the unification problem, namely the Fixed Point (FP) problem, Common Term (CT) problem and the Common Equation (CE) problem for string rewriting systems. Our main motivation is computing fixed points in systems, such as loop invariants in programming languages. We show that the fixed point (FP) problem is reducible to the common term problem. Our new results are: (i) the fixed point problem is undecidable for finite convergent string rewriting systems whereas it is decidable in polynomial time for finite, convergent and dwindling string rewriting systems, (ii) the common term problem is undecidable for the class of dwindling string rewriting systems, and (iii) for the class of finite, monadic and convergent systems, the common equation problem is decidable in polynomial time but for the class of dwindling string rewriting systems, common equation problem is undecidable.