Computing Weak Periods of Partial Words

Fine and Wilf’s well-known theorem states that any word having periods p, q and length at least p + q − gcd(p, q) also has gcd(p, q), the greatest common divisor of p and q, as a period. Moreover, the length p + q − gcd(p, q) is critical since counterexamples can be provided for shorter words. This result has since been extended to partial words, or finite sequences that may contain a number of “do not know” symbols or “holes.” More precisely, any partial word u with H holes having weak periods p, q and length at least the so-denoted lH(p, q) also has strong period gcd(p, q) provided u is not (H,(p, q))-special. This extension was done for one hole by Berstel and Boasson (where the class of (1,(p, q))-special partial words is empty), for two or three holes by Blanchet-Sadri and Hegstrom, and for an arbitrary number of holes by Blanchet-Sadri. In this paper, we further extend these results, allowing an arbitrary number of weak periods. In addition to speciality, the concepts of intractable period sets and interference between periods play a role.