On Non-Complete Sets and Restivo's Conjecture

TL;DR

This paper constructs an infinite series of non-complete word sets with uncompletable words longer than the bounds proposed by Restivo's conjecture, providing counterexamples to the conjecture.

Contribution

It introduces a new series of non-complete sets with uncompletable words exceeding the conjectured length bounds, disproving Restivo's conjecture.

Findings

Counterexamples to Restivo's conjecture with uncompletable words longer than 2k^2

Explicit formulas for minimal uncompletable word lengths

Infinite series of non-complete sets with specific properties

Abstract

A finite set S of words over the alphabet A is called non-complete if Fact(S*) is different from A*. A word w in A* - Fact(S*) is said to be uncompletable. We present a series of non-complete sets S_k whose minimal uncompletable words have length 5k^2 - 17k + 13, where k > 3 is the maximal length of words in S_k. This is an infinite series of counterexamples to Restivo's conjecture, which states that any non-complete set possesses an uncompletable word of length at most 2k^2.