Generalized pattern matching and the complexity of unavoidability testing

Abstract

We formulate the GENERALIZED PATTERN MATCHING problem, a natural extension of string searching capturing regularities across scale. The special case of UNAVOIDABILITY TESTING is obtained for pure generalized patterns by fixing an appropriate family of text strings - the Zimin words. We investigate the complexity of this restricted decision problem. Although the efficiency of standard string searching is well-known, determining the occurrence of generalized patterns in Zimin words does not appear so tractable. We provide an exponential lower bound on any algorithmic decision procedure relying exclusively on the equivalent deletion sequence characterization of unavoidable patterns. We also demonstrate that the four other known necessary conditions are not sufficient to decide pattern unavoidability.