Many Concepts and Two Logics of Algorithmic Reduction
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Within the program of finding axiomatizations for various parts of computability logic, it was proven earlier that the logic of interactive Turing reduction is exactly the implicative fragment of Heyting’s intuitionistic calculus. That sort of reduction permits unlimited reusage of the computational resource represented by the antecedent. An at least equally basic and natural sort of algorithmic reduction, however, is the one that does not allow such reusage. The present article shows that turning the logic of the first sort of reduction into the logic of the second sort of reduction takes nothing more than just deleting the contraction rule from its Gentzen-style axiomatization. The first (Turing) sort of interactive reduction is also shown to come in three natural versions. While those three versions are very different from each other, their logical behaviors (in isolation) turn out to be indistinguishable, with that common behavior being precisely captured by implicative intuitionistic logic. Among the other contributions of the present article is an informal introduction of a series of new — finite and bounded — versions of recurrence operations and the associated reduction operations.
KeywordsComputability logic Intuitionistic logic Affine logic Linear logic Interactivecomputation Game semantics
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- 2.Blass A.: ‘Degrees of indeterminacy of games’. Fundamenta Mathematicae 77, 151–166 (1972)Google Scholar
- 4.Blass, A., ‘Resource consciousness in classical logic’, in G. Mints and R. Muskens (eds.), Games, Logic, and Constructive Sets (Proceedings of LLC9, the 9th conference on Logic, Language, and Computation, held at CSLI), 2003, pp. 61–74.Google Scholar
- 11.Japaridze, G., ‘Computability logic: a formal theory of interaction’, in D. Goldin, S. Smolka and P. Wegner (eds.), Interactive Computation: The New Paradigm, Springer Verlag, 2006, pp. 183–223.Google Scholar
- 14.Japaridze G.: ‘Intuitionistic computability logic’. Acta Cybernetica 18, 77–113 (2007)Google Scholar
- 18.Japaridze, G., ‘In the beginning was game semantics’, in O. Majer, A.-V. Pietarinen and T. Tulenheimo (eds.), Games: Unifying Logic, Language and Philosophy, Springer Verlag, 2009, pp. 249–250.Google Scholar
- 19.Japaridze, G., ‘Towards applied theories based on computability logic’, Preprint is available at http://arxiv.org/abs/0805.3521
- 20.Vereshchagin, N., ‘Japaridze’s computability logic and intuitionistic propositional calculus’, Moscow State University preprint (Russian), 2006. Available at http://lpcs.math.msu.su/?ver/papers/japaridze.ps