A new PAC bound for intersection-closed concept classes

For hyper-rectangles in $\mathbb{R}^{d}$ Auer (1997) proved a PAC bound of $O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$ , where $\varepsilon$ and $\delta$ are the accuracy and confidence parameters. It is still an open question whether one can obtain the same bound for intersection-closed concept classes of VC-dimension $d$ in general. We present a step towards a solution of this problem showing on one hand a new PAC bound of $O(\frac{1}{\varepsilon}(d\log d + \log \frac{1}{\delta}))$ for arbitrary intersection-closed concept classes, complementing the well-known bounds $O(\frac{1}{\varepsilon}(\log \frac{1}{\delta}+d\log \frac{1}{\varepsilon}))$ and $O(\frac{d}{\varepsilon}\log \frac{1}{\delta})$ of Blumer et al. and (1989) and Haussler, Littlestone and Warmuth (1994). Our bound is established using the closure algorithm, that generates as its hypothesis the intersection of all concepts that are consistent with the positive training examples. On the other hand, we show that many intersection-closed concept classes including e.g. maximum intersection-closed classes satisfy an additional combinatorial property that allows a proof of the optimal bound of $O(\frac{1}{\varepsilon}(d+\log \frac{1}{\delta}))$ . For such improved bounds the choice of the learning algorithm is crucial, as there are consistent learning algorithms that need $\Omega(\frac{1}{\varepsilon}(d\log\frac{1}{\varepsilon} +\log\frac{1}{\delta}))$ examples to learn some particular maximum intersection-closed concept classes.     

Continuous Monitoring of Distributed Data Streams over a Time-Based Sliding Window

In this paper we extend the study of algorithms for monitoring distributed data streams from whole data streams to a time-based sliding window. The concern is how to minimize the communication between individual streams and the root, while allowing the root, at any time, to report the global statistics of all streams within a given error bound. This paper presents communication-efficient algorithms for three classical statistics, namely, basic counting, frequent items and quantiles. The worst-case communication cost over a window is $O(\frac{k}{\varepsilon} \log\frac{\varepsilon N}{k})$ bits for basic counting, $O(\frac{k}{\varepsilon} \log\frac{N}{k})$ words for frequent items and $O(\frac{k}{\varepsilon^{2}} \log\frac{N}{k})$ words for quantiles, where k is the number of distributed data streams, N is the total number of items in the streams that arrive or expire in the window, and ε<1 is the given error bound. The performance of our algorithms matches and nearly matches the corresponding lower bounds. We also show how to generalize these results to streams with out-of-order data.     

Pseudorandom generators for combinatorial checkerboards

We define a combinatorial checkerboard to be a function f : {1, . . . , m} ^d → {1,?1} of the form ${f(u_1,\ldots,u_d)=\prod_{i=1}^df_i(u_i)}$ for some functions f _i : {1, . . . , m} → {1,?1}. This is a variant of combinatorial rectangles, which can be defined in the same way but using {0, 1} instead of {1,?1}. We consider the problem of constructing explicit pseudorandom generators for combinatorial checkerboards. This is a generalization of small-bias generators, which correspond to the case m = 2. We construct a pseudorandom generator that ${\epsilon}$ -fools all combinatorial checkerboards with seed length ${O\bigl(\log m+\log d\cdot\log\log d+\log^{3/2} \frac{1}{\epsilon}\bigr)}$ . Previous work by Impagliazzo, Nisan, and Wigderson implies a pseudorandom generator with seed length ${O\bigl(\log m+\log^2d+\log d\cdot\log\frac{1}{\epsilon}\bigr)}$ . Our seed length is better except when ${\frac{1}{\epsilon}\geq d^{\omega(\log d)}}$ .     

