P versus NP and computability theoretic constructions in complexity theory over algebraic structures

Abstract

We show that there is a structure of countably infinite signature with P=N₂P and a structure of finite signature with P=N₁P and N₁P ≠ N₂P. We give a further example of a structure of finite signature with P ≠ N₁P and N₁P ≠ N₂P. Together with a result from [Koiran] this implies that for each possibility of P versus NP over structures there is an example of countably infinite signature. Then we show that for some finite ℒ the class of ℒ-structures with P=N₁P is not closed under ultraproducts and obtain as corollaries that this class is not Δ-elementary and that the class of ℒ-structures with P ≠ N₁P is not elementary. Finally we prove that for all f dominating all polynomials there is a structure of finite signature with the following properties: P ≠ N₁P, N₁P ≠ N₂P, the levels N₂TIME(nⁱ) of N₂P and the levels N₁TIME(nⁱ) of N₁P are different for different i, indeed DTIME(n^i’) ⊈ N₂TIME(nⁱ) if i’>i, DTIME(f) ⊈ N₂P, and N₂P ⊈ DEC. DEC is the class of recognizable sets with recognizable complements. So this is an example where the internal structure of N₂P is analyzed in a more detailed way. In our proofs we use methods in the style of classical computability theory to construct structures except for one use of ultraproducts.