858 research outputs found

    Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates

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    The class FORMULA[s]GFORMULA[s] \circ \mathcal{G} consists of Boolean functions computable by size-ss de Morgan formulas whose leaves are any Boolean functions from a class G\mathcal{G}. We give lower bounds and (SAT, Learning, and PRG) algorithms for FORMULA[n1.99]GFORMULA[n^{1.99}]\circ \mathcal{G}, for classes G\mathcal{G} of functions with low communication complexity. Let R(k)(G)R^{(k)}(\mathcal{G}) be the maximum kk-party NOF randomized communication complexity of G\mathcal{G}. We show: (1) The Generalized Inner Product function GIPnkGIP^k_n cannot be computed in FORMULA[s]GFORMULA[s]\circ \mathcal{G} on more than 1/2+ε1/2+\varepsilon fraction of inputs for s=o ⁣(n2(k4kR(k)(G)log(n/ε)log(1/ε))2). s = o \! \left ( \frac{n^2}{ \left(k \cdot 4^k \cdot {R}^{(k)}(\mathcal{G}) \cdot \log (n/\varepsilon) \cdot \log(1/\varepsilon) \right)^{2}} \right). As a corollary, we get an average-case lower bound for GIPnkGIP^k_n against FORMULA[n1.99]PTFk1FORMULA[n^{1.99}]\circ PTF^{k-1}. (2) There is a PRG of seed length n/2+O(sR(2)(G)log(s/ε)log(1/ε))n/2 + O\left(\sqrt{s} \cdot R^{(2)}(\mathcal{G}) \cdot\log(s/\varepsilon) \cdot \log (1/\varepsilon) \right) that ε\varepsilon-fools FORMULA[s]GFORMULA[s] \circ \mathcal{G}. For FORMULA[s]LTFFORMULA[s] \circ LTF, we get the better seed length O(n1/2s1/4log(n)log(n/ε))O\left(n^{1/2}\cdot s^{1/4}\cdot \log(n)\cdot \log(n/\varepsilon)\right). This gives the first non-trivial PRG (with seed length o(n)o(n)) for intersections of nn half-spaces in the regime where ε1/n\varepsilon \leq 1/n. (3) There is a randomized 2nt2^{n-t}-time #\#SAT algorithm for FORMULA[s]GFORMULA[s] \circ \mathcal{G}, where t=Ω(nslog2(s)R(2)(G))1/2.t=\Omega\left(\frac{n}{\sqrt{s}\cdot\log^2(s)\cdot R^{(2)}(\mathcal{G})}\right)^{1/2}. In particular, this implies a nontrivial #SAT algorithm for FORMULA[n1.99]LTFFORMULA[n^{1.99}]\circ LTF. (4) The Minimum Circuit Size Problem is not in FORMULA[n1.99]XORFORMULA[n^{1.99}]\circ XOR. On the algorithmic side, we show that FORMULA[n1.99]XORFORMULA[n^{1.99}] \circ XOR can be PAC-learned in time 2O(n/logn)2^{O(n/\log n)}

    An Introduction to Quantum Complexity Theory

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    We give a basic overview of computational complexity, query complexity, and communication complexity, with quantum information incorporated into each of these scenarios. The aim is to provide simple but clear definitions, and to highlight the interplay between the three scenarios and currently-known quantum algorithms.Comment: 28 pages, LaTeX, 11 figures within the text, to appear in "Collected Papers on Quantum Computation and Quantum Information Theory", edited by C. Macchiavello, G.M. Palma, and A. Zeilinger (World Scientific
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