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March 2004 P versus NP and computability theoretic constructions in complexity theory over algebraic structures
Gunther Mainhardt
J. Symbolic Logic 69(1): 39-64 (March 2004). DOI: 10.2178/jsl/1080938824

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 the class of -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 -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.

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Gunther Mainhardt. "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

Information

Published: March 2004
First available in Project Euclid: 2 April 2004

zbMATH: 1067.03051
MathSciNet: MR2039344
Digital Object Identifier: 10.2178/jsl/1080938824

Rights: Copyright © 2004 Association for Symbolic Logic

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Vol.69 • No. 1 • March 2004
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