## ＜学術雑誌論文＞COMBINATORY LOGIC AND $lambda$-CALCULUS FOR CLASSICAL LOGIC

作成者 作成者名 所属機関 所属機関名 Graduate School of Information Science and Electrical Engineering, Kyushu University 九州大学大学院システム情報科学府 作成者名 所属機関 所属機関名 Department of Information Science, Kyoto University 京都大学情報学科 著者識別子 作成者名 所属機関 所属機関名 Computing and Communications Center, Kyushu University 九州大学情報基盤センター 英語 Research Association of Statistical Sciences 統計科学研究会 2000-12 32 2 105 122 Version of Record open access https://doi.org/10.5109/13496 Bulletin of informatics and cybernetics || 32(2) || p105-122 http://bic.math.kyushu-u.ac.jp/ Since Griffin's work in 1990, classical logic has been an attractive target for extracting computational contents. However, the classical principle used in Griffin's type system is the double-negation...-elimination rule, which prevents one to analyze the intuitionistic part and the purely classical part separately. By formulating a calculus with $mathrm{J}$ (for the elimination rule of falsehood) and $mathrm{P}$ (for Peirce formula which is concerned with purely classical reasoning) combinators, we can separate these two parts. This paper studies the $lambda mathrm{PJ}$ calculus with $mathrm{P}$ and $mathrm{J}$ combinators and the $lambda mathrm{C}$ calculus with $mathrm{C}$ combinator(for the double-negation-elimination rule). We also propose two $lambda$-calculi which correspond to $lambda mathrm{PJ}$ and $lambda mathrm{C}$. We give four classes of reduction rules for each calculus, and systematically study their relationship by simulating reduction rules in one calculus by the corresponding one in the other. It is shown that, by restricting the type of $P$, simulation succeeds for several choices of reduction rules, but that simulating the full calculus $lambda mathrm{PJ}$ in $lambda mathrm{C}$ succeeds only for one class. Some programming examples of our calculi such as encoding of conjunction and disjunction are also given.続きを見る

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PISSN 0286-522X AA10634475 13496 査読有 学術雑誌論文 2009.04.22 2022.01.24