Finite-State Complexity and the Size of Transducers

TL;DR

This paper explores the hierarchy of finite-state complexity based on the size of minimal finite transducers needed to describe strings, proving that this hierarchy is infinite under standard encoding.

Contribution

It demonstrates that the state-size hierarchy in finite-state complexity is infinite, extending the understanding of complexity measures in finite automata theory.

Findings

The state-size hierarchy with standard encoding is infinite.

Hierarchies from more general computable encodings are also considered.

Finite-state complexity provides a variant of algorithmic information theory.

Abstract

Finite-state complexity is a variant of algorithmic information theory obtained by replacing Turing machines with finite transducers. We consider the state-size of transducers needed for minimal descriptions of arbitrary strings and, as our main result, we show that the state-size hierarchy with respect to a standard encoding is infinite. We consider also hierarchies yielded by more general computable encodings.