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Weak theories of concatenation and minimal essentially undecidable theories

An encounter of \({\mathsf{WTC}}\) and \({\mathsf{S2S}}\)

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Abstract

We consider weak theories of concatenation, that is, theories for strings or texts. We prove that the theory of concatenation \({\mathsf{WTC}^{-\varepsilon}}\), which is a weak subtheory of Grzegorczyk’s theory \({\mathsf{TC}^{-\varepsilon}}\), is a minimal essentially undecidable theory, that is, the theory \({\mathsf{WTC}^{-\varepsilon}}\) is essentially undecidable and if one omits an axiom scheme from \({\mathsf{WTC}^{-\varepsilon}}\), then the resulting theory is no longer essentially undecidable. Moreover, we give a positive answer to Grzegorczyk and Zdanowski’s conjecture that ‘The theory \({\mathsf{TC}^{-\varepsilon}}\) is a minimal essentially undecidable theory’. For the alternative theories \({\mathsf{WTC}}\) and \({\mathsf{TC}}\) which have the empty string, we also prove that the each theory without the neutrality of \({\varepsilon}\) is to be such a theory too.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, Chiba University, 1-33 Yayoi-cho, Inage, Chiba, 263-8522, Japan

    Kojiro Higuchi

  2. Yonago National College of Technology, 4448, Hikona, Yonago, Tottori, Japan

    Yoshihiro Horihata

Authors
  1. Kojiro Higuchi
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  2. Yoshihiro Horihata
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Correspondence to Yoshihiro Horihata.

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We would like to dedicate our study to Professor Andrzej Grzegorczyk.

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Higuchi, K., Horihata, Y. Weak theories of concatenation and minimal essentially undecidable theories. Arch. Math. Logic 53, 835–853 (2014). https://doi.org/10.1007/s00153-014-0391-x

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  • DOI: https://doi.org/10.1007/s00153-014-0391-x

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