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Zapiski Nauchnykh Seminarov POMI, 2012, Volume 399, Pages 109–127 (Mi znsl5223)  

Diophantine hierarchy

A. A. Knop

Saint-Petersburg State University, Saint-Petersburg, Russia
References:
Abstract: Adelman and Manders (1975) defined the class $\mathrm D$ of “non-deterministic diophantine” languages. A language $\mathrm L$ is in $\mathrm D$ iff there are polynomials $p$ and $q$ such that $x\in\mathrm L\Leftrightarrow\exists\,y_1,\dots,y_ n<2^{q(|x|)}\ p(x,y_1,\dots,y_n)=0$ (in this formula, bit strings are treated as natural numbers). While clearly $\mathrm D$ is a subset of $\mathrm{NP}$, it is unknown whether these classes are equal.
The well-known polynomial hierarchy PH consists of complexity classes constructed on the basis of the class $\mathrm{NP}$. We consider a hierarchy constructed on the basis of $\mathrm D$ in a similar way. We prove that $\mathrm D$ is in the second level of the polynomial hierarchy, and hence all the classes of the two hierarchies are successively contained in each other.
Key words and phrases: Diophantine complexity.
Received: 12.04.2012
English version:
Journal of Mathematical Sciences (New York), 2013, Volume 188, Issue 1, Pages 59–69
DOI: https://doi.org/10.1007/s10958-012-1106-7
Bibliographic databases:
Document Type: Article
UDC: 510.52+519.6
Language: Russian
Citation: A. A. Knop, “Diophantine hierarchy”, Computational complexity theory. Part X, Zap. Nauchn. Sem. POMI, 399, POMI, St. Petersburg, 2012, 109–127; J. Math. Sci. (N. Y.), 188:1 (2013), 59–69
Citation in format AMSBIB
\Bibitem{Kno12}
\by A.~A.~Knop
\paper Diophantine hierarchy
\inbook Computational complexity theory. Part~X
\serial Zap. Nauchn. Sem. POMI
\yr 2012
\vol 399
\pages 109--127
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl5223}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=2945002}
\transl
\jour J. Math. Sci. (N. Y.)
\yr 2013
\vol 188
\issue 1
\pages 59--69
\crossref{https://doi.org/10.1007/s10958-012-1106-7}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-84871936998}
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  • https://www.mathnet.ru/eng/znsl5223
  • https://www.mathnet.ru/eng/znsl/v399/p109
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