Abstract
The paper presents two situations where unit-cost complexity results are closely related with results from the classical computability. • In Section 2 we study an important theorem by Koiran and Fournier from an axiomatic point of view. It is proved that the algebraic Knapsack problem belongs to P over some ordered abelian semi-group iff P = NP classically. In this case there would exist a unit-cost machine solving the algebraic Knapsack problem over all ordered abelian semi-groups in some uniform polynomial time. • In Section 3 we apply the theorem of Matiyasevich in order to construct a ring with P ≠ NBP ≠ NP and such that its polynomial hierarchy does not collapse at any level.
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