## Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

If you need an accessible version of this item please contact JSTOR User Support

# P versus NP and Computability Theoretic Constructions in Complexity Theory over Algebraic Structures

Gunther Mainhardt
The Journal of Symbolic Logic
Vol. 69, No. 1 (Mar., 2004), pp. 39-64
Stable URL: http://www.jstor.org/stable/30041706
Page Count: 26
If you need an accessible version of this item please contact JSTOR User Support
Preview not available

## Abstract

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.

• 39
• 40
• 41
• 42
• 43
• 44
• 45
• 46
• 47
• 48
• 49
• 50
• 51
• 52
• 53
• 54
• 55
• 56
• 57
• 58
• 59
• 60
• 61
• 62
• 63
• 64