Forbidden Factors in Finite and Infinite Words

Abstract

We study minimal forbidden factors in finite and infinite words. In the case of a finite word w we consider two parameters: the first counts the minimal forbidden factors of w and the second gives the length of the longest minimal forbidden factor of w. We prove sharp upper and lower bounds for both parameters. We prove also that the second parameter is related to the minimal period of w. In the case of an infinite word w we consider the following two functions: gw(n) that counts the allowed factors of w of length n and fw (n) that counts the minimal forbidden factors of w of length n. We address the following general problem: which informations about the structure of w can be derived from the pair (gw,fw)? We prove that these two functions characterize, up to the automorphism exchanging the two letters, the language of factors of any infinite Sturmian word.