Deenability and Compression

A compression algorithm takes a nite structure of a class K as input and produces a nite structure of a diierent class K' as output. Given a property P on the class K deened in a logic L, we study the deenability of property P on the class K'. We consider two compression schemas on unary ordered structures (words), a naive compression and the classical Lempel-Ziv. First-order properties of strings are rst-order on naively compressed strings, but this fails for images, i.e. 2-dimensional strings. We present simple rst-order properties of strings which are not rst-order deenable on strings compressed with the Lempel-Ziv compression schema. We show that all properties of strings that are rst-order deenable on strings are deenable on Lempel-Ziv compressed strings in FO(TC), the extension of rst-order logic with the transitive closure operator. We deene a subclass C of the rst-order properties of strings such that if L is deened by a property in C, it is also rst-order deenable on the Lempel-Ziv compressed strings.