Leader election using loneliness detection

We consider the problem of leader election (LE) in single-hop radio networks with synchronized time slots for transmitting and receiving messages. We assume that the actual number n of processes is unknown, while the size u of the ID space is known, but is possibly much larger. We consider two types of collision detection: strong (SCD), whereby all processes detect collisions, and weak (WCD), whereby only non-transmitting processes detect collisions. We introduce loneliness detection (LD) as a key subproblem for solving LE in WCD systems. LD informs all processes whether the system contains exactly one process or more than one. We show that LD captures the difference in power between SCD and WCD, by providing an implementation of SCD over WCD and LD. We present two algorithms that solve deterministic and probabilistic LD in WCD systems with time costs of ${\mathcal{O}(\log \frac{u}{n})}$ and ${\mathcal{O}(\min( \log \frac{u}{n}, \frac{\log (1/\epsilon)}{n}))}$ , respectively, where ${\epsilon}$ is the error probability. We also provide matching lower bounds. Assuming LD is solved, we show that SCD systems can be emulated in WCD systems with factor-2 overhead in time. We present two algorithms that solve deterministic and probabilistic LE in SCD systems with time costs of ${\mathcal{O}(\log u)}$ and ${\mathcal{O}(\min ( \log u, \log \log n + \log (\frac{1}{\epsilon})))}$ , respectively, where ${\epsilon}$ is the error probability. We provide matching lower bounds.     

Property Testing on k-Vertex-Connectivity of?Graphs

We present an algorithm for testing the k-vertex-connectivity of graphs with the given maximum degree. The time complexity of the algorithm is independent of the number of vertices and edges of graphs. Fixed degree bound d, a graph G with n vertices and a maximum degree at most d is called ε-far from k-vertex-connectivity when at least $\frac{\epsilon dn}{2}$ edges must be added to or removed from G to obtain a k-vertex-connected graph with a maximum degree at most d. The algorithm always accepts every graph that is k-vertex-connected and rejects every graph that is ε-far from k-vertex-connectivity with a probability of at least 2/3. The algorithm runs in $O(d(\frac{c}{\epsilon d})^{k}\log\frac {1}{\epsilon d})$ time (c>1 is a constant) for (k?1)-vertex-connected graphs, and in $O(d(\frac{ck}{\epsilon d})^{k}\log\frac{k}{\epsilon d})$ time (c>1 is a constant) for general graphs. It is the first constant-time k-vertex-connectivity testing algorithm for general k≥4.     

On reducing factorization to the discrete logarithm problem modulo a composite

The discrete logarithm problem modulo a composite??abbreviate it as DLPC??is the following: given a (possibly) composite integer n??? 1 and elements ${a, b \in \mathbb{Z}_n^*}$ , determine an ${x \in \mathbb{N}}$ satisfying a ^x ?=?b if one exists. The question whether integer factoring can be reduced in deterministic polynomial time to the DLPC remains open. In this paper we consider the problem ${{\rm DLPC}_\varepsilon}$ obtained by adding in the DLPC the constraint ${x\le (1-\varepsilon)n}$ , where ${\varepsilon}$ is an arbitrary fixed number, ${0 < \varepsilon\le\frac{1}{2}}$ . We prove that factoring n reduces in deterministic subexponential time to the ${{\rm DLPC}_\varepsilon}$ with ${O_\varepsilon((\ln n)^2)}$ queries for moduli less or equal to n.     

Delayed theory combination vs. Nelson-Oppen for satisfiability modulo theories: a comparative analysis

