Intractability of the Reductive Decision Procedure for Unavoidability Testing, a Special Case of Generalized Pattern Matching

We formulate the Generalized Pattern Matching decision 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 the known methods for solving this restricted decision problem. Although the efficiency of standard string searching is wellknown, 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.