We show that there is a structure of countably infinite signature with $P = N_{2}P$ and a structure of finite signature with $P = N_{1}P$ and $N_{1}P \neq N_{2}P$ . We give a further example of a structure of finite signature with $P \neq N_{1}P$ and $N_{1}P \neq N_{2}P$ . 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 ℒ the class of ℒ-structures with $P = N_{1}P$ is not closed under ultraproducts and obtain as corollaries that this class is not $\delta$ -elementary and that the class of ᵍ-structures with $P \neq N_{1}P$ is not elementary. Finally we prove that for all f dominating all polynomials there is a structure of finite signature with the following properties: $P \neq N_{1}P$ . $N_{1}P \neq N_{2}P$ , the levels $N_{2}TIME(n^{i})$ of $N_{2}P$ and the levels $N_{1}TIME(n^{i})$ of $N_{1}P$ are different for different i, indeed $DTIME(n^{i'}) \nsubseteq N_{2}TIME(n^{i})$ if $i' \textgreater i$ ; $DTIME(f) \nsubseteq N_{2}P$ , and $N_{2}P \nsubseteq DEC$ . DEC is the class of recognizable sets with recognizable complements. So this is an example where the internal structure of $N_{2}P$ 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. (shrink)

Let a trace be a computably enumerable set of natural numbers such that V[m]={n:〈n,m〉∈V}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V^{[m]} = \{n : \langle n, m\rangle \in V \}}$$\end{document} is finite for all m, where 〈.,.〉\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\langle^{.},^{.}\rangle}$$\end{document} denotes an appropriate pairing function. After looking at some basic properties of traces like that there is no uniform enumeration of all traces, we prove varied results on traceability and variants thereof, where (...) a function f:N→N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${f : \mathbb{N} \rightarrow \mathbb{N}}$$\end{document} is traceable via a trace V if for all m,〈f(m),m〉∈V.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${m, \langle f(m), m\rangle \in V.}$$\end{document} Then we turn to lattices Ltr(V)=({W:V⊆WandWatrace},⊆),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\textit{\textbf{L}}_{tr}(V) = (\{W : V \subseteq W \, {\rm and} \, W \, {\rm a} \, {\rm trace}\}, \, \subseteq),$$\end{document}V a trace. Here, we study the close relationship to E=({A:A⊆Nc.e.},⊆)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{E} = (\{A : A \subseteq \mathbb{N} \quad c.e.\}, \subseteq)}$$\end{document}, automorphisms, isomorphisms, and isomorphic embeddings. (shrink)