State-Size Hierarchy for Finite-State Complexity

Finite-state complexity is a variant of algorithmic information theory obtained by replacing Turing machines with finite transducers. We consider the number of states needed for transducers used in minimal descriptions of arbitrary strings and, as our main result, show that the state-size hierarchy with respect to a standard encoding is infinite. We consider corresponding hierarchies yielded by more general computable encodings and establish that for a suitably chosen computable encoding every level of the state-size hierarchy can be strict.