72,110 research outputs found

    Support Sets in Exponential Families and Oriented Matroid Theory

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    The closure of a discrete exponential family is described by a finite set of equations corresponding to the circuits of an underlying oriented matroid. These equations are similar to the equations used in algebraic statistics, although they need not be polynomial in the general case. This description allows for a combinatorial study of the possible support sets in the closure of an exponential family. If two exponential families induce the same oriented matroid, then their closures have the same support sets. Furthermore, the positive cocircuits give a parameterization of the closure of the exponential family.Comment: 27 pages, extended version published in IJA

    On the Complexity of Quantum ACC

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    For any q>1q > 1, let \MOD_q be a quantum gate that determines if the number of 1's in the input is divisible by qq. We show that for any q,t>1q,t > 1, \MOD_q is equivalent to \MOD_t (up to constant depth). Based on the case q=2q=2, Moore \cite{moore99} has shown that quantum analogs of AC(0)^{(0)}, ACC[q][q], and ACC, denoted QACwf(0)^{(0)}_{wf}, QACC[2][2], QACC respectively, define the same class of operators, leaving q>2q > 2 as an open question. Our result resolves this question, proving that QACwf(0)=^{(0)}_{wf} = QACC[q]=[q] = QACC for all qq. We also develop techniques for proving upper bounds for QACC in terms of related language classes. We define classes of languages EQACC, NQACC and BQACC_{\rats}. We define a notion log\log-planar QACC operators and show the appropriately restricted versions of EQACC and NQACC are contained in P/poly. We also define a notion of log\log-gate restricted QACC operators and show the appropriately restricted versions of EQACC and NQACC are contained in TC(0)^{(0)}. To do this last proof, we show that TC(0)^{(0)} can perform iterated addition and multiplication in certain field extensions. We also introduce the notion of a polynomial-size tensor graph and show that families of such graphs can encode the amplitudes resulting from apply an arbitrary QACC operator to an initial state.Comment: 22 pages, 4 figures This version will appear in the July 2000 Computational Complexity conference. Section 4 has been significantly revised and many typos correcte

    Software Engineering and Complexity in Effective Algebraic Geometry

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    We introduce the notion of a robust parameterized arithmetic circuit for the evaluation of algebraic families of multivariate polynomials. Based on this notion, we present a computation model, adapted to Scientific Computing, which captures all known branching parsimonious symbolic algorithms in effective Algebraic Geometry. We justify this model by arguments from Software Engineering. Finally we exhibit a class of simple elimination problems of effective Algebraic Geometry which require exponential time to be solved by branching parsimonious algorithms of our computation model.Comment: 70 pages. arXiv admin note: substantial text overlap with arXiv:1201.434

    On Symmetric Circuits and Fixed-Point Logics

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    We study properties of relational structures such as graphs that are decided by families of Boolean circuits. Circuits that decide such properties are necessarily invariant to permutations of the elements of the input structures. We focus on families of circuits that are symmetric, i.e., circuits whose invariance is witnessed by automorphisms of the circuit induced by the permutation of the input structure. We show that the expressive power of such families is closely tied to definability in logic. In particular, we show that the queries defined on structures by uniform families of symmetric Boolean circuits with majority gates are exactly those definable in fixed-point logic with counting. This shows that inexpressibility results in the latter logic lead to lower bounds against polynomial-size families of symmetric circuits.Comment: 22 pages. Full version of a paper to appear in STACS 201

    Non-Malleable Codes for Small-Depth Circuits

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    We construct efficient, unconditional non-malleable codes that are secure against tampering functions computed by small-depth circuits. For constant-depth circuits of polynomial size (i.e. AC0\mathsf{AC^0} tampering functions), our codes have codeword length n=k1+o(1)n = k^{1+o(1)} for a kk-bit message. This is an exponential improvement of the previous best construction due to Chattopadhyay and Li (STOC 2017), which had codeword length 2O(k)2^{O(\sqrt{k})}. Our construction remains efficient for circuit depths as large as Θ(log(n)/loglog(n))\Theta(\log(n)/\log\log(n)) (indeed, our codeword length remains nk1+ϵ)n\leq k^{1+\epsilon}), and extending our result beyond this would require separating P\mathsf{P} from NC1\mathsf{NC^1}. We obtain our codes via a new efficient non-malleable reduction from small-depth tampering to split-state tampering. A novel aspect of our work is the incorporation of techniques from unconditional derandomization into the framework of non-malleable reductions. In particular, a key ingredient in our analysis is a recent pseudorandom switching lemma of Trevisan and Xue (CCC 2013), a derandomization of the influential switching lemma from circuit complexity; the randomness-efficiency of this switching lemma translates into the rate-efficiency of our codes via our non-malleable reduction.Comment: 26 pages, 4 figure
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