共查询到20条相似文献,搜索用时 390 毫秒
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Andrej Muchnik Alexander Shen Mikhail Ustinov Nikolai Vereshchagin Michael Vyugin 《Theoretical computer science》2007
Assume that a program p on input a outputs b. We are looking for a shorter program q having the same property (q(a)=b). In addition, we want q to be simple conditional to p (this means that the conditional Kolmogorov complexity K(q|p) is negligible). In the present paper, we prove that sometimes there is no such program q, even in the case when the complexity of p is much bigger than K(b|a). We give three different constructions that use the game approach, probabilistic arguments and algebraic arguments, respectively. 相似文献
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This paper is a sequel to “Computing diagonal form and Jacobson normal form of a matrix using Gröbner bases” (Levandovskyy and Schindelar, 2011). We present a new fraction-free algorithm for the computation of a diagonal form of a matrix over a certain non-commutative Euclidean domain over a computable field with the help of Gröbner bases. This algorithm is formulated in a general constructive framework of non-commutative Ore localizations of G-algebras (OLGAs). We use the splitting of the computation of a normal form for matrices over Ore localizations into the diagonalization and the normalization processes. Both of them can be made fraction-free. For a given matrix M over an OLGA R, we provide a diagonalization algorithm to compute U,V and D with fraction-free entries such that UMV=D holds and D is diagonal. The fraction-free approach allows to obtain more information on the associated system of linear functional equations and its solutions, than the classical setup of an operator algebra with coefficients in rational functions. In particular, one can handle distributional solutions together with, say, meromorphic ones. We investigate Ore localizations of common operator algebras over K[x] and use them in the unimodularity analysis of transformation matrices U,V. In turn, this allows to lift the isomorphism of modules over an OLGA Euclidean domain to a smaller polynomial subring of it. We discuss the relation of this lifting with the solutions of the original system of equations. Moreover, we prove some new results concerning normal forms of matrices over non-simple domains. Our implementation in the computer algebra system Singular:Plural follows the fraction-free strategy and shows impressive performance, compared with methods which directly use fractions. In particular, we experience a moderate swell of coefficients and obtain simple transformation matrices. Thus the method we propose is well suited for solving nontrivial practical problems. 相似文献
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We consider orthogonal drawings of a plane graph G with specified face areas. For a natural number k, a k-gonal drawing of G is an orthogonal drawing such that the boundary of G is drawn as a rectangle and each inner face is drawn as a polygon with at most k corners whose area is equal to the specified value. In this paper, we show that every slicing graph G with a slicing tree T and a set of specified face areas admits a 10-gonal drawing D such that the boundary of each slicing subgraph that appears in T is also drawn as a polygon with at most 10 corners. Such a drawing D can be found in linear time. 相似文献
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Solomonoff’s central result on induction is that the prediction of a universal semimeasure M converges rapidly and with probability 1 to the true sequence generating predictor μ, if the latter is computable. Hence, M is eligible as a universal sequence predictor in the case of unknown μ. Despite some nearby results and proofs in the literature, the stronger result of convergence for all (Martin-Löf) random sequences remained open. Such a convergence result would be particularly interesting and natural, since randomness can be defined in terms of M itself. We show that there are universal semimeasures M which do not converge to μ on all μ-random sequences, i.e. we give a partial negative answer to the open problem. We also provide a positive answer for some non-universal semimeasures. We define the incomputable measure D as a mixture over all computable measures and the enumerable semimeasure W as a mixture over all enumerable nearly measures. We show that W converges to D and D to μ on all random sequences. The Hellinger distance measuring closeness of two distributions plays a central role. 相似文献
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A real x is called h-bounded computable , for some function h:N→N, if there is a computable sequence (xs) of rational numbers which converges to x such that, for any n∈N, at most h(n) non-overlapping pairs of its members are separated by a distance larger than 2-n. In this paper we discuss properties of h-bounded computable reals for various functions h. We will show a simple sufficient condition for a class of functions h such that the corresponding h-bounded computable reals form an algebraic field. A hierarchy theorem for h-bounded computable reals is also shown. Besides we compare semi-computability and weak computability with the h-bounded computability for special functions h. 相似文献
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Let F(x,y) be a polynomial over a field K and m a nonnegative integer. We call a polynomial g over K an m-near solution of F(x,y) if there exists a c∈K such that F(x,g)=cxm, and the number c is called an m-value of F(x,y) corresponding to g. In particular, c can be 0. Hence, by viewing F(x,y)=0 as a polynomial equation over K[x] with variable y, every solution of the equation F(x,y)=0 in K[x] is also an m-near solution. We provide an algorithm that gives all m-near solutions of a given polynomial F(x,y) over K, and this algorithm is polynomial time reducible to solving one variable equations over K. We introduce approximate solutions to analyze the algorithm. We also give some interesting properties of approximate solutions. 相似文献
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We define an operation called transposition on words of fixed length. This operation arises naturally when the letters of a word are considered as entries of a matrix. Words that are invariant with respect to transposition are of special interest. It turns out that transposition invariant words have a simple interpretation by means of elementary group theory. This leads us to investigate some properties of the ring of integers modulo n and primitive roots. In particular, we show that there are infinitely many prime numbers p with a primitive root dividing p+1 and infinitely many prime numbers p without a primitive root dividing p+1. We also consider the orbit of a word under transposition. 相似文献
