Definability and compression

  • Authors:
  • Foto Afrati;Hans Leiß;Michel de Rougemont

  • Affiliations:
  • National Technical University, Athens, Greece;Universität München, CIS, D-80538München, Germany;Université Paris-II & LRI, Bââtiment 490, F-91405 Orsay Cedex, France

  • Venue:
  • Fundamenta Informaticae - Special issue on computing patterns in strings
  • Year:
  • 2002

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Abstract

A compression algorithm takes a finite structure of a class K as input and produces a finite structure of a different class K' as output. Given a property P on the class K defined in a logic L, we study the definability of property P on the class K'. We consider two compression schemes on unary ordered structures (strings), compression by run-length encoding and the classical Lempel-Ziv-78 scheme.First-order properties of strings are first-order on run-length compressed strings, but this fails for images, i.e. 2-dimensional strings. We present simple first-order properties of strings which are not first-order definable on strings compressed with the Lempel-Ziv-78 compression scheme.We show that all properties of strings that are first-order definable on strings are definable on Lempel-Ziv compressed strings in FO(TC), the extension of first-order logic with the transitive closure operator. We define a subclass F of the first-order properties of strings such that if P is a property in F, it is also first-order definable on the Lempel-Ziv compressed strings. Monadic second-order properties of strings, i.e. regular languages, are dyadic second-order definable on Lempel-Ziv compressed strings.