Computational Flows in Arithmetic
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A computational flow is a pair consisting of a sequence of computational problems of a certain sort and a sequence of computational reductions among them. In this paper we will develop a theory for these computational flows and we will use it to make a sound and complete interpretation for bounded theories of arithmetic. This property helps us to decompose a first order arithmetical proof to a sequence of computational reductions by which we can extract the computational content of low complexity statements in some bounded theories of arithmetic such as IΔ_0, T^k_n, IΔ_0+EXP and PRA. In the last section, by generalizing term-length flows to ordinal-length flows, we will extend our investigation from bounded theories to strong unbounded ones such as IΣ_n and PA+TI(α) and we will capture their total NP search problems as a consequence.
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