Minimal Forbidden Factors of Circular Words

TL;DR

This paper explores the properties and computation of minimal forbidden factors in circular words, extending existing algorithms and definitions to this context, and analyzing specific cases like circular Fibonacci words.

Contribution

It generalizes the algorithm for minimal forbidden factors from linear words to circular words and introduces the factor automaton for circular words.

Findings

Extended Crochemore's algorithm to circular words

Defined the factor automaton for circular words

Analyzed minimal forbidden factors of circular Fibonacci words

Abstract

Minimal forbidden factors are a useful tool for investigating properties of words and languages. Two factorial languages are distinct if and only if they have different (antifactorial) sets of minimal forbidden factors. There exist algorithms for computing the minimal forbidden factors of a word, as well as of a regular factorial language. Conversely, Crochemore et al. [IPL, 1998] gave an algorithm that, given the trie recognizing a finite antifactorial language $M$, computes a DFA recognizing the language whose set of minimal forbidden factors is $M$. In the same paper, they showed that the obtained DFA is minimal if the input trie recognizes the minimal forbidden factors of a single word. We generalize this result to the case of a circular word. We discuss several combinatorial properties of the minimal forbidden factors of a circular word. As a byproduct, we obtain a formal definition of the factor automaton of a circular word. Finally, we investigate the case of minimal forbidden factors of the circular Fibonacci words.