Most state-of-the-art approaches for Satisfiability Modulo Theories $(SMT(\mathcal{T}))$ rely on the integration between a SAT solver and a decision procedure for sets of literals in the background theory $\mathcal{T} (\mathcal{T}{\text {-}}solver)$ . Often $\mathcal{T}$ is the combination $\mathcal{T}_1 \cup \mathcal{T}_2$ of two (or more) simpler theories $(SMT(\mathcal{T}_1 \cup \mathcal{T}_2))$ , s.t. the specific ${\mathcal{T}_i}{\text {-}}solvers$ must be combined. Up to a few years ago, the standard approach to $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ was to integrate the SAT solver with one combined $\mathcal{T}_1 \cup \mathcal{T}_2{\text {-}}solver$ , obtained from two distinct ${\mathcal{T}_i}{\text {-}}solvers$ by means of evolutions of Nelson and Oppen’s (NO) combination procedure, in which the ${\mathcal{T}_i}{\text {-}}solvers$ deduce and exchange interface equalities. Nowadays many state-of-the-art SMT solvers use evolutions of a more recent $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ procedure called Delayed Theory Combination (DTC), in which each ${\mathcal{T}_i}{\text {-}}solver$ interacts directly and only with the SAT solver, in such a way that part or all of the (possibly very expensive) reasoning effort on interface equalities is delegated to the SAT solver itself. In this paper we present a comparative analysis of DTC vs. NO for $SMT(\mathcal{T}_1 \cup \mathcal{T}_2)$ . On the one hand, we explain the advantages of DTC in exploiting the power of modern SAT solvers to reduce the search. On the other hand, we show that the extra amount of Boolean search required to the SAT solver can be controlled. In fact, we prove two novel theoretical results, for both convex and non-convex theories and for different deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ , which relate the amount of extra Boolean search required to the SAT solver by DTC with the number of deductions and case-splits required to the ${\mathcal{T}_i}{\text {-}}solvers$ by NO in order to perform the same tasks: (i) under the same hypotheses of deduction capabilities of the ${\mathcal{T}_i}{\text {-}}solvers$ required by NO, DTC causes no extra Boolean search; (ii) using ${\mathcal{T}_i}{\text {-}}solvers$ with limited or no deduction capabilities, the extra Boolean search required can be reduced down to a negligible amount by controlling the quality of the $\mathcal{T}$ -conflict sets returned by the ${\mathcal{T}_i}{\text {-}}solvers$ .     

Tight Bounds for Adopt-Commit Objects

We give matching upper and lower bounds of \(\varTheta(\min(\frac{\log m}{\log \log m},\, n))\) for the individual step complexity of a wait-free m-valued adopt-commit object implemented using multi-writer registers for n anonymous processes. While the upper bound is deterministic, the lower bound holds for randomized adopt-commit objects as well. Our results are based on showing that adopt-commit objects are equivalent, up to small additive constants, to a simpler class of objects that we call conflict detectors. Our anonymous lower bound also applies to the individual step complexity of m-valued wait-free anonymous consensus, even for randomized algorithms with global coins against an oblivious adversary. The upper bound can be used to slightly improve the cost of randomized consensus against an oblivious adversary. For deterministic non-anonymous implementations of adopt-commit objects, we show a lower bound of \(\varOmega(\min(\frac{\log m}{\log \log m}, \frac{\sqrt{\log n}}{\log \log n}))\) and an upper bound of \(O(\min(\frac{\log m}{\log \log m},\, \log n))\) on the worst-case individual step complexity. For randomized non-anonymous implementations, we show that any execution contains at least one process whose steps exceed the deterministic lower bound.     

Stackelberg Network Pricing Games

We study a multi-player one-round game termed Stackelberg Network Pricing Game, in which a leader can set prices for a subset of m priceable edges in a graph. The other edges have a fixed cost. Based on the leader’s decision one or more followers optimize a polynomial-time solvable combinatorial minimization problem and choose a minimum cost solution satisfying their requirements based on the fixed costs and the leader’s prices. The leader receives as revenue the total amount of prices paid by the followers for priceable edges in their solutions. Our model extends several known pricing problems, including single-minded and unit-demand pricing, as well as Stackelberg pricing for certain follower problems like shortest path or minimum spanning tree. Our first main result is a tight analysis of a single-price algorithm for the single follower game, which provides a (1+ε)log?m-approximation. This can be extended to provide a (1+ε)(log?k+log?m)-approximation for the general problem and k followers. The problem is also shown to be hard to approximate within $\mathcal{O}(\log^{\varepsilon}k + \log^{\varepsilon}m)$ for some ε>0. If followers have demands, the single-price algorithm provides an $\mathcal{O}(m^{2})$ -approximation, and the problem is hard to approximate within $\mathcal{O}(m^{\epsilon})$ for some ε>0. Our second main result is a polynomial time algorithm for revenue maximization in the special case of Stackelberg bipartite vertex-cover, which is based on non-trivial max-flow and LP-duality techniques. This approach can be extended to provide constant-factor approximations for any constant number of followers.     

