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The number of proofs for a BCK-formula

Published online by Cambridge University Press:  12 March 2014

Yuichi Komori  and
Sachio Hirokawa
Yuichi Komori
Affiliation:
Department of Mathematics, Shizuoka University, Shizuoka 422, Japan
Sachio Hirokawa
Affiliation:
Department of Computer Science, College of General Education, Kyushu University, Fukuoka 810, Japan, E-mail: hirokawa@ec.kyushu-u.ac.jp
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Extract

In this note, we give a necessary and sufficient condition for a BCK-formula to have the unique normal form proof.

We call implicational propositional formulas formulas for short. BCK-formulas are the formulas which are derivable from axioms B = (a → b) → (c → a) → c → b, C = (a→b→c)→b→a→c, and K = a→b→a by substitution and modus ponens. It is known that the property of being a BCK-formula is decidable (Jaskowski [11, Theorem 6.5], Ben-Yelles [3, Chapter 3, Theorem 3.22], Komori [12, Corollary 6]). The set of BCK-formulas is identical to the set of provable formulas in the natural deduction system with the following two inference rules.

Here γ occurs at most once in (→I). By the formulae-as-types correspondence [10], this set is identical to the set of type-schemes of closed BCK-λ-terms. (See [5].) A BCK-λ-term is a λ-term in which no variable occurs twice. Basic notion concerning the type assignment system can be found [4]. Uniqueness of normal form proofs has been known for balanced formulas. (See [2,14].) It is related to the coherence theorem in cartesian closed categories. A formula is balanced when no variable occurs more than twice in it. It was shown in [8] that the proofs of balanced formulas are BCK-proofs. Relevantly balanced formulas were defined in [9], and it was proved that such formulas have unique normal form proofs. Balanced formulas are included in the set of relevantly balanced formulas.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 58 , Issue 2 , June 1993 , pp. 626 - 628
Copyright
Copyright © Association for Symbolic Logic 1993

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References

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