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

Petr Kiyashko, Alexey Talambutsa

TL;DR

A new approach is applied 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 $r\geq 2$.

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.

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

TL;DR

A new approach is applied based on the explicit basis expansion and summation of weighted rectangles, which allows us to construct a much simpler algorithm with time complexity for any .

Abstract

In this note we propose a new algorithm for checking whether two counting functions on a free monoid of rank are equivalent modulo a bounded function. The previously known algorithm has time complexity for all ranks , but for it was estimated only to be . 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 for any . We work in the multi-tape Turing machine model with non-constant-time arithmetic operations.
Paper Structure (10 sections, 23 theorems, 63 equations, 1 figure)

This paper contains 10 sections, 23 theorems, 63 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary results
  3. Algorithm for $M_2$
  4. Basis decomposition
  5. A naive $O(n^3)$ algorithm
  6. Shorthand basis
  7. Basis proof
  8. The main algorithm
  9. Generalization for $M_r$, $r \geq 2$
  10. Acknowledgements

Key Result

Theorem 1.1

There exists an algorithm that takes as input two counting functions $f$ and $g$ represented by linear combinations of elementary counting functions over the monoid $M_r$ and checks whether they are equivalent. Furthermore, this algorithm has time complexity $O(r^3 n)$ for integer coefficients and $

Figures (1)

  • Figure 1: Histogram example for the function $\sigma_k^m$.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1: Theorem 1.3 in ggd, Theorem 1.4 in alt
  • Definition 2.2
  • Theorem 2.3: Theorem 1.5 in ggd, Theorem 1.5 in alt
  • Lemma 2.4: Lemma 4.2 in ht-sbornik
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 32 more