Self assembly of rectangular shapes on concentration programming and probabilistic tile assembly models

Efficient tile sets for self assembling rectilinear shapes is of critical importance in algorithmic self assembly. A lower bound on the tile complexity of any deterministic self assembly system for an n?×?n square is $\Upomega(\frac{\log(n)}{\log(\log(n))})$ (inferred from the Kolmogrov complexity). Deterministic self assembly systems with an optimal tile complexity have been designed for squares and related shapes in the past. However designing $\Uptheta(\frac{\log(n)}{\log(\log(n))})$ unique tiles specific to a shape is still an intensive task in the laboratory. On the other hand copies of a tile can be made rapidly using PCR (polymerase chain reaction) experiments. This led to the study of self assembly on tile concentration programming models. We present two major results in this paper on the concentration programming model. First we show how to self assemble rectangles with a fixed aspect ratio (??:??), with high probability, using $\Uptheta(\alpha+\beta)$ tiles. This result is much stronger than the existing results by Kao et?al. (Randomized self-assembly for approximate shapes, LNCS, vol 5125. Springer, Heidelberg, 2008) and Doty (Randomized self-assembly for exact shapes. In: proceedings of the 50th annual IEEE symposium on foundations of computer science (FOCS), IEEE, Atlanta. pp 85?C94, 2009)??which can only self assembly squares and rely on tiles which perform binary arithmetic. On the other hand, our result is based on a technique called staircase sampling. This technique eliminates the need for sub-tiles which perform binary arithmetic, reduces the constant in the asymptotic bound, and eliminates the need for approximate frames (Kao et?al. Randomized self-assembly for approximate shapes, LNCS, vol 5125. Springer, Heidelberg, 2008) . Our second result applies staircase sampling on the equimolar concentration programming model (The tile complexity of linear assemblies. In: proceedings of the 36th international colloquium automata, languages and programming: Part I on ICALP ??09, Springer-Verlag, pp 235?C253, 2009), to self assemble rectangles (of fixed aspect ratio) with high probability. The tile complexity of our algorithm is $\Uptheta(\log(n))$ and is optimal on the probabilistic tile assembly model (PTAM)??n being an upper bound on the dimensions of a rectangle.     

An Improved Approximation Scheme for Variable-Sized Bin Packing

The Variable-Sized Bin Packing Problem (abbreviated as VSBPP or VBP) is a well-known generalization of the NP-hard Bin Packing Problem (BP) where the items can be packed in bins of M given sizes. The objective is to minimize the total capacity of the bins used. We present an asymptotic approximation scheme (AFPTAS) for VBP and BP with performance guarantee \(A_{\varepsilon }(I) \leq (1+ \varepsilon )OPT(I) + \mathcal {O}\left ({\log ^{2}\left (\frac {1}{\varepsilon }\right )}\right )\) for any problem instance I and any ε>0. The additive term is much smaller than the additive term of already known AFPTAS. The running time of the algorithm is \(\mathcal {O}\left ({ \frac {1}{\varepsilon ^{6}} \log \left ({\frac {1}{\varepsilon }}\right ) + \log \left ({\frac {1}{\varepsilon }}\right ) n}\right )\) for BP and \(\mathcal {O}\left ({ \frac {1}{{\varepsilon }^{6}} \log ^{2}\left ({\frac {1}{\varepsilon }}\right ) + M + \log \left ({\frac {1}{\varepsilon }}\right )n}\right )\) for VBP, which is an improvement to previously known algorithms. Many approximation algorithms have to solve subproblems, for example instances of the Knapsack Problem (KP) or one of its variants. These subproblems - like KP - are in many cases NP-hard. Our AFPTAS for VBP must in fact solve a generalization of KP, the Knapsack Problem with Inversely Proportional Profits (KPIP). In this problem, one of several knapsack sizes has to be chosen. At the same time, the item profits are inversely proportional to the chosen knapsack size so that the largest knapsack in general does not yield the largest profit. We introduce KPIP in this paper and develop an approximation scheme for KPIP by extending Lawler’s algorithm for KP. Thus, we are able to improve the running time of our AFPTAS for VBP.     

