Efficient computations with counting functions on free groups and free monoids

TL;DR

This paper introduces efficient algorithms for analyzing counting functions and quasimorphisms on free groups and monoids, achieving linear and near-linear time complexities for key decision problems.

Contribution

It provides the first algorithms with proven linear and near-linear time complexities for deciding boundedness and cohomology of counting functions on free groups and monoids.

Findings

Algorithms operate in linear space and time for integer coefficients with rank ≥ 3.

Rational coefficient algorithms run in O(N log N) time, matching addition complexity.

The methods build on previous characterizations of bounded counting functions.

Abstract

We present efficient algorithms to decide whether two given counting functions on non-abelian free groups or monoids are at bounded distance from each other and to decide whether two given counting quasimorphisms on non-abelian free groups are cohomologous. We work in the multi-tape Turing machine model with non-constant time arithmetic operations. In the case of integer coefficients we construct an algorithm of linear space and time complexity (assuming that the rank is at least $3$ in the monoid case). In the case of rational coefficients we prove that the time complexity is $O(N\log N)$, where $N$ denotes the size of the input, i.e. it is as fast as addition of rational numbers (implemented using the Harvey--van der Hoeven algorithm for integer multiplication). These algorithms are based on our previous work which characterizes bounded counting functions.