Searcharxiv⌕ Search

arXiv subjects

Sergey V. Yakhontov

Publications and source records attributed to Sergey V. Yakhontov.

6 recordsLinked to original sources
arXiv2017

P = NP

The present work proves that P=NP. The proof, presented in this work, is a constructive one: the program of a polynomial time deterministic multi-tape Turing machine M_ExistsAcceptingPath, that determines if there exists an accepting computational path of a polynomial time non-deterministic single-tape Turing machine M_NP, is constructed (machine M_ExistsAcceptingPath is different for each Turing machine M_NP). Machine M_ExistsAcceptingPath is based on reduction to problem LP (linear programming) instead of reduction to problem 3-CNF-SAT which is commonly used. In more detail, machine M_AcceptingPath uses a reduction of the initial string problem to another string problem TCPE (defined in the paper) that is NP-complete and decidable in polynomial time. The time complexity of machine M_ExistsAcceptingPath is O(t(n)^{272}) wherein t(n) is an upper bound of the time complexity of machine M_NP.

cs.CC↗
arXiv2014

Notes on space complexity of integration of computable real functions in Ko-Friedman model

In the present paper it is shown that real function $g(x)=\int_{0}^{x}f(t)dt$ is a linear-space computable real function on interval $[0,1]$ if $f$ is a linear-space computable $C^2[0,1]$ real function on interval $[0,1]$, and this result does not depend on any open question in the computational complexity theory. The time complexity of computable real functions and integration of computable real functions is considered in the context of Ko-Friedman model which is based on the notion of Cauchy functions computable by Turing machines. In addition, a real computable function $f$ is given such that $\int_{0}^{1}f\in FDSPACE(n^2)_{C[a,b]}$ but $\int_{0}^{1}f\notin FP_{C[a,b]}$ if $FP\ne#P$.

cs.CC↗
arXiv2014

FP//LINSPACE computability of Riemann zeta function in Ko-Friedman model

In the present paper, we construct an algorithm for the evaluation of real Riemann zeta function $ζ(s)$ for all real $s$, $s>1$, in polynomial time and linear space on Turing machines in Ko-Friedman model. The algorithms is based on a series expansion of real Riemann zeta function $ζ(s)$ (the series globally convergents) and uses algorithms for the evaluation of real function $(1+x)^h$ and hypergeometric series in polynomial time and linear space. The algorithm from the present paper modified in an obvious way to work with the complex numbers can be used to evaluate complex Riemann zeta function $ζ(s)$ for $s=σ+\mathbf{i}t$, $σ\ne 1$ (so, also for the case of $σ<1$), in polynomial time and linear space in $n$ wherein $2^{-n}$ is a precision of the computation; the modified algorithm will be also polynomial time and linear space in $\lceil \log_2(t)\rceil$ and exponential time and exponential space in $\lceil \log_2(σ)\rceil$.

cs.CC↗
arXiv2014

Computable real function F such that F is not polynomial time computable on [0,1]

A computable real function F on [0,1] is constructed such that there exists an exponential time algorithm for the evaluation of the function on [0,1] on Turing machine but there does not exist any polynomial time algorithm for the evaluation of the function on [0,1] on Turing machine (moreover, it holds for any rational point on (0,1))

cs.CC↗
arXiv2012

Time- and space-efficient evaluation of the complex exponential function using series expansion

An algorithm for the evaluation of the complex exponential function is proposed which is quasi-linear in time and linear in space. This algorithm is based on a modified binary splitting method for the hypergeometric series and a modified Karatsuba method for the fast evaluation of the exponential function. The time complexity of this algorithm is equal to that of the ordinary algorithm for the evaluation of the exponential function based on the series expansion: O(M(n)log(n)^2).

cs.DS↗