An efficient simulation algorithm on Kripke structures

A number of algorithms for computing the simulation preorder (and equivalence) on Kripke structures are available. Let $\varSigma $ denote the state space, ${\rightarrow }$ the transition relation and $P_{\mathrm {sim}}$ the partition of $\varSigma $ induced by simulation equivalence. While some algorithms are designed to reach the best space bounds, whose dominating additive term is $|P_{\mathrm {sim}}|^2$ , other algorithms are devised to attain the best time complexity $O(|P_{\mathrm {sim}}||{\rightarrow }|)$ . We present a novel simulation algorithm which is both space and time efficient: it runs in $O(|P_ {\mathrm {sim}}|^2 \log |P_{\mathrm {sim}}| + |\varSigma |\log |\varSigma |)$ space and $O(|P_{\mathrm {sim}}||{\rightarrow }|\log |\varSigma |)$ time. Our simulation algorithm thus reaches the best space bounds while closely approaching the best time complexity.     

Effective Randomness of Unions and Intersections

We investigate the ??-randomness of unions and intersections of random sets under various notions of randomness corresponding to different probability measures. For example, the union of two relatively Martin-Löf random sets is not Martin-Löf random but is random with respect to the Bernoulli measure $\lambda_{\frac{3}{4}}$ under which any number belongs to the set with probability $\frac{3}{4}$ . Conversely, any $\lambda_{\frac{3}{4}}$ random set is the union of two Martin-Löf random sets. Unions and intersections of random closed sets are also studied.     

Fully Smoothed ℓ 1-TV Models: Bounds for the Minimizers and Parameter Choice

We consider a class of convex functionals that can be seen as $\mathcal{C}^{1}$ smooth approximations of the ? ₁-TV model. The minimizers of such functionals were shown to exhibit a qualitatively different behavior compared to the nonsmooth ? ₁-TV model (Nikolova et al. in Exact histogram specification for digital images using a variational approach, 2012). Here we focus on the way the parameters involved in these functionals determine the features of the minimizers  $\hat{u}$ . We give explicit relationships between the minimizers and these parameters. Given an input digital image f, we prove that the error $\|\hat{u}- f\| _{\infty}$ obeys $b-\varepsilon\leq\|\hat{u}-f\|_{\infty}\leq b$ where b is a constant independent of the input image. Further we can set the parameters so that ε>0 is arbitrarily close to zero. More precisely, we exhibit explicit formulae relating the model parameters, the input image f and the values b and ε. Conversely, we can fix the parameter values so that the error $\|\hat{u}- f\|_{\infty}$ meets some prescribed b,ε. All theoretical results are confirmed using numerical tests on natural digital images of different sizes with disparate content and quality.     

Plaintext awareness in identity-based key encapsulation

The notion of plaintext awareness ( ${\mathsf{PA}}$ ) has many applications in public key cryptography: it offers unique, stand-alone security guarantees for public key encryption schemes, has been used as a sufficient condition for proving indistinguishability against adaptive chosen-ciphertext attacks ( ${\mathsf{IND}\hbox {-}{\mathsf{CCA}}}$ ), and can be used to construct privacy-preserving protocols such as deniable authentication. Unlike many other security notions, plaintext awareness is very fragile when it comes to differences between the random oracle and standard models; for example, many implications involving ${\mathsf{PA}}$ in the random oracle model are not valid in the standard model and vice versa. Similarly, strategies for proving ${\mathsf{PA}}$ of schemes in one model cannot be adapted to the other model. Existing research addresses ${\mathsf{PA}}$ in detail only in the public key setting. This paper gives the first formal exploration of plaintext awareness in the identity-based setting and, as initial work, proceeds in the random oracle model. The focus is laid mainly on identity-based key encapsulation mechanisms (IB-KEMs), for which the paper presents the first definitions of plaintext awareness, highlights the role of ${\mathsf{PA}}$ in proof strategies of ${\mathsf{IND}\hbox {-}{\mathsf{CCA}}}$ security, and explores relationships between ${\mathsf{PA}}$ and other security properties. On the practical side, our work offers the first, highly efficient, general approach for building IB-KEMs that are simultaneously plaintext-aware and ${\mathsf{IND}\hbox {-}{\mathsf{CCA}}}$ -secure. Our construction is inspired by the Fujisaki-Okamoto (FO) transform, but demands weaker and more natural properties of its building blocks. This result comes from a new look at the notion of $\gamma $ -uniformity that was inherent in the original FO transform. We show that for IB-KEMs (and PK-KEMs), this assumption can be replaced with a weaker computational notion, which is in fact implied by one-wayness. Finally, we give the first concrete IB-KEM scheme that is ${\mathsf{PA}}$ and ${\mathsf{IND}\hbox {-}{\mathsf{CCA}}}$ -secure by applying our construction to a popular IB-KEM and optimizing it for better performance.     

