Definability and compression

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 schemas on unary ordered structures (words), compression by runlength encoding and the classical Lempel-Ziv. First-order properties of strings are first-order on runlength 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 compression schema. 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 L is defined by a property in ℱ, it is also first-order definable on the Lempel-Ziv compressed strings. Monadic second-order properties of strings are dyadic second order definable on Lempel-Ziv compressed strings