Many concepts and two logics of algorithmic reduction

TL;DR

This paper explores two types of algorithmic reductions in computability logic, showing how removing the contraction rule transforms the logic of unlimited reusage into that of no reusage, and introduces new bounded recurrence operations.

Contribution

It demonstrates that removing contraction from the logic captures the second reduction type and introduces finite, bounded recurrence operations in computability logic.

Findings

Removing contraction yields the no-reusage reduction logic.

Different natural versions of Turing reduction share the same logical behavior.

Introduces finite and bounded recurrence operations.

Abstract

Within the program of finding axiomatizations for various parts of computability logic, it was proved 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. An online source on computability logic can be found at http://www.cis.upenn.edu/~giorgi/cl.html