On the Exact Complexity of Evaluating Quantified k -CNF

We relate the exponential complexities 2 ^s(k)n of $\textsc {$k$-sat}$ and the exponential complexity $2^{s(\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf}))n}$ of $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ (the problem of evaluating quantified formulas of the form $\forall\vec{x} \exists\vec{y} \textsc {F}(\vec {x},\vec{y})$ where F is a 3-cnf in $\vec{x}$ variables and $\vec{y}$ variables) and show that s(∞) (the limit of s(k) as k→∞) is at most $s(\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf}))$ . Therefore, if we assume the Strong Exponential-Time Hypothesis, then there is no algorithm for $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ running in time 2 ^cn with c<1. On the other hand, a nontrivial exponential-time algorithm for $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ would provide a $\textsc {$k$-sat}$ solver with better exponent than all current algorithms for sufficiently large k. We also show several syntactic restrictions of the evaluation problem $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ have nontrivial algorithms, and provide strong evidence that the hardest cases of $\textsc {eval}(\mathrm {\varPi }_{2} 3\textsc {-cnf})$ must have a mixture of clauses of two types: one universally quantified literal and two existentially quantified literals, or only existentially quantified literals. Moreover, the hardest cases must have at least n?o(n) universally quantified variables, and hence only o(n) existentially quantified variables. Our proofs involve the construction of efficient minimally unsatisfiable $\textsc {$k$-cnf}$ s and the application of the Sparsification lemma.     

Derandomized Parallel Repetition Theorems for Free Games

Raz’s parallel repetition theorem (SIAM J Comput 27(3):763–803, 1998) together with improvements of Holenstein (STOC, pp 411–419, 2007) shows that for any two-prover one-round game with value at most ${1- \epsilon}$ 1 - ? (for ${\epsilon \leq 1/2}$ ? ≤ 1 / 2 ), the value of the game repeated n times in parallel on independent inputs is at most ${(1- \epsilon)^{\Omega(\frac{\epsilon^2 n}{\ell})}}$ ( 1 - ? ) Ω ( ? 2 n ? ) , where ? is the answer length of the game. For free games (which are games in which the inputs to the two players are uniform and independent), the constant 2 can be replaced with 1 by a result of Barak et al. (APPROX-RANDOM, pp 352–365, 2009). Consequently, ${n=O(\frac{t \ell}{\epsilon})}$ n = O ( t ? ? ) repetitions suffice to reduce the value of a free game from ${1- \epsilon}$ 1 - ? to ${(1- \epsilon)^t}$ ( 1 - ? ) t , and denoting the input length of the game by m, it follows that ${nm=O(\frac{t \ell m}{\epsilon})}$ n m = O ( t ? m ? ) random bits can be used to prepare n independent inputs for the parallel repetition game. In this paper, we prove a derandomized version of the parallel repetition theorem for free games and show that O(t(m + ?)) random bits can be used to generate correlated inputs, such that the value of the parallel repetition game on these inputs has the same behavior. That is, it is possible to reduce the value from ${1- \epsilon}$ 1 - ? to ${(1- \epsilon)^t}$ ( 1 - ? ) t while only multiplying the randomness complexity by O(t) when m = O(?). Our technique uses strong extractors to “derandomize” a lemma of Raz and can be also used to derandomize a parallel repetition theorem of Parnafes et al. (STOC, pp 363–372, 1997) for communication games in the special case that the game is free.     

