More on the Size of Higman-Haines Sets: Effective Constructions

  • Authors:

  • Hermann Gruber;Markus Holzer;Martin Kutrib

  • Affiliations:

  • (Correspd.) Institut für Informatik, Ludwig-Maximilians-Universität München, Oettingenstr. 67, 80538 München, Germany. gruberh@tcs.ifi.lmu.de;Institut für Informatik, Technische Universität München, Boltzmannstr. 3, 85748 Garching bei München, Germany. holzer@in.tum.de;Institut für Informatik, Universität Giessen, Arndtstr. 2, 35392 Giessen, Germany. kutrib@informatik.uni-giessen.de

  • Venue:
  • Fundamenta Informaticae - Machines, Computations and Universality, Part I
  • Year:
  • 2009

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Abstract

A not so well-known result in formal language theory is that the Higman-Haines sets for any language are regular [11, Theorem 4.4]. It is easily seen that these sets cannot be effectively computed in general. The Higman-Haines sets are the languages of all scattered subwords of a given language as well as the sets of all words that contain some word of a given language as a scattered subword. Recently, the exact level of unsolvability of Higman-Haines sets was studied in [8]. Here we focus on language families whose Higman-Haines sets are effectively constructible. In particular, we study the size of descriptions of Higman-Haines sets for the lower classes of the Chomsky hierarchy, namely for the family of regular, linear context-free, and context-free languages. We prove upper and lower bounds on the size of descriptions of these sets for general and unary languages.