A simple algorithm for checking equivalence of counting functions on free monoids

TL;DR

This paper introduces a new, simplified algorithm for verifying the equivalence of counting functions on free monoids, achieving linear time complexity for all ranks $r \,\geq\, 2$, improving over previous methods.

Contribution

It presents a novel algorithm based on explicit basis expansion and weighted rectangles, reducing the time complexity to $O(n)$ for all ranks $r\geq 2$.

Findings

Achieves $O(n)$ time complexity for all ranks $r\geq 2$

Simplifies the process of checking function equivalence

Works within the multi-tape Turing machine model with non-constant-time arithmetic

Abstract

In this note we propose a new algorithm for checking whether two counting functions on a free monoid $M_r$ of rank $r$ are equivalent modulo a bounded function. The previously known algorithm has time complexity $O(n)$ for all ranks $r>2$, but for $r=2$ it was estimated only to be $O(n^2)$. We apply a new approach based on the explicit basis expansion and summation of weighted rectangles, which allows us to construct a much simpler algorithm with time complexity $O(n)$ for any $r\geq 2$. We work in the multi-tape Turing machine model with non-constant-time arithmetic operations.