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On Polynomial Time Computable Numbers [article]

Tetsushi Matsui
2006 arXiv   pre-print
It will be shown that the polynomial time computable numbers form a field, and especially an algebraically closed field.  ...  Computable number means that it can be approximated by a computable function, and polynomial time computable number by a polynomial time computable function.  ...  The Field of Polynomial Time Computable Numbers In this section, we show that the whole polynomial time computable numbers C P is a field.  ... 
arXiv:cs/0608067v1 fatcat:xjpavj7rofaqvbsbtnc2rugeba

Polynomial time quantum algorithm for the computation of the unit group of a number field

Arthur Schmidt, Ulrich Vollmer
2005 Proceedings of the thirty-seventh annual ACM symposium on Theory of computing - STOC '05  
This algorithm is applied to compute the unit group of finite extensions of Q. Execution time for fixed field degree over Q is polynomial in the discriminant of the field.  ...  We present a quantum algorithm for the computation of the irrational period lattice of a function on Z n which is periodic in a relaxed sense.  ...  His algorithm executes in polynomial time provided function values can be computed within time polynomial in the period and the size of the arguments.  ... 
doi:10.1145/1060590.1060661 dblp:conf/stoc/SchmidtV05 fatcat:t4zx37j7tjhzrgdkh7dugkhllm

Market Equilibria in Polynomial Time for Fixed Number of Goods or Agents

Nikhil R. Devanur, Ravi Kannan
2008 2008 49th Annual IEEE Symposium on Foundations of Computer Science  
A final note on computing approximate 4 equilibria: for constant number of goods, there are various algorithms that compute an approximate equilibria in time exponential in the number of goods, such as  ...  equilibrium in polynomial time.  ...  Running Time: If m is a constant, then the number of polynomials and their degree in each step are polynomials. And the number of variables is a constant. Hence the running time is polynomial.  ... 
doi:10.1109/focs.2008.30 dblp:conf/focs/DevanurK08 fatcat:er6ei5fxxbcdfonwrz7laqqn3q

A polynomial time algorithm for computing the HNF of a module over the integers of a number field

Jean-François Biasse, Claus Fieker
2012 Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation - ISSAC '12  
The modular strategy was conjectured to run in polynomial time by Cohen, but so far, no such proof was available in the literature.  ...  We present a variation of the modular algorithm for computing the Hermite Normal Form of an OK -module presented by Cohen [2], where OK is the ring of integers of a number field K.  ...  This result is significant since other applications rely on the possibility of computing the HNF of an OK -module in polynomial time.  ... 
doi:10.1145/2442829.2442844 dblp:conf/issac/BiasseF12 fatcat:e6kym36d35h2fgxn6kpp5finby

A polynomial-time algorithm to approximately count contingency tables when the number of rows is constant

Mary Cryan, Martin Dyer
2002 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing - STOC '02  
In this case, we show that the solution lies in approximating the volume of a single convex body, a problem which is known to be solvable in polynomial time [5] .  ...  We show that the number of contingency tables can be expressed as the weighted sum of a polynomial number of new instances of the problem, where each instance consists of some new row sums and the original  ...  By dynamic programming, we can count the number of contingency tables on the small columns for any given list of partial row sums in polynomial time.  ... 
doi:10.1145/509907.509946 dblp:conf/stoc/CryanD02 fatcat:d7mfmzwftfgnfnk3nwk3raurju

A Polynomial Time Presburger Criterion and Synthesis for Number Decision Diagrams

J. Leroux
20th Annual IEEE Symposium on Logic in Computer Science (LICS' 05)  
In this paper, we show how to determine in polynomial time whether a NDD represents a Presburger-definable set, and we provide in this positive case a polynomial time algorithm that constructs from the  ...  Number Decision Diagrams (NDD) are the automatabased symbolic representation for manipulating sets of integer vectors encoded as strings of digit vectors (least or most significant digit first).  ...  Acknowledgment: We thank Pierre McKenzie for his support and for his interesting remarks on so many versions of this paper.  ... 
doi:10.1109/lics.2005.2 dblp:conf/lics/Leroux05 fatcat:euvwnjx64ffdblzvvfk6wjhqpm

A polynomial-time algorithm for the perfect phylogeny problem when the number of character states is fixed

R. Agarwala, D. Fernandez-Baca
Proceedings of 1993 IEEE 34th Annual Foundations of Computer Science  
C1) S c V ( T ) c A1 x . . . x A" W e present a polynomial-time algorithm f o r determining whether a set of species, described by the characters they exhibit, has a perfect phylogeny, assuming the maximum  ...  number of possible states f o r a charact e r is fixed.  ...  Checking whether a split (GI, G2) is of type I can be achieved O(nm) time as follows. First, compute M(G1) and the common state between G1 and G2 on each c E M(G1).  ... 
doi:10.1109/sfcs.1993.366873 dblp:conf/focs/AgarwalaF93 fatcat:tmrkc7pevvfmvaajvpoeift33i

