On the computational complexity of algebraic numbers: the Hartmanis–Stearns problem revisited
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- by Boris Adamczewski, Julien Cassaigne and Marion Le Gonidec PDF
- Trans. Amer. Math. Soc. 373 (2020), 3085-3115 Request permission
Abstract:
We consider the complexity of integer base expansions of algebraic irrational numbers from a computational point of view. A major contribution in this area is that the base-$b$ expansion of algebraic irrational real numbers cannot be generated by finite automata. Our aim is to provide two natural generalizations of this theorem. Our main result is that the base-$b$ expansion of algebraic irrational real numbers cannot be generated by deterministic pushdown automata. Incidentally, this completely solves the Hartmanis–Stearns problem for the class of multistack machines. We also confirm an old claim of Cobham from 1968 proving that such real numbers cannot be generated by tag machines with dilation factor larger than one. In order to stick with the modern terminology, we also show that the latter generate the same class of real numbers as morphisms with exponential growth.References
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Additional Information
- Boris Adamczewski
- Affiliation: Université Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, F-69622 Villeurbanne Cedex, France
- MR Author ID: 704234
- Email: Boris.Adamczewski@math.cnrs.fr
- Julien Cassaigne
- Affiliation: CNRS, Aix-Marseille Université, Institut de Mathématiques de Marseille, case 907, 163 avenue de Luminy, 13288 Marseille Cedex 9, France
- MR Author ID: 338907
- Email: Julien.Cassaigne@math.cnrs.fr
- Marion Le Gonidec
- Affiliation: Université de la Réunion, Laboratoire d’Informatique et de Mathématiques, Parc technologique universitaire, 2 rue Joseph Wetzell, 97490 Sainte-Clotilde, Réunion France
- MR Author ID: 806099
- Email: marion.le-gonidec@univ-reunion.fr
- Received by editor(s): October 3, 2017
- Received by editor(s) in revised form: October 31, 2018, and March 13, 2019
- Published electronically: February 20, 2020
- Additional Notes: This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under the Grant Agreement No 648132.
- © Copyright 2020 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 3085-3115
- MSC (2010): Primary 11B85, 11J81, 68Q45, 68R15, 11J87
- DOI: https://doi.org/10.1090/tran/7915
- MathSciNet review: 4082234