Results 1  10
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41
Monotone Complexity
, 1990
"... We give a general complexity classification scheme for monotone computation, including monotone spacebounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple ..."
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Cited by 2359 (12 self)
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We give a general complexity classification scheme for monotone computation, including monotone spacebounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions. We show that mNL (monotone nondeterministic logspace) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = coNL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for stconnectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC 1 , motivated by Barrington's result [Bar89] that BWBP = NC 1 . Although we cannot answer t...
Structure and Importance of LogspaceMODClasses
, 1992
"... . We refine the techniques of Beigel, Gill, Hertrampf [4] who investigated polynomial time counting classes, in order to make them applicable to the case of logarithmic space. We define the complexity classes MOD k L and demonstrate their significance by proving that all standard problems of linear ..."
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Cited by 41 (1 self)
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. We refine the techniques of Beigel, Gill, Hertrampf [4] who investigated polynomial time counting classes, in order to make them applicable to the case of logarithmic space. We define the complexity classes MOD k L and demonstrate their significance by proving that all standard problems of linear algebra over the finite rings Z/kZ are complete for these classes. We then define new complexity classes LogFew and LogFewNL and identify them as adequate logspace versions of Few and FewP. We show that LogFewNL is contained in MODZ k L and that LogFew is contained in MOD k L for all k. Also an upper bound for L #L in terms of computation of integer determinants is given from which we conclude that all logspace counting classes are contained in NC 2 . 1 Introduction Valiant [21] defined the class #P of functions f such that there is a nondeterministic polynomial time Turing machine which, on input x, has exactly f(x) accepting computation paths. Many complexity classes in the area betw...
Fast Parallel Absolute Irreducibility Testing
 J. Symbolic Comput
, 1985
"... We present a fast parallel deterministic algorithm for testing multivariate integral polynomials for absolute irreducibility, that is irreducibility over the complex numbers. More precisely, we establish that the set of absolutely irreducible integral polynomials belongs to the complexity class NC o ..."
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Cited by 31 (7 self)
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We present a fast parallel deterministic algorithm for testing multivariate integral polynomials for absolute irreducibility, that is irreducibility over the complex numbers. More precisely, we establish that the set of absolutely irreducible integral polynomials belongs to the complexity class NC of Boolean circuits of polynomial size and logarithmic depth. Therefore it also belongs to the class of sequentially polynomialtime problems. Our algorithm can be extended to compute in parallel one irreducible complex factor of a multivariate integral polynomial. However, the coefficients of the computed factor are only represented modulo a not necessarily irreducible polynomial specifying a splitting field. A consequence of our algoithm is that multivariate polynomials over finite fields can be tested for absolute irreducibility in deterministic sequential polynomial time in the size of the input. We also obtain a sharp bound for the last prime p for which, when taking an absolutely irreducible integral polynomial modulo p, the polynomial's irreducibility in the algebraic closure of the finite field order p is not preserved.
Fast parallel computation of Hermite and Smith forms of polynomial matrices
 SIAM J. Alg. Discrete Meth
, 1987
"... Abstract. Boolean circuits of polynomial size and polylogarithmic depth are given for computing the Hermite and Smith normal forms of polynomial matrices over finite fields and the field of rational numbers. The circuits for the Smith normal form computation are probabilistic ones and also determin ..."
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Cited by 30 (2 self)
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Abstract. Boolean circuits of polynomial size and polylogarithmic depth are given for computing the Hermite and Smith normal forms of polynomial matrices over finite fields and the field of rational numbers. The circuits for the Smith normal form computation are probabilistic ones and also determine very efficient sequential algorithms. Furthermore, we give a polynomialtime deterministic sequential algorithm for the Smith normal form over the rationals. The Smith normal form algorithms are applied to the Rational canonical form of matrices over finite fields and the field of rational numbers. Ke ywords: Parallel algorithm, Hermite normal form, Smith normal form, polynomialtime complexity. 1.
Factoring Rational Polynomials over the Complex Numbers
, 1989
"... eskeleton on the surface (P = 0) whose number of connected components is precisely the number of connected components of P =0minus its singular points. The connectivity of this curveskeleton is constructed symbolically using Sturm sequences associated with the various polynomials de ning these ..."
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Cited by 19 (2 self)
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eskeleton on the surface (P = 0) whose number of connected components is precisely the number of connected components of P =0minus its singular points. The connectivity of this curveskeleton is constructed symbolically using Sturm sequences associated with the various polynomials de ning these maps. Given the number of irreducible factors and their degree, the actual factors can be reconstructed using the recent result of Ne [22] on nding zeroes of one variable polynomials in NC. 1 Introduction Factoring polynomials is a basic problem in symbolic computation with applications as diverse as theorem proving and computeraided design. Our goal is to approximate the factors, irreducible over the complex numbers, of a multivariable polynomial with rational coecients in deterministic NC with respect to the polynomial's degree and coecient size, assuming that the number of variables is xed. Further if the number of variables is not xed, we will nd the number of irreducible facto
Computing Popov and Hermite forms of polynomial matrices
 In International Symposium on Symbolic and Algebmic Computation, Zutich, .%isse
, 1996
"... For a polynomial matrix P(z) of degree d in M~,~(K[z]) where K is a commutative field, a reduction to the Hermite normal form can be computed in O (ndM(n) + M(nd)) arithmetic operations if M(n) is the time required to multiply two n x n matrices over K. Further, a reduction can be computed using O(l ..."
