1,768 research outputs found

    A Nearly Optimal Lower Bound on the Approximate Degree of AC0^0

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    The approximate degree of a Boolean function f ⁣:{1,1}n{1,1}f \colon \{-1, 1\}^n \rightarrow \{-1, 1\} is the least degree of a real polynomial that approximates ff pointwise to error at most 1/31/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function ff with approximate degree dd into a function FF on O(npolylog(n))O(n \cdot \operatorname{polylog}(n)) variables with approximate degree at least D=Ω(n1/3d2/3)D = \Omega(n^{1/3} \cdot d^{2/3}). In particular, if d=n1Ω(1)d= n^{1-\Omega(1)}, then DD is polynomially larger than dd. Moreover, if ff is computed by a polynomial-size Boolean circuit of constant depth, then so is FF. By recursively applying our transformation, for any constant δ>0\delta > 0 we exhibit an AC0^0 function of approximate degree Ω(n1δ)\Omega(n^{1-\delta}). This improves over the best previous lower bound of Ω(n2/3)\Omega(n^{2/3}) due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of nn that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: * For any constant δ>0\delta > 0, an Ω(n1δ)\Omega(n^{1-\delta}) lower bound on the quantum communication complexity of a function in AC0^0. * A Boolean function ff with approximate degree at least C(f)2o(1)C(f)^{2-o(1)}, where C(f)C(f) is the certificate complexity of ff. This separation is optimal up to the o(1)o(1) term in the exponent. * Improved secret sharing schemes with reconstruction procedures in AC0^0.Comment: 40 pages, 1 figur

    Imitation Games and Computation

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    TAn imitation game is a finite two person normal form game in which the two players have the same set of pure strategies and the goal of the second player is to choose the same pure strategy as the first player. Gale et al. (1950) gave a way of passing from a given two person game to a symmetric game whose symmetric Nash equilibria are in oneto-one correspondence with the Nash equilibria of the given game. We give a way of passing from a given symmetric two person game to an imitation game whose Nash equilibria are in one-to-one correspondence with the symmetric Nash equilibria of the given symmetric game. Lemke (1965) portrayed the Lemke-Howson algorithm as a special case of the Lemke paths algorithm. Using imitation games, we show how Lemke paths may be obtained by projecting Lemke-Howson paths.

    High rate locally-correctable and locally-testable codes with sub-polynomial query complexity

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    In this work, we construct the first locally-correctable codes (LCCs), and locally-testable codes (LTCs) with constant rate, constant relative distance, and sub-polynomial query complexity. Specifically, we show that there exist binary LCCs and LTCs with block length nn, constant rate (which can even be taken arbitrarily close to 1), constant relative distance, and query complexity exp(O~(logn))\exp(\tilde{O}(\sqrt{\log n})). Previously such codes were known to exist only with Ω(nβ)\Omega(n^{\beta}) query complexity (for constant β>0\beta > 0), and there were several, quite different, constructions known. Our codes are based on a general distance-amplification method of Alon and Luby~\cite{AL96_codes}. We show that this method interacts well with local correctors and testers, and obtain our main results by applying it to suitably constructed LCCs and LTCs in the non-standard regime of \emph{sub-constant relative distance}. Along the way, we also construct LCCs and LTCs over large alphabets, with the same query complexity exp(O~(logn))\exp(\tilde{O}(\sqrt{\log n})), which additionally have the property of approaching the Singleton bound: they have almost the best-possible relationship between their rate and distance. This has the surprising consequence that asking for a large alphabet error-correcting code to further be an LCC or LTC with exp(O~(logn))\exp(\tilde{O}(\sqrt{\log n})) query complexity does not require any sacrifice in terms of rate and distance! Such a result was previously not known for any o(n)o(n) query complexity. Our results on LCCs also immediately give locally-decodable codes (LDCs) with the same parameters

    On the Limits of Depth Reduction at Depth 3 Over Small Finite Fields

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    Recently, Gupta et.al. [GKKS2013] proved that over Q any nO(1)n^{O(1)}-variate and nn-degree polynomial in VP can also be computed by a depth three ΣΠΣ\Sigma\Pi\Sigma circuit of size 2O(nlog3/2n)2^{O(\sqrt{n}\log^{3/2}n)}. Over fixed-size finite fields, Grigoriev and Karpinski proved that any ΣΠΣ\Sigma\Pi\Sigma circuit that computes DetnDet_n (or PermnPerm_n) must be of size 2Ω(n)2^{\Omega(n)} [GK1998]. In this paper, we prove that over fixed-size finite fields, any ΣΠΣ\Sigma\Pi\Sigma circuit for computing the iterated matrix multiplication polynomial of nn generic matrices of size n×nn\times n, must be of size 2Ω(nlogn)2^{\Omega(n\log n)}. The importance of this result is that over fixed-size fields there is no depth reduction technique that can be used to compute all the nO(1)n^{O(1)}-variate and nn-degree polynomials in VP by depth 3 circuits of size 2o(nlogn)2^{o(n\log n)}. The result [GK1998] can only rule out such a possibility for depth 3 circuits of size 2o(n)2^{o(n)}. We also give an example of an explicit polynomial (NWn,ϵ(X)NW_{n,\epsilon}(X)) in VNP (not known to be in VP), for which any ΣΠΣ\Sigma\Pi\Sigma circuit computing it (over fixed-size fields) must be of size 2Ω(nlogn)2^{\Omega(n\log n)}. The polynomial we consider is constructed from the combinatorial design. An interesting feature of this result is that we get the first examples of two polynomials (one in VP and one in VNP) such that they have provably stronger circuit size lower bounds than Permanent in a reasonably strong model of computation. Next, we prove that any depth 4 ΣΠ[O(n)]ΣΠ[n]\Sigma\Pi^{[O(\sqrt{n})]}\Sigma\Pi^{[\sqrt{n}]} circuit computing NWn,ϵ(X)NW_{n,\epsilon}(X) (over any field) must be of size 2Ω(nlogn)2^{\Omega(\sqrt{n}\log n)}. To the best of our knowledge, the polynomial NWn,ϵ(X)NW_{n,\epsilon}(X) is the first example of an explicit polynomial in VNP such that it requires 2Ω(nlogn)2^{\Omega(\sqrt{n}\log n)} size depth four circuits, but no known matching upper bound
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