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Oleg Golubitsky Marina Kondratieva Marc Moreno Maza Alexey Ovchinnikov 《Journal of Symbolic Computation》2008
We consider the Rosenfeld–Gröbner algorithm for computing a regular decomposition of a radical differential ideal generated by a set of ordinary differential polynomials in n indeterminates. For a set of ordinary differential polynomials F, let M(F) be the sum of maximal orders of differential indeterminates occurring in F. We propose a modification of the Rosenfeld–Gröbner algorithm, in which for every intermediate polynomial system F, the bound M(F)?(n−1)!M(F0) holds, where F0 is the initial set of generators of the radical ideal. In particular, the resulting regular systems satisfy the bound. Since regular ideals can be decomposed into characterizable components algebraically, the bound also holds for the orders of derivatives occurring in a characteristic decomposition of a radical differential ideal. 相似文献
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We formalize paper fold (origami) by graph rewriting. Origami construction is abstractly described by a rewriting system (O,?), where O is the set of abstract origamis and ? is a binary relation on O, that models fold . An abstract origami is a structure (Π,∽,?), where Π is a set of faces constituting an origami, and ∽ and ? are binary relations on Π, each representing adjacency and superposition relations between the faces. 相似文献
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Let f(X,Y)∈Z[X,Y] be an irreducible polynomial over Q. We give a Las Vegas absolute irreducibility test based on a property of the Newton polytope of f, or more precisely, of f modulo some prime integer p. The same idea of choosing a p satisfying some prescribed properties together with LLL is used to provide a new strategy for absolute factorization of f(X,Y). We present our approach in the bivariate case but the techniques extend to the multivariate case. Maple computations show that it is efficient and promising as we are able to construct the algebraic extension containing one absolute factor of a polynomial of degree up to 400. 相似文献
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We investigate the group key management problem for broadcasting applications. Previous work showed that, in handling key updates, batch rekeying can be more cost effective than individual rekeying. One model for batch rekeying is to assume that every user has probability p of being replaced by a new user during a batch period with the total number of users unchanged. Under this model, it was recently shown that an optimal key tree can be constructed in linear time when p is a constant and in O(n4) time when p→0. In this paper, we investigate more efficient algorithms for the case p→0, i.e., when membership changes are sparse. We design an O(n) heuristic algorithm for the sparse case and show that it produces a nearly 2-approximation to the optimal key tree. Simulation results show that its performance is even better in practice. We also design a refined heuristic algorithm and show that it achieves an approximation ratio of 1+? for any fixed ?>0 and n, as p→0. Finally, we give another approximation algorithm for any p∈(0,0.693) which is shown to be quite good by our simulations. 相似文献
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We consider time-space tradeoffs for static data structure problems in the cell probe model with word size 1 (the bit probe model). In this model, the goal is to represent n-bit data with s=n+r bits such that queries (of a certain type) about the data can be answered by reading at most t bits of the representation. Ideally, we would like to keep both s and t small, but there are tradeoffs between the values of s and t that limit the possibilities of keeping both parameters small. 相似文献
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A. Abouelaoualim K.Ch. Das L. Faria Y. Manoussakis C. Martinhon R. Saad 《Theoretical computer science》2008
This paper deals with the existence and search for properly edge-colored paths/trails between two, not necessarily distinct, vertices s and t in an edge-colored graph from an algorithmic perspective. First we show that several versions of the s−t path/trail problem have polynomial solutions including the shortest path/trail case. We give polynomial algorithms for finding a longest properly edge-colored path/trail between s and t for a particular class of graphs and characterize edge-colored graphs without properly edge-colored closed trails. Next, we prove that deciding whether there exist k pairwise vertex/edge disjoint properly edge-colored s−t paths/trails in a c-edge-colored graph Gc is NP-complete even for k=2 and c=Ω(n2), where n denotes the number of vertices in Gc. Moreover, we prove that these problems remain NP-complete for c-edge-colored graphs containing no properly edge-colored cycles and c=Ω(n). We obtain some approximation results for those maximization problems together with polynomial results for some particular classes of edge-colored graphs. 相似文献
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The aim of this paper is to investigate the possibility of solving a linear differential equation of degree n by means of differential equations of degree less than or equal to a fixed d, 1≤d<n. This paper recovers and extends work of G. Fano, M. F. Singer and E. Compoint. Representations of algebraic Lie algebras are the main tool. 相似文献
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The claw finding problem has been studied in terms of query complexity as one of the problems closely connected to cryptography. Given two functions, f and g, with domain sizes N and M(N≤M), respectively, and the same range, the goal of the problem is to find x and y such that f(x)=g(y). This problem has been considered in both quantum and classical settings in terms of query complexity. This paper describes an optimal algorithm that uses quantum walk to solve this problem. Our algorithm can be slightly modified to solve the more general problem of finding a tuple consisting of elements in the two function domains that has a prespecified property. It can also be generalized to find a claw of k functions for any constant integer k>1, where the domain sizes of the functions may be different. 相似文献