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 ob- tained 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 pat- terns in Zimin words does not appear so tractable. We provide an expo- nential lower bound on any algorithmic decision procedure relying exclu- sively on the equivalent deletion sequence characterization of unavoid- able patterns. We also demonstrate that the four other known necessary conditions are not sufficient to decide pattern unavoidability.