Abstract: State feedback stabilization for Boolean control networks is investigated in this technical note. Based on the algebraic representation of logical dynamics in terms of the semi-tensor product of matrices, a necessary and sufficient condition is derived for the existence of a globally stabilizing state feedback controller, and a general control design approach is proposed when global stabilization is feasible via state feedback. Instead of designing the logical form of a stabilizing feedback law directly, we first construct its algebraic representation and then convert the algebraic representation back to the logical form. An example is worked out to illustrate the proposed design procedure.

Abstract: We show, under natural assumptions for qubit systems, that measurement-based quantum computations (MBQCs) which compute a nonlinear Boolean function with a high probability are contextual The class of contextual MBQCs includes an example which is of practical interest and has a superpolynomial speedup over the best-known classical algorithm, namely, the quantum algorithm that solves the ``discrete log'' problem

Abstract: The analysis of large-scale Boolean network dynamics is of great importance in understanding complex phenomena where systems are characterized by a large number of components The computational cost to reveal the number of attractors and the period of each attractor increases exponentially as the number of nodes in the networks increases This paper presents an efficient algorithm to find attractors for medium to large-scale networks This is achieved by analyzing subnetworks within the network in a way that allows to reveal the attractors of the full network with little computational cost In particular, for each subnetwork modeled as a Boolean control network, the input-state cycles are found and they are composed to reveal the attractors of the full network The proposed algorithm reduces the computational cost significantly, especially in finding attractors of short period, or any periods if the aggregation network is acyclic Also, this paper shows that finding the best acyclic aggregation is equivalent to finding the strongly connected components of the network graph Finally, the efficiency of the algorithm is demonstrated on two biological systems, namely a T-cell receptor network and an early flower development network

Abstract: Given oracle access to a Boolean function f:{0,1}n -> {0,1}, we design a randomized tester that takes as input a parameter e>0, and outputs Yes if the function is monotonically non-increasing, and outputs No with probability >2/3, if the function is e-far from being monotone, that is, f needs to be modified at e-fraction of the points to make it monotone. Our non-adaptive, one-sided tester makes ~O(n5/6e-5/3) queries to the oracle.

Abstract: Sequencing problems are among the most widely studied problems in operations research. Specific variations of sequencing problems include single machine scheduling, the traveling salesman problem with time windows, and precedence-constrained machine scheduling. In this work we propose a new approach for solving sequencing problems based on multivalued decision diagrams (MDDs). Decision diagrams are compact graphical representations of Boolean functions, originally introduced for applications in circuit design by Lee [7], and widely studied and applied in computer science. They have been recently used to represent the feasible set of discrete optimization problems, as demonstrated in [2] and [3, 4]. This is done by perceiving the constraints of a problem as a Boolean function f(x) representing whether a solution x is feasible. Nonetheless, such MDDs can grow exponentially large, which makes any practical computation prohibitive in general.

Abstract: Stochastic Boolean Function Evaluation is the problem of determining the value of a given Boolean function f on an unknown input x, when each bit of x_i of x can only be determined by paying an associated cost c_i. The assumption is that x is drawn from a given product distribution, and the goal is to minimize the expected cost. This problem has been studied in Operations Research, where it is known as "sequential testing" of Boolean functions. It has also been studied in learning theory in the context of learning with attribute costs. We consider the general problem of developing approximation algorithms for Stochastic Boolean Function Evaluation. We give a 3-approximation algorithm for evaluating Boolean linear threshold formulas. We also present an approximation algorithm for evaluating CDNF formulas (and decision trees) achieving a factor of O(log kd), where k is the number of terms in the DNF formula, and d is the number of clauses in the CNF formula. In addition, we present approximation algorithms for simultaneous evaluation of linear threshold functions, and for ranking of linear functions. Our function evaluation algorithms are based on reductions to the Stochastic Submodular Set Cover (SSSC) problem. This problem was introduced by Golovin and Krause. They presented an approximation algorithm for the problem, called Adaptive Greedy. Our main technical contribution is a new approximation algorithm for the SSSC problem, which we call Adaptive Dual Greedy. It is an extension of the Dual Greedy algorithm for Submodular Set Cover due to Fujito, which is a generalization of Hochbaum's algorithm for the classical Set Cover Problem. We also give a new bound on the approximation achieved by the Adaptive Greedy algorithm of Golovin and Krause.

Abstract: Quantum satisfiability is a constraint satisfaction problem that generalizes classical boolean satisfiability. In the quantum k-SAT problem, each constraint is specified by a k-local projector and is satisfied by any state in its nullspace. Bravyi showed that quantum 2-SAT can be solved efficiently on a classical computer and that quantum k-SAT with k ≥ 4 is QMA1-complete [4]. Quantum 3-SAT was known to be contained in QMA1 [4], but its computational hardness was unknown until now. We prove that quantum 3-SAT is QMA1-hard, and therefore complete for this complexity class.

Abstract: We consider the partial equivalence checking problem (PEC), i. e., checking whether a given partial implementation of a combinational circuit can (still) be extended to a complete design that is equivalent to a given full specification. To solve PEC, we give a linear transformation from PEC to the question whether a dependency quantified Boolean formula (DQBF) is satisfied. Our novel algorithm to solve DQBF based on quantifier elimination can therefore be applied to solve PEC.We also present first experimental results showing the feasibility of our approach and the inaccuracy of QBF approximations, which are usually used for deciding the PEC so far.

