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The logic of interactive Turing reduction. (English) Zbl 1161.03015
The most notable attempts to explain and develop the constructivistic nature of intuitionistic logic, by the corresponding semantics, were Kleene’s realizability, Gödel’s Dialectica interpretation and Medvedev’s finite-problem semantics. In this paper, by considering the intuitionistic implication as interactive algorithmic reduction, the author obtains a soundness and completeness theorem for the implicative fragment of intuitionistic calculus with respect to the semantics of computability logic. This associated concept of reducibility is equivalent to Turing reducibility when restricted to the traditional, two-step, input/output sorts of problems. Familiarity with the author’s recent paper [G. Japaridze, “In the beginning was game semantics?” in: O. Majer et al. (eds.), Games: Unifying logic, language, and philosophy. Berlin: Springer. Logic, Epistemology, and the Unity of Science 15, 249–350 (2009; Zbl 1171.03015)] is a necessary pre-condition for a complete understanding of this paper.

MSC:
03B70 Logic in computer science
03B20 Subsystems of classical logic (including intuitionistic logic)
03F50 Metamathematics of constructive systems
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