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

Sergey V. Yakhontov
2014 arXiv   pre-print
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))  ...  Theorem 3 . 3 Real function F is not a polynomial time computable real function on [0, 1] of real numbers.  ... 
arXiv:1404.7053v1 fatcat:dr5axojtu5akjir2usxolkpkvm

Computational complexity of real functions

Ker-I. Ko, Harvey Friedman
1982 Theoretical Computer Science  
Recursive analysis, the theory of computation of functions on real numbers, has been studied from various aspects.  ...  We study the complexity of the roots and the differentiability of polynomial time computable real functions.  ...  We will see later that all algebraic numbers are polynomial time computable. e and 'IT are polynomial time computable. However, not every recursive real number is polynomial time computable.  ... 
doi:10.1016/s0304-3975(82)80003-0 fatcat:njl6vcb7kjhx3owlluzvmis4ei

Polynomial running times for polynomial-time oracle machines [article]

Akitoshi Kawamura, Florian Steinberg
2017 arXiv   pre-print
Thereby making it possible to consider functions on the natural numbers as running times of oracle Turing machines and avoiding second-order polynomials, which are notoriously difficult to handle.  ...  The new notion is named "strong polynomial-time computability", and proven to be a strictly stronger requirement than polynomial-time computability.  ...  polynomial-time computability of a functional on Σ * * means.  ... 
arXiv:1704.01405v2 fatcat:heacap7njrbjbey7otxmrwij34

Computing power series in polynomial time

Ker-I Ko, Harvey Friedman
1988 Advances in Applied Mathematics  
It is shown that if a real-valued function f is polynomial-time computable on [a, b], with a < 0 < b, and is analytic at 0, then the Taylor coefficients off at 0, as a sequence of real numbers, is polynomial-time  ...  computable. 0 19x8 Academic PESS.  ...  Our main result shows that the derivatives of a polynomial-time computable, analytic function form a polynomial-time computable sequence of real numbers.  ... 
doi:10.1016/0196-8858(88)90006-1 fatcat:ic26g2wldng4fbebjwkajxzgd4

New non-uniform segmentation technique for software function evaluation

Justine Bonnot, Erwan Nogues, Daniel Menard
2016 2016 IEEE 27th International Conference on Application-specific Systems, Architectures and Processors (ASAP)  
Indeed, polynomial approximation methods allow to find a trade-off between accuracy and computation time. Software implementation of polynomial approximation in fixed-point processors is considered.  ...  This paper presents a method to compute the values of a function on I using non-uniform segmentation, and polynomial approximation.  ...  be computed or the computation time, knowing the size of the indexing table T and of the coefficients table P as well as the equation giving the computation time depending on the degree N l and the number  ... 
doi:10.1109/asap.2016.7760782 dblp:conf/asap/BonnotNM16 fatcat:7z7aiimbbfbvbl76zw2ixie2ae

On the complexity of finding circumscribed rectangles and squares for a two-dimensional domain

Fuxiang Yu, Arthur Chou, Ker-I Ko
2006 Journal of Complexity  
We study this problem in the polynomial-time complexity theory of real functions based on the oracle Turing machine model.  ...  We show that for any polynomial-time computable Jordan curve , there must exist at least one computable circumscribed square (not necessarily of the minimum area), but this square may have arbitrarily  ...  We say a rectangle R is polynomial-time computable if its four corners are polynomial-time computable complex numbers.  ... 
doi:10.1016/j.jco.2006.05.005 fatcat:g2m3rioevffavo5kemkrrykgbi

Some negative results on the computational complexity of total variation and differentiation

Ko Ker-I
1982 Information and Control  
There exists a polynomial time computable function f on [0, 1] such that f' exists and is continuous on [0, 1] burr' is not polynomial time computable.  ...  A real number x is polynomial time computable if there is a sequence {dn} of dyadic rationals binary converging to x and the function 20"[d,] is polynomial time computable.  ...  Note added in proof The author has obtained the following stronger version of Theorem4: there exists a polynomial time computable function f on [0, 1] having a continuous derivativef' on [0, 1] such that  ... 
doi:10.1016/s0019-9958(82)91083-x fatcat:7lpmmgf5ofhdvly66lsy46vhfu

Computing a Solution of Feigenbaum's Functional Equation in Polynomial Time

Peter Hertling, Christoph Spandl, Matthias Schröder
2014 Logical Methods in Computer Science  
We show that this solution is a polynomial time computable function. This implies in particular that the so-called first Feigenbaum constant is a polynomial time computable real number.  ...  The proof is based on a number of claims in Lanford's paper [10] .  ...  We intend to see how many digits of the first Feigenbaum constant we can compute with a correctness guarantee, using the exact real number arithmetic package iRRAM by Müller [12] .  ... 
doi:10.2168/lmcs-10(4:7)2014 fatcat:kmkas7f7izatrmgse2hqa5mt2e
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