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(ni’) ⊈ 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.
"P versus NP and computability theoretic constructions in complexity theory over algebraic structures." J. Symbolic Logic 69 (1) 39 - 64, March 2004. https://doi.org/10.2178/jsl/1080938824