On weak growing context-sensitive grammars

Abstract

We introduce the weak growing context-sensitive grammars (WGCSG): grammars that are growing with respect to a valuation of the positions inside a rule as well as the symbols. We evaluate a string by summing up every product of a symbol value with its position value. For every position valuation which does not agree with the beginning of any exponential function, the corresponding class of WGCSGs characterizes CSL. Such a valuation function is called unsteady. On the other hand all WGCSG related to steady position valuations are linearly inclusion-ordered and characterize the hierarchy of exponential time bounded languages in CSL. This hierarchy collapses iff. any class defined by an unsteady position valuation has a normal form of order 2 (e. g. Cremers' or Kuroda's normal form) any class defined by a steady position valuation is closed under inverse homomorphisms.