Approximating Threshold Circuits by Rational Functions

AbstractMotivated by the problem of understanding the limitations of threshold networks for representing boolean functions, we consider size-depth trade-offs for threshold circuits that compute the parity function. Using a fundamental result in the theory of rational approximation, we show how to approximate small threshold circuits by rational functions of low degree. We apply this result to establish an almost optimal lower bound of Ω(n2/ln2n) on the number of edges of any depth-2 threshold circuit with polynomially bounded weights that computes the parity function. We also prove that any depth-3 threshold circuit with polynomially bounded weights requires Ω(n1.2/ln5/3n) edges to compute parity. On the other hand, we give a construction of a depth d threshold circuit that computes parity with n1+1/Θ(φd) edges where φ = (1 + √5)/2 is the golden ratio. We conjecture that there are no linear size bounded depth threshold circuits for computing parity

Unbounded-Error Classical and Quantum Communication Complexity

Since the seminal work of Paturi and Simon \cite[FOCS'84 & JCSS'86]{PS86}, the unbounded-error classical communication complexity of a Boolean function has been studied based on the arrangement of points and hyperplanes. Recently, \cite[ICALP'07]{INRY07} found that the unbounded-error {\em quantum} communication complexity in the {\em one-way communication} model can also be investigated using the arrangement, and showed that it is exactly (without a difference of even one qubit) half of the classical one-way communication complexity. In this paper, we extend the arrangement argument to the {\em two-way} and {\em simultaneous message passing} (SMP) models. As a result, we show similarly tight bounds of the unbounded-error two-way/one-way/SMP quantum/classical communication complexities for {\em any} partial/total Boolean function, implying that all of them are equivalent up to a multiplicative constant of four. Moreover, the arrangement argument is also used to show that the gap between {\em weakly} unbounded-error quantum and classical communication complexities is at most a factor of three.Comment: 11 pages. To appear at Proc. ISAAC 200

Partition-function zeros of spherical spin glasses and their relevance to chaos

We investigate partition-function zeros of the many-body interacting spherical spin glass, the so-called $p$-spin spherical model, with respect to the complex temperature in the thermodynamic limit. We use the replica method and extend the procedure of the replica symmetry breaking ansatz to be applicable in the complex-parameter case. We derive the phase diagrams in the complex-temperature plane and calculate the density of zeros in each phase. Near the imaginary axis away from the origin, there is a replica symmetric phase having a large density. On the other hand, we observe no density in the spin-glass phases, irrespective of the replica symmetry breaking. We speculate that this suggests the absence of the temperature chaos. To confirm this, we investigate the multiple many-body interacting case which is known to exhibit the chaos effect. The result shows that the density of zeros actually takes finite values in the spin-glass phase, even on the real axis. These observations indicate that the density of zeros is more closely connected to the chaos effect than the replica symmetry breaking.Comment: 22 pages, 8 figure

Unbounded-error One-way Classical and Quantum Communication Complexity

This paper studies the gap between quantum one-way communication complexity $Q(f)$ and its classical counterpart $C(f)$, under the {\em unbounded-error} setting, i.e., it is enough that the success probability is strictly greater than 1/2. It is proved that for {\em any} (total or partial) Boolean function $f$, $Q(f)=\lceil C(f)/2 \rceil$, i.e., the former is always exactly one half as large as the latter. The result has an application to obtaining (again an exact) bound for the existence of $(m,n,p)$-QRAC which is the $n$-qubit random access coding that can recover any one of $m$ original bits with success probability $\geq p$. We can prove that $(m,n,>1/2)$-QRAC exists if and only if $m\leq 2^{2n}-1$. Previously, only the construction of QRAC using one qubit, the existence of $(O(n),n,>1/2)$-RAC, and the non-existence of $(2^{2n},n,>1/2)$-QRAC were known.Comment: 9 pages. To appear in Proc. ICALP 200

Fluprostenol-Induced MAPK Signaling is Independent of Aging in Fischer 344/NNiaHSd x Brown Norway/BiNia Rat Aorta

The factors that regulate vascular mechanotransduction and how this process may be altered with aging are poorly understood and have not been widely studied. Recent data suggest that increased tissue loading can result in the release of prostaglandin F2 alpha (PGF2α) and other reports indicate that aging diminishes the ability of the aged aorta to activate mitogen activated protein kinase (MAPK) signaling in response to increased loading. Using ex vivo incubations, here we investigate whether aging affects the ability of the aorta to induce phosphorylation of extracellular signal-regulated kinase 1/2 (ERK½-MAPK), p38-MAPK, and Jun N-terminal kinase (JNK-MAPK) activation following stimulation with a PGF2α analog, fluprostenol. Compared to aortas from 6-mo animals, the amounts of ERK½- and p38-MAPK remained unchanged with aging, while the level of JNK-MAPK protein increased by 135% and 100% at 30- and 36-mo, respectively. Aging increased the basal phosphorylation of ERK½ (115% and 47%) and JNK (29% and 69%) (p \u3c0.05) in 30- and 36-mo aortas, while p38 phosphorylation levels remained unaltered. Compared to age-matched controls, fluprostenol induced phosphorylation of ERK½ (310%, 286%, and 554%), p38-MAPK (unchanged, 48%, and 148%), and JNK (78%, 88%, and 95%) in 6-, 30- and 36-mo aortas, respectively. These findings suggest that aging does not affect the ability of the rat aorta to activate ERK½-, p38-MAPK, and JNK-MAPK phosphorylation in response to PGF2α stimulation

