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**P versus NP and computability theoretic constructions in complexity theory over algebraic structures.**
*(English)*
Zbl 1067.03051

Summary: We show that there is a structure of countably infinite signature with \({\text P} = {\text N}_2{\text P}\) and a structure of finite signature with \({\text P} = {\text N}_1{\text P}\) and \({\text N}_1{\text P} \neq {\text N}_2{\text P}\). We give a further example of a structure of finite signature with \({\text P}\neq {\text N}_1{\text P}\) and \({\text N}_1{\text P}\neq {\text N}_2{\text P}\). Together with a result of P. Koiran [Theor. Comput. Sci. 133, 35–47 (1994; Zbl 0822.68028)] this implies that for each possibility of \({\text P}\) versus \(\text{NP}\) over structures there is an example of countably infinite signature. Then we show that for some finite \(\mathcal L\) the class of \(\mathcal L\)-structures with \({\text P} = {\text N}_1 {\text P}\) is not closed under ultraproducts and obtain as corollaries that this class is not \(\Delta\)-elementary and that the class of \(\mathcal L\)-structures with \({\text P}\neq {\text N}_1{\text P}\) is not elementary. Finally we prove that for all \(f\) dominating all polynomials there is a structure of finite signature with the following properties: \({\text P}\neq {\text N}_1{\text P}\), \({\text N}_1{\text P}\neq {\text N}_2{\text P}\), the levels \({\text N}_2\text{TIME}(n^i)\) of \({\text N}_2{\text P}\) and the levels \({\text N}_1\text{TIME}(n^i)\) of \({\text N}_1{\text P}\) are different for different \(i\), indeed \(\text{DTIME}(n^{i'})\nsubseteq {\text N}_2\text{TIME}(n^i)\) if \(i' > i\); \(\text{DTIME}(f)\nsubseteq {\text N}_2{\text P}\), and \({\text N}_2{\text P}\subseteq \text{DEC}\). \(\text{DEC}\) is the class of recognizable sets with recognizable complements. So this is an example where the internal structure of \({\text N}_2{\text P}\) 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.

### MSC:

03D15 | Complexity of computation (including implicit computational complexity) |

03C10 | Quantifier elimination, model completeness, and related topics |

68Q15 | Complexity classes (hierarchies, relations among complexity classes, etc.) |

### Keywords:

complexity classes### Citations:

Zbl 0822.68028
Full Text:
DOI

### References:

[1] | Model theory (1990) |

[2] | Models and ultraproducts (1969) · Zbl 0179.31402 |

[3] | Structural complexity 1, 2 (1988) |

[4] | DOI: 10.1137/0204037 · Zbl 0323.68033 |

[5] | Recursively enumerable sets and degrees (1987) |

[6] | A model-theoretic proof for P NP over all infinite abelian groups 67 pp 235– (2002) · Zbl 1014.03040 |

[7] | Mathematical logic (1984) |

[8] | DOI: 10.1002/1521-3870(200101)47:1<67::AID-MALQ67>3.0.CO;2-V · Zbl 0967.03034 |

[9] | DOI: 10.1016/0304-3975(93)00063-B · Zbl 0822.68028 |

[10] | DOI: 10.1002/malq.19980440311 · Zbl 0918.03024 |

[11] | DOI: 10.1002/malq.19980440102 · Zbl 0911.03023 |

[12] | Computability and complexity over structures of finite type (1995) |

[13] | On structures with proper polynomial time hierarchy (2002) |

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