On Problems Dual to Unification

TL;DR

This paper explores the dual of the unification problem, focusing on the common term problem in string rewriting systems, demonstrating its undecidability in certain cases and its relevance to computing fixed points like loop invariants.

Contribution

It introduces the common term problem for string rewriting systems, proves its undecidability for dwindling systems, and links it to fixed point computations in programming languages.

Findings

The fixed point problem reduces to the common term problem.

The common term problem is undecidable for dwindling systems.

Implications for loop invariants in programming languages.

Abstract

In this paper, we investigate a problem dual to the unification problem, namely the Common Term (CT) problem for string rewriting systems. Our main motivation was in computing fixed points in systems, such as loop invariants in programming languages. We show that the fixed point problem is reducible to the common term problem. We also prove that the common term problem is undecidable for dwindling string rewriting systems.