On Searching Zimin Patterns

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. Althoug...

Observations On Compressed Pattern-Matching with Ranked Variables in Zimin Words

Zimin words are very special finite words which are closely related to the pattern-avoidability problem. This problem consists in testing if an instance of a given pattern with variables occurs in almost all words over any finite alphabet. The problem is not well understood, no polynomial time algorithm is known and its NP-hardness is also not known. The pattern-avoidability problem is equivale...

On Long Words Avoiding Zimin Patterns

A pattern is encountered in a word if some infix of the word is the image of the pattern under some non-erasing morphism. A pattern p is unavoidable if, over every finite alphabet, every sufficiently long word encounters p. A theorem by Zimin and independently by Bean, Ehrenfeucht and McNulty states that a pattern over n distinct variables is unavoidable if, and only if, p itself is encountered...

Tower-type bounds for unavoidable patterns in words

A word w is said to contain the pattern P if there is a way to substitute a nonempty word for each letter in P so that the resulting word is a subword of w. Bean, Ehrenfeucht and McNulty and, independently, Zimin characterised the patterns P which are unavoidable, in the sense that any sufficiently long word over a fixed alphabet contains P . Zimin’s characterisation says that a pattern is unav...

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