887 research outputs found

    Ultimate Polynomial Time

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    The class UP\mathcal{UP} of `ultimate polynomial time' problems over C\mathbb C is introduced; it contains the class P\mathcal P of polynomial time problems over C\mathbb C. The τ\tau-Conjecture for polynomials implies that UP\mathcal{UP} does not contain the class of non-deterministic polynomial time problems definable without constants over C\mathbb C. This latest statement implies that P≠NP\mathcal P \ne \mathcal{NP} over C\mathbb C. A notion of `ultimate complexity' of a problem is suggested. It provides lower bounds for the complexity of structured problems

    Discovering the roots: Uniform closure results for algebraic classes under factoring

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    Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots rr is small but the multiplicities are exponentially large. Our method sets up a linear system in rr unknowns and iteratively builds the roots as formal power series. For an algebraic circuit f(x1,…,xn)f(x_1,\ldots,x_n) of size ss we prove that each factor has size at most a polynomial in: ss and the degree of the squarefree part of ff. Consequently, if f1f_1 is a 2Ξ©(n)2^{\Omega(n)}-hard polynomial then any nonzero multiple ∏ifiei\prod_{i} f_i^{e_i} is equally hard for arbitrary positive eie_i's, assuming that βˆ‘ideg(fi)\sum_i \text{deg}(f_i) is at most 2O(n)2^{O(n)}. It is an old open question whether the class of poly(nn)-sized formulas (resp. algebraic branching programs) is closed under factoring. We show that given a polynomial ff of degree nO(1)n^{O(1)} and formula (resp. ABP) size nO(log⁑n)n^{O(\log n)} we can find a similar size formula (resp. ABP) factor in randomized poly(nlog⁑nn^{\log n})-time. Consequently, if determinant requires nΞ©(log⁑n)n^{\Omega(\log n)} size formula, then the same can be said about any of its nonzero multiples. As part of our proofs, we identify a new property of multivariate polynomial factorization. We show that under a random linear transformation Ο„\tau, f(Ο„xβ€Ύ)f(\tau\overline{x}) completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. This with allRootsNI are the techniques that help us make progress towards the old open problems, supplementing the large body of classical results and concepts in algebraic circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \& Burgisser, FOCS 2001).Comment: 33 Pages, No figure

    VPSPACE and a transfer theorem over the complex field

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    We extend the transfer theorem of [KP2007] to the complex field. That is, we investigate the links between the class VPSPACE of families of polynomials and the Blum-Shub-Smale model of computation over C. Roughly speaking, a family of polynomials is in VPSPACE if its coefficients can be computed in polynomial space. Our main result is that if (uniform, constant-free) VPSPACE families can be evaluated efficiently then the class PAR of decision problems that can be solved in parallel polynomial time over the complex field collapses to P. As a result, one must first be able to show that there are VPSPACE families which are hard to evaluate in order to separate P from NP over C, or even from PAR.Comment: 14 page

    Kolmogorov Complexity Theory over the Reals

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    Kolmogorov Complexity constitutes an integral part of computability theory, information theory, and computational complexity theory -- in the discrete setting of bits and Turing machines. Over real numbers, on the other hand, the BSS-machine (aka real-RAM) has been established as a major model of computation. This real realm has turned out to exhibit natural counterparts to many notions and results in classical complexity and recursion theory; although usually with considerably different proofs. The present work investigates similarities and differences between discrete and real Kolmogorov Complexity as introduced by Montana and Pardo (1998)

    A complex analogue of Toda's Theorem

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    Toda \cite{Toda} proved in 1989 that the (discrete) polynomial time hierarchy, PH\mathbf{PH}, is contained in the class \mathbf{P}^{#\mathbf{P}}, namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #\mathbf{P}. This result, which illustrates the power of counting is considered to be a seminal result in computational complexity theory. An analogous result (with a compactness hypothesis) in the complexity theory over the reals (in the sense of Blum-Shub-Smale real machines \cite{BSS89}) was proved in \cite{BZ09}. Unlike Toda's proof in the discrete case, which relied on sophisticated combinatorial arguments, the proof in \cite{BZ09} is topological in nature in which the properties of the topological join is used in a fundamental way. However, the constructions used in \cite{BZ09} were semi-algebraic -- they used real inequalities in an essential way and as such do not extend to the complex case. In this paper, we extend the techniques developed in \cite{BZ09} to the complex projective case. A key role is played by the complex join of quasi-projective complex varieties. As a consequence we obtain a complex analogue of Toda's theorem. The results contained in this paper, taken together with those contained in \cite{BZ09}, illustrate the central role of the Poincar\'e polynomial in algorithmic algebraic geometry, as well as, in computational complexity theory over the complex and real numbers -- namely, the ability to compute it efficiently enables one to decide in polynomial time all languages in the (compact) polynomial hierarchy over the appropriate field.Comment: 31 pages. Final version to appear in Foundations of Computational Mathematic
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