Notes on the Polish Algorithm

TL;DR

This paper investigates the Polish Algorithm's behavior on algebraic laws, providing partial results on its termination for some laws and demonstrating divergence for others, with computational assistance.

Contribution

It offers new insights into the termination properties of the Polish Algorithm for specific algebraic laws, including partial results and divergence proofs.

Findings

Partial results on algorithm termination for left distributivity

Proof of divergence for certain algebraic laws

Use of computational methods to analyze algorithm behavior

Abstract

We study, with the help of a computer program, the Polish Algorithm for finite terms satisfying various algebraic laws, e.g., left distributivity a(bc) = (ab)(ac). While the termination of the algorithm for left distributivity remains open in general, we can establish some partial results, which might be useful towards a positive solution. In contrast, we show the divergence of the algorithm for the laws a(bc) = (ab)(cc) and a(bc) = (ab)(a(ac)).