Degrees of Infinite Words, Polynomials, and Atoms (Extended Version)

TL;DR

This paper investigates the hierarchy of infinite words under finite-state transducers, revealing an infinite structure of minimal degrees using linear algebra and analysis, advancing understanding of automata-based transformations.

Contribution

It introduces the first proof of infinitely many atoms in the hierarchy of transducer degrees for infinite words, using novel mathematical methods.

Findings

Existence of infinitely many atoms in transducer degrees.

Development of linear algebra and analysis techniques for automata theory.

Enhanced understanding of the complexity hierarchy of infinite words.

Abstract

We study finite-state transducers and their power for transforming infinite words. Infinite sequences of symbols are of paramount importance in a wide range of fields, from formal languages to pure mathematics and physics. While finite automata for recognising and transforming languages are well-understood, very little is known about the power of automata to transform infinite words. The word transformation realised by finite-state transducers gives rise to a complexity comparison of words and thereby induces equivalence classes, called (transducer) degrees, and a partial order on these degrees. The ensuing hierarchy of degrees is analogous to the recursion-theoretic degrees of unsolvability, also known as Turing degrees, where the transformational devices are Turing machines. However, as a complexity measure, Turing machines are too strong: they trivialise the classification problem by identifying all computable words. Finite-state transducers give rise to a much more fine-grained, discriminating hierarchy. In contrast to Turing degrees, hardly anything is known about transducer degrees, in spite of their naturality. We use methods from linear algebra and analysis to show that there are infinitely many atoms in the transducer degrees, that is, minimal non-trivial degrees.