A1 Refereed original research article in a scientific journal
Degrees of Infinite Words, Polynomials and Atoms
Authors: Endrullis J, Karhumaki J, Klop JW, Saarela A
Publisher: WORLD SCIENTIFIC PUBL CO PTE LTD
Publication year: 2018
Journal: International Journal of Foundations of Computer Science
Journal name in source: INTERNATIONAL JOURNAL OF FOUNDATIONS OF COMPUTER SCIENCE
Journal acronym: INT J FOUND COMPUT S
Volume: 29
Issue: 05
First page : 825
Last page: 843
Number of pages: 19
ISSN: 0129-0541
eISSN: 1793-6373
DOI: https://doi.org/10.1142/S0129054118420066
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/35741209
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.
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