Local enumeration and majority lower bounds

Depth-3 circuit lower bounds and k-SAT algorithms are intimately related; the state-of-the-art Σ^k_3-circuit lower bound (Or-And-Or circuits with bottom fan-in at most k) and the k-SAT algorithm of Paturi, Pudlák, Saks, and Zane (J. ACM'05) are based on the same combinatorial theorem regarding k-CNFs. In this paper we define a problem which reveals new interactions between the two, and suggests a concrete approach to significantly stronger circuit lower bounds and improved k-SAT algorithms. For a natural number k and a parameter t, we consider the Enum(k, t) problem defined as follows: given an n-variable k-CNF and an initial assignment α, output all satisfying assignments at Hamming distance t(n) of α, assuming that there are no satisfying assignments of Hamming distance less than t(n) of α. We observe that an upper bound b(n, k, t) on the complexity of Enum(k, t) simultaneously implies depth-3 circuit lower bounds and k-SAT algorithms: - Depth-3 circuits: Any Σ^k_3 circuit computing the Majority function has size at least binom(n,n/2)/b(n, k, n/2). - k-SAT: There exists an algorithm solving k-SAT in time O(∑_{t=1}^{n/2}b(n, k, t)). A simple construction shows that b(n, k, n/2) ≥ 2^{(1 - O(log(k)/k))n}. Thus, matching upper bounds for b(n, k, n/2) would imply a Σ^k_3-circuit lower bound of 2^Ω(log(k)n/k) and a k-SAT upper bound of 2^{(1 - Ω(log(k)/k))n}. The former yields an unrestricted depth-3 lower bound of 2^ω(√n) solving a long standing open problem, and the latter breaks the Super Strong Exponential Time Hypothesis. In this paper, we propose a randomized algorithm for Enum(k, t) and introduce new ideas to analyze it. We demonstrate the power of our ideas by considering the first non-trivial instance of the problem, i.e., Enum(3, n/2). We show that the expected running time of our algorithm is 1.598ⁿ, substantially improving on the trivial bound of 3^{n/2} ≃ 1.732ⁿ. This already improves Σ^3_3 lower bounds for Majority function to 1.251ⁿ. The previous bound was 1.154ⁿ which follows from the work of Håstad, Jukna, and Pudlák (Comput. Complex.'95). By restricting ourselves to monotone CNFs, Enum(k, t) immediately becomes a hypergraph Turán problem. Therefore our techniques might be of independent interest in extremal combinatorics

The interpretation of the field angle dependence of the critical current in defect-engineered superconductors

We apply the vortex path model of critical currents to a comprehensive analysis of contemporary data on defect-engineered superconductors, showing that it provides a consistent and detailed interpretation of the experimental data for a diverse range of materials. We address the question of whether electron mass anisotropy plays a role of any consequence in determining the form of this data and conclude that it does not. By abandoning this false interpretation of the data, we are able to make significant progress in understanding the real origin of the observed behavior. In particular, we are able to explain a number of common features in the data including shoulders at intermediate angles, a uniform response over a wide angular range and the greater discrimination between individual defect populations at higher fields. We also correct several misconceptions including the idea that a peak in the angular dependence of the critical current is a necessary signature of strong correlated pinning, and conversely that the existence of such a peak implies the existence of correlated pinning aligned to the particular direction. The consistency of the vortex path model with the principle of maximum entropy is introduced.Comment: 14 pages, 7 figure

Nested quantum search and NP-complete problems

A quantum algorithm is known that solves an unstructured search problem in a number of iterations of order $\sqrt{d}$, where $d$ is the dimension of the search space, whereas any classical algorithm necessarily scales as $O(d)$. It is shown here that an improved quantum search algorithm can be devised that exploits the structure of a tree search problem by nesting this standard search algorithm. The number of iterations required to find the solution of an average instance of a constraint satisfaction problem scales as $\sqrt{d^\alpha}$, with a constant $\alpha<1$ depending on the nesting depth and the problem considered. When applying a single nesting level to a problem with constraints of size 2 such as the graph coloring problem, this constant $\alpha$ is estimated to be around 0.62 for average instances of maximum difficulty. This corresponds to a square-root speedup over a classical nested search algorithm, of which our presented algorithm is the quantum counterpart.Comment: 18 pages RevTeX, 3 Postscript figure

Altered Regulation of Contraction-Induced Akt/mTOR/p70S6k Pathway Signaling in Skeletal Muscle of the Obese Zucker Rat

Increased muscle loading results in the phosphorylation of the 70 kDa ribosomal S6 kinase (p70S6k), and this event is strongly correlated with the degree of muscle adaptation following resistance exercise. Whether insulin resistance or the comorbidities associated with this disorder may affect the ability of skeletal muscle to activate p70S6k signaling following an exercise stimulus remains unclear. Here, we compare the contraction-induced activation of p70S6k signaling in the plantaris muscles of lean and insulin resistant obese Zucker rats following a single bout of increased contractile loading. Compared to lean animals, the basal phosphorylation of p70S6k (Thr389; 37.2% and Thr421/Ser424; 101.4%), Akt (Thr308; 25.1%), and mTOR (Ser2448; 63.0%) was higher in obese animals. Contraction increased the phosphorylation of p70S6k (Thr389), Akt (Ser473), and mTOR (Ser2448) in both models however the magnitude and kinetics of activation differed between models. These results suggest that contraction-induced activation of p70S6k signaling is altered in the muscle of the insulin resistant obese Zucker rat