Arithmetizing Classes Around \textsf{NC} 1 and \textsf{L}

The parallel complexity class $\textsf{NC}$ ¹ has many equivalent models such as polynomial size formulae and bounded width branching programs. Caussinus et al. (J. Comput. Syst. Sci. 57:200–212, 1992) considered arithmetizations of two of these classes, $\textsf{\#NC}$ ¹ and $\textsf{\#BWBP}$ . We further this study to include arithmetization of other classes. In particular, we show that counting paths in branching programs over visibly pushdown automata is in $\textsf{FLogDCFL}$ , while counting proof-trees in logarithmic width formulae has the same power as $\textsf{\#NC}$ ¹. We also consider polynomial-degree restrictions of $\textsf{SC}$ ⁱ , denoted $\textsf{sSC}$ ⁱ , and show that the Boolean class $\textsf{sSC}$ ¹ is sandwiched between $\textsf{NC}$ ¹ and $\textsf{L}$ , whereas $\textsf{sSC}$ ⁰ equals $\textsf{NC}$ ¹. On the other hand, the arithmetic class $\textsf{\#sSC}$ ⁰ contains $\textsf{\#BWBP}$ and is contained in $\textsf{FL}$ , and $\textsf{\#sSC}$ ¹ contains $\textsf{\#NC}$ ¹ and is in $\textsf{SC}$ ². We also investigate some closure properties of the newly defined arithmetic classes.     

Pseudorandom generators for CC0[p] and the Fourier spectrum of low-degree polynomials over finite fields

In this paper, we give the first construction of a pseudorandom generator, with seed length O(log n), for CC⁰[p], the class of constant-depth circuits with unbounded fan-in MOD _p gates, for some prime p. More accurately, the seed length of our generator is O(log n) for any constant error ${\epsilon > 0}$ . In fact, we obtain our generator by fooling distributions generated by low-degree polynomials, over ${\mathbb{F}_p}$ , when evaluated on the Boolean cube. This result significantly extends previous constructions that either required a long seed (Luby et al. 1993) or could only fool the distribution generated by linear functions over ${\mathbb{F}_p}$ , when evaluated on the Boolean cube (Lovett et al. 2009; Meka & Zuckerman 2009). En route of constructing our PRG, we prove two structural results for low-degree polynomials over finite fields that can be of independent interest.

1. Let f be an n-variate degree d polynomial over ${\mathbb{F}_p}$ . Then, for every ${\epsilon > 0}$ , there exists a subset ${S \subset [n]}$ , whose size depends only on d and ${\epsilon}$ , such that ${\sum_{\alpha \in \mathbb{F}_p^n: \alpha \ne 0, \alpha_S=0}|\hat{f}(\alpha)|^2 \leq \epsilon}$ . Namely, there is a constant size subset S such that the total weight of the nonzero Fourier coefficients that do not involve any variable from S is small.
2. Let f be an n-variate degree d polynomial over ${\mathbb{F}_p}$ . If the distribution of f when applied to uniform zero–one bits is ${\epsilon}$ -far (in statistical distance) from its distribution when applied to biased bits, then for every ${\delta > 0}$ , f can be approximated over zero–one bits, up to error δ, by a function of a small number (depending only on ${\epsilon,\delta}$ and d) of lower degree polynomials.

     

Improved direct product theorems for randomized query complexity

The ??direct product problem?? is a fundamental question in complexity theory which seeks to understand how the difficulty in computing a function on each of k independent inputs scales with k. We prove the following direct product theorem (DPT) for query complexity: if every T-query algorithm has success probability at most ${1 - \varepsilon}$ in computing the Boolean function f on input distribution???, then for ?? ?? 1, every ${\alpha \varepsilon Tk}$ -query algorithm has success probability at most ${(2^{\alpha \varepsilon}(1-\varepsilon))^k}$ in computing the k-fold direct product ${f^{\otimes k}}$ correctly on k independent inputs from???. In light of examples due to Shaltiel, this statement gives an essentially optimal trade-off between the query bound and the error probability. Using this DPT, we show that for an absolute constant ?? > 0, the worst-case success probability of any ?? R ₂(f) k-query randomized algorithm for ${f^{\otimes k}}$ falls exponentially with k. The best previous statement of this type, due to Klauck, ?palek, and de Wolf, required a query bound of O(bs(f) k). Our proof technique involves defining and analyzing a collection of martingales associated with an algorithm attempting to solve ${f^{\otimes k}}$ . Our method is quite general and yields a new XOR lemma and threshold DPT for the query model, as well as DPTs for the query complexity of learning tasks, search problems, and tasks involving interaction with dynamic entities. We also give a version of our DPT in which decision tree size is the resource of interest.