# The Frobenius and factor universality problems of the Kleene star of a finite set of words

## TL;DR

This paper investigates the computational complexity of the Frobenius and factor universality problems for the Kleene star of finite word sets, proving both are PSPACE-complete and providing bounds on related word lengths.

## Contribution

It establishes the PSPACE-completeness of the Frobenius and factor universality problems and introduces set rewriting systems as a new analytical tool.

## Key findings

- Both problems are PSPACE-complete.
- Exponential bounds on the length of certain words related to the problems.
- Negative resolution of Restivo's longstanding conjecture.

## Abstract

We solve open problems concerning the Kleene star $L^*$ of a finite set $L$ of words over an alphabet $\Sigma$. The \emph{Frobenius monoid} problem is the question for a given finite set of words $L$, whether the language $L^*$ is cofinite. We show that it is PSPACE-complete. We also exhibit an infinite family of sets $L$ such that the length of the longest words not in $L^*$ (when $L^*$ is cofinite) is exponential in the length of the longest words in $L$ and subexponential in the sum of the lengths of words in $L$. The \emph{factor universality} problem is the question for a given finite set of words $L$, whether every word over $\Sigma$ is a factor (substring) of some word from $L^*$. We show that it is also PSPACE-complete. Besides that, we exhibit an infinite family of sets $L$ such that the length of the shortest words not being a factor of any word in $L^*$ is exponential in the length of the longest words in $L$ and subexponential in the sum of the lengths of words in $L$. This essentially settles in the negative the longstanding Restivo's conjecture (1981) and its weak variations. All our solutions base on one shared construction, and as an auxiliary general tool, we introduce the concept of \emph{set rewriting systems}. Finally, we complement the results with upper bounds.