Many Concepts and Two Logics of Algorithmic Reduction
 Giorgi Japaridze
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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 Gentzenstyle 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.
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 Title
 Many Concepts and Two Logics of Algorithmic Reduction
 Journal

Studia Logica
Volume 91, Issue 1 , pp 124
 Cover Date
 20090201
 DOI
 10.1007/s1122500991647
 Print ISSN
 00393215
 Online ISSN
 15728730
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Computability logic
 Intuitionistic logic
 Affine logic
 Linear logic
 Interactivecomputation
 Game semantics
 Authors

 Giorgi Japaridze ^{(1)} ^{(2)}
 Author Affiliations

 1. Institute of Artificial Intelligence, Xiamen University, Xiamen, Fujian, China
 2. Department of Computing Sciences, Villanova University, 800 Lancaster Avenue, Villanova, PA, 19085, USA