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Cited by 19 (10 self)
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For a polynomial matrix P(z) of degree d in M~,~(K[z]) where K is a commutative field, a reduction to the Hermite normal form can be computed in O (ndM(n) + M(nd)) arithmetic operations if M(n) is the time required to multiply two n x n matrices over K. Further, a reduction can be computed using O(log~+ ’ (ml)) pamlel arithmetic steps and O(L(nd) ) processors if the same processor bound holds with time O (logX (rid)) for determining the lexicographically first maximal linearly independent subset of the set of the columns of an nd x nd matrix over K. These results are obtamed by applying in the matrix case, the techniques used in the scalar case of the gcd of polynomials.
On the Existence of Hard Sparse Sets under Weak Reductions
, 1996
"... Recently a 1978 conjecture by Hartmanis [Har78] was resolved [CS95], following progress made by [Ogi95]. It was shown that there is no sparse set that is hard for P under logspace manyone reductions, unless P = LOGSPACE. We extend the results to the case of sparse sets that are hard under more gene ..."
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Cited by 18 (4 self)
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Recently a 1978 conjecture by Hartmanis [Har78] was resolved [CS95], following progress made by [Ogi95]. It was shown that there is no sparse set that is hard for P under logspace manyone reductions, unless P = LOGSPACE. We extend the results to the case of sparse sets that are hard under more general reducibilities. Our main results are as follows. (1) If there exists a sparse set that is hard for P under bounded truthtable reductions, then P = NC 2 . (2) If there exists a sparse set that is hard for P under randomized logspace reductions with onesided error, then P = Randomized LOGSPACE. (3) If there exists an NPhard sparse set under randomized polynomialtime reductions with onesided error, then NP = RP. (4) If there exists a 2 (log n) O(1) sparse hard set for P under truthtable reductions, then P ` DSPACE[(logn) O(1) ]. As a byproduct of (4), we obtain a uniform O(log 2 n log log n) time parallel algorithm for computing the rank of a 2 log 2 n \Theta n matrix o...
Some Complexity Results for Polynomial Ideals
, 1997
"... In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)tuple P = ( f, g1, g2,.. ..."
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Cited by 16 (0 self)
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In this paper, we survey some of our new results on the complexity of a number of problems related to polynomial ideals. We consider multivariate polynomials over some ring, like the integers or the rationals. For instance, a polynomial ideal membership problem is a (w + 1)tuple P = ( f, g1, g2,..., gw) where f and the gi are multivariate polynomials, and the problem is to determine whether f is in the ideal generated by the gi. For polynomials over the integers or rationals, this problem is known to be exponential space complete. We discuss further complexity results for problems related to polynomial ideals, like the word and subword problems for commutative semigroups, a quantitative version of Hilbert’s Nullstellensatz in a complexity theoretic version, and problems concerning the computation of reduced polynomials and Gröbner bases.
The Resolution of a Hartmanis Conjecture
, 1995
"... Building on the recent breakthrough by Ogihara, we resolve a conjecture made by Hartmanis in 1978 regarding the (non) existence of sparse sets complete for P under logspace manyone reductions. We show that if there exists a sparse hard set for P under logspace manyone reductions, then P = LOGSPACE ..."
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Cited by 13 (4 self)
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Building on the recent breakthrough by Ogihara, we resolve a conjecture made by Hartmanis in 1978 regarding the (non) existence of sparse sets complete for P under logspace manyone reductions. We show that if there exists a sparse hard set for P under logspace manyone reductions, then P = LOGSPACE. We further prove that if P has a sparse hard set under manyone reductions computable in NC 1 , then P collapses to NC 1 . 1 Introduction A set S is called sparse if there are at most a polynomial number of strings in S up to length n. Sparse sets have been the subject of study in complexity theory for the past 20 years, as they reveal inherent structure and limitations of computation [BH77, HOW92, You92a, You92b]. For instance, it is well known that the class of languages polynomial time Turing reducible (i.e. by Cook reductions) to a sparse set is precisely the class of languages with polynomial size circuits. One major motivation for the study of sparse sets, and various reducib...
Arithmetic Circuits: a survey of recent results and open questions
"... A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last fi ..."
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Cited by 12 (3 self)
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A large class of problems in symbolic computation can be expressed as the task of computing some polynomials; and arithmetic circuits form the most standard model for studying the complexity of such computations. This algebraic model of computation attracted a large amount of research in the last five decades, partially due to its simplicity and elegance. Being a more structured model than Boolean circuits, one could hope that the fundamental problems of theoretical computer science, such as separating P from NP, will be easier to solve for arithmetic circuits. However, in spite of the appearing simplicity and the vast amount of mathematical tools available, no major breakthrough has been seen. In fact, all the fundamental questions are still open for this model as well. Nevertheless, there has been a lot of progress in the area and beautiful results have been found, some in the last few years. As examples we mention the connection between polynomial identity testing and lower bounds of Kabanets and Impagliazzo, the lower bounds of Raz for multilinear formulas, and two new approaches for proving lower bounds: Geometric Complexity Theory and Elusive Functions. The goal of this monograph is to survey the field of arithmetic circuit complexity, focusing mainly on what we find to be the most interesting and accessible research directions. We aim to cover the main results and techniques, with an emphasis on works from the last two decades. In particular, we