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Balanced formulas, BCK-minimal formulas and their proofs

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Logical Foundations of Computer Science — Tver '92 (LFCS 1992)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 620))

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Abstract

The structure of normal form proof figures are investigated for implicatonal formulas in BCK-logic, in which each assumption can be used at most once. Proof figures are identified with λ-terms. A formula is balanced iff no type variable occurs more than twice in it. It is known that proof figure in βη-normal is unique for balanced formulas. In this paper, it is shown that closed λ-terms in β-normal form having balanced types are BCK-λ-terms in which each variable occurs at most once. A formula is BCK-minimal iff it is BCK-provable and it is not a non-trivial substitution instance of other BCK-provable formula. It is also shown that the set BCK-minimal formulas is identical to the set of principal type-schemes of BCK-λ-terms in βη-normal form.

Supported by a Grant-in-Aid for Encouragement of Young Scientists No.02740115 and No.03750298 of the Ministry of Education

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Authors and Affiliations

  1. Department of Computer Science College of General Education, Kyushu University, Ropponmatsu 4-1-1, 810, Fukuoka, Japan

    Sachio Hirokawa

Authors
  1. Sachio Hirokawa
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Anil Nerode Mikhail Taitslin

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© 1992 Springer-Verlag Berlin Heidelberg

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Hirokawa, S. (1992). Balanced formulas, BCK-minimal formulas and their proofs. In: Nerode, A., Taitslin, M. (eds) Logical Foundations of Computer Science — Tver '92. LFCS 1992. Lecture Notes in Computer Science, vol 620. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0023874

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  • DOI: https://doi.org/10.1007/BFb0023874

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-55707-4

  • Online ISBN: 978-3-540-47276-6

  • eBook Packages: Springer Book Archive

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