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Incompleteness Theorems, Large Cardinals, and Automata over Finite Words

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Theory and Applications of Models of Computation (TAMC 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10185))

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Abstract

We prove that one can construct various kinds of automata over finite words for which some elementary properties are actually independent from strong set theories like \(T_n =:\mathbf{ZFC} +\) “There exist (at least) n inaccessible cardinals”, for integers \(n\ge 0\). In particular, we prove independence results for languages of finite words generated by context-free grammars, or accepted by 2-tape or 1-counter automata. Moreover we get some independence results for weighted automata and for some related finitely generated subsemigroups of the set \(\mathbb {Z}^{3\times 3}\) of 3-3 matrices with integer entries. Some of these latter results are independence results from the Peano axiomatic system PA.

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Authors and Affiliations

  1. Institut de Mathématiques de Jussieu - Paris Rive Gauche, CNRS et Université Paris 7, Paris, France

    Olivier Finkel

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  1. Olivier Finkel
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Correspondence to Olivier Finkel .

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Editors and Affiliations

  1. Anna University, Chennai, India

    T.V. Gopal

  2. Universität Bern , Bern, Switzerland

    Gerhard Jäger

  3. Universität Bern , Bern, Switzerland

    Silvia Steila

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Finkel, O. (2017). Incompleteness Theorems, Large Cardinals, and Automata over Finite Words. In: Gopal, T., Jäger , G., Steila, S. (eds) Theory and Applications of Models of Computation. TAMC 2017. Lecture Notes in Computer Science(), vol 10185. Springer, Cham. https://doi.org/10.1007/978-3-319-55911-7_17

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  • DOI: https://doi.org/10.1007/978-3-319-55911-7_17

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  • Online ISBN: 978-3-319-55911-7

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