Hirokawa, Sachio

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TitleNumber of Proofs for ImplicationalFormulas(MATHEMATICAL LOGIC AND ITSAPPLICATIONS)Author(s)Hirokawa, SachioCitation数理解析研究所講究録 (1991), 772: 72-74Issue Date1991-12URL http://hdl.handle.net/2433/82383RightType Departmental Bulletin PaperTextversionpublisherKyoto University72Number of Proofs for Implicational Formulas(Sachio Hirokawa)*An algorithm is shown which determines the number $0,1,$ $\cdots,$ $\infty$ of normal form proofsfor implicational formulas. The number of proofs had not been studied well. Concerningto BCK-logic, it is proved by Komori and Hirokawa [3] that the number is identical to thenumber of BCK-minimal formulas of $\alpha$ . For general implicational formulas in intuitionisticlogic, Ben-Yelles [1] showed an algorithm which enumerates all the normal form proofs for$\alpha$ when $\alpha$ has finitely many proofs. But we cannot use the algorithm to decide whether$\alpha$ has infinitely many proofs or not. We show a limit of proof search to decide whether $\alpha$has infinitely many proofs.Given an implicational formula $\alpha$ , we denote by $|\alpha|$ the number of occurrences ofpropositional variables and the implicational symbol $‘arrow$ . We consider proof figures inthe intuitionistic logic in Natural Deduction System (NJ) [4]. We denote by proof $(\alpha)$the set of normal form proofs of $\alpha$ . The cardinality of proof $(\alpha)$ is denoted by $\# P^{roof(\alpha)}$ .The depth of a thread in a proof $\pi$ is the number of minimum formula occurrences in thethread. The depth of $\pi$ , denoted by depth $(\pi)$ , is the maximal depth among all the threadsin $\pi$ . According to the formulae-as-types correspondence [2], a normal form proof $\pi$ canbe represented by a closed A-term $\Lambda l$ in $\beta$-normal form. Then the depth $(7\ulcorner)$ is identicalto the depth of Bohm-tree of M.Theorem 1 Given an implicational formula $\alpha$ ,$\# proof(\alpha)=\infty$’Department of Computer Science, College of General Education, Kyushu University, Fukuoka 810JAPAN. ( $e$ -mail: $hirokawa\Phi ec$ . kyushu-u. ac. jp) Supported by a Grant-in-Aid for Encouragement ofYoung Scientists No.02740115 of the Ministry of Education.1772 1991 72-7473iff there is a normal form proof $\pi\in proof(\alpha)$ such that(1) depth $(\pi)\leq|\alpha|2^{|\alpha|+1}$ and(2) $\pi$ contains a thread in which a formula $\xi$ occurs twice as minimum formula occur-rence.$\pi\{$$\dot{\xi}\dot{\xi}\dot{\alpha}\}\pi_{1}I^{\pi_{2}}$Outline of proof. If-part is trivial. In fact, we can replace $\pi_{1}$ by $\pi_{2}$ . We can applythis rewriting successively. Thus we $have\# proof(\alpha)=\infty$ . To prove only-if-part, assumethat $\# proof(\alpha)=\infty$ . Then there is a proof $\pi\in proof(\alpha)$ which contains a thread withdepth $\geq 2d$ , where $d=|\alpha|2^{|\alpha|}$ . Then the thread contains more than 2 $d$ ninimumformula occurrences. Let $\xi$ be an arbitrary minimum formula occurrence in the threadand $\{\delta_{1}, \cdots, \delta_{n}\}$ the assumption set for the sub-proof for $\xi$ . By the sub-formula property,all of $\xi,$ $\delta_{1},$ $\cdots,$ $\delta_{n}$ are sub-formulas of $\alpha$ . So we have at most $d$ such pairs $(\xi, \{\delta_{1}, \cdots, \delta_{n}\})$ .Since the depth of the thread is longer than 2 $d$ , it contains three occurrences of the sameminimum formula occurrence $\xi$ with the same assumption set $\{\delta_{1}, \cdots, \delta_{n}\}$ . Let $\pi_{1},$ $\pi_{2}$ ,and $\pi_{3}$ be sub-proof for such occurrences of $\xi$ which $\pi$; appears above $\pi_{i+1}(i=1,2)$ . Thenwe can replace $\pi_{2}$ by $\pi_{1}$ obtaining a smaller proof of $\alpha$ . We can apply this transformationuntil we obtain a proof of $\alpha$ with depth $\leq 2d$ . ITheorem 2 There is an algorithm which determines $\# proof(\alpha)$ for implicational formula$\alpha$ ,Proof. Consider the set of normal form proofs of $\alpha$ with depth $\leq|\alpha|2^{|\alpha|+1}$ . Notethat the set is finite. If this set contains some $\pi$ which satisfies (2) of Theorem 1, then$\#P^{roof(\alpha)}=\infty$ . Otherwise $\# proof(\alpha)$ is finite. I274Theorem 1 without (1) is proved in Ben-Yelles [1]. Proof of Theorem 1 would remindsome readers the similarity to the proof of uvwxy-theorem and infinity test for context freelanguages. Further work shall be necessary on this similarity.References[1] Ben-Yelles, C.-B., Type-assignment in the lambda-calculus, PhD thesis, UniversityCollege, Swansea, 1979.[2] Howard, W.A., The formulae-as-types notion of construction, in: Hindley and SeldinEd., To H.B. Curry, Essays on Combinatory Logic, Lambda Calculus and Formalism(Academic Press, 1980) 479-490.[3] Komori,Y., Hirokawa, S., Number of proofs for BCK-formulas, preprint.[4] Prawitz, D., Natural Deduction (Almqvist and Wiksell, Stockholm, 1965)3

Number of Proofs for Implicational Formulas(MATHEMATICAL LOGIC AND ITS APPLICATIONS)

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Number of Proofs for Implicational Formulas(MATHEMATICAL LOGIC AND ITS APPLICATIONS)

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