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

We show that there is a structure of countably infinite signature with P = N2P and a structure of finite signature with P = N1 P and N1 P N2P. We give a further example of a structure of finite signature with P : NIP and N1 P $ N2P. Together with a result from [10] 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 2 the class of 2-structures with P = NI P is not closed under ultraproducts and obtain as corollaries that this class is not A-elementary and that the class of 2-structures with P $ N1 P 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 / NiP, NIP N2P, the levels N2TIME(ni) of N2P and the levels Ni TIME(ni) of NIP are different for different i, indeed DTIME(n') = 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. s