Abstract: Criteria based on the analysis of the properties of vectorial Boolean functions for selection of substitutions (S-boxes) for symmetric cryptographic primitives are given. We propose an improved gradient descent method for increasing performance of nonlinear vectorial Boolean functions generation with optimal cryptographic properties. Substitutions are generated by proposed method for the most common 8-bits input and output messages have nonlinearity 104, 8-uniformity and algebraic immunity 3.

Abstract: Dataflow programming models are well-suited to program many-core streaming applications. However, many streaming applications have a dynamic behavior. To capture this behavior, parametric dataflow models have been introduced over the years. Still, such models do not allow the topology of the dataflow graph to change at runtime, a feature that is also required to program modern streaming applications. To overcome these restrictions, we propose a new model of computation, the Boolean Parametric Data Flow (BPDF) model which combines integer parameters (to express dynamic rates) and boolean parameters (to express the activation and deactivation of communication channels). High dynamism is provided by integer parameters which can change at each basic iteration and boolean parameters which can even change within the iteration. The major challenge with such dynamic models is to guarantee liveness and boundedness. We present static analyses which ensure statically the liveness and the boundedness of BDPF graphs. We also introduce a scheduling methodology to implement our model on highly parallel platforms and demonstrate our approach using a video decoder case study.

Abstract: We provide a general approach for the design of a response Boolean network (BN) to achieve complete synchronization with a given drive BN. The approach is based on the algebraic representation of BNs in terms of the semi-tensor product of matrices. Instead of designing the logical dynamic equations of a response BN directly, we first construct its algebraic representation and then convert the algebraic representation back to the logical form. The results are applied to a three-neuron network in order to illustrate the effectiveness of the proposed approach.

Abstract: This paper explores the evolution of Boolean functions for a cryptographic usage, with genetic algorithms and genetic programming. We also experiment with a new mutation operator and a new kind of initialization process. Results obtained show that those modifications can help in obtaining better solutions. The results indicate that it is possible to obtain high quality Boolean functions with algorithms that are not tailor-made for this purpose. Additionally, among the algorithms tested, the best performance was obtained with variations of genetic programming.

Abstract: We give three new algorithms to solve the “isomorphism of polynomial” problem, which was underlying the hardness of recovering the secret-key in some multivariate trapdoor one-way functions. In this problem, the adversary is given two quadratic functions, with the promise that they are equal up to linear changes of coordinates. Her objective is to compute these changes of coordinates, a task which is known to be harder than Graph-Isomorphism. Our new algorithm build on previous work in a novel way. Exploiting the birthday paradox, we break instances of the problem in time q 2n/3 (rigorously) and q n/2 (heuristically), where q n is the time needed to invert the quadratic trapdoor function by exhaustive search. These results are obtained by turning the algebraic problem into a combinatorial one, namely that of recovering partial information on an isomorphism between two exponentially large graphs. These graphs, derived from the quadratic functions, are new tools in multivariate cryptanalysis.

Abstract: In hardware, substitution boxes for block ciphers can be saved already masked in the implementation. The masks must be chosen under two constraints: their number is determined by the implementation area and their properties should allow to deny high-order zero-offset attacks of highest degree. First, we show that this problem translates into a known trade-off in Boolean functions, namely finding correlation-immune functions of lowest weight. For instance, this allows to prove that a byte-oriented block cipher such as AES can be protected with only 16 mask values against zero-offset correlation power attacks of orders 1, 2 and 3. Second, we study dth-order correlation-immune Boolean functions F2 → F2 of low-weight and exhibit such functions of minimal weight found by a satisfiability modulo theory tool. In particular, we give the minimal weight for n ≤ 10. Some of these results were not known previously, such as the minimal weight for (n = 9, d = 4) and (n = 10, d ∈ {4, 5}). These results set new bounds for the minimal number of lines of binary orthogonal arrays. In particular, we point out that the minimal weight wn,d of a dth-order correlation-immune function might not be increasing with the number of variables n.

Abstract: The e-approximate degree of a Boolean function f: {−1, 1}n→{−1, 1} is the minimum degree of a real polynomial that approximates f to within e in the l∞ norm. We prove several lower bounds on this important complexity measure by explicitly constructing solutions to the dual of an appropriate linear program. Our first result resolves the e-approximate degree of the two-level AND-OR tree for any constant e>0. We show that this quantity is $\Theta(\sqrt{n})$, closing a line of incrementally larger lower bounds [3,11,21,30,32]. The same lower bound was recently obtained independently by Sherstov using related techniques [25]. Our second result gives an explicit dual polynomial that witnesses a tight lower bound for the approximate degree of any symmetric Boolean function, addressing a question of Spalek [34]. Our final contribution is to reprove several Markov-type inequalities from approximation theory by constructing explicit dual solutions to natural linear programs. These inequalities underly the proofs of many of the best-known approximate degree lower bounds, and have important uses throughout theoretical computer science.

Abstract: We show that quantum query complexity satisfies a strong direct product theorem. This means that computing k copies of a function with fewer than k times the quantum queries needed to compute one copy of the function implies that the overall success probability will be exponentially small in k. For a boolean function f, we also show an XOR lemma—computing the parity of k copies of f with fewer than k times the queries needed for one copy implies that the advantage over random guessing will be exponentially small. We do this by showing that the multiplicative adversary method, which inherently satisfies a strong direct product theorem, characterizes bounded-error quantum query complexity. In particular, we show that the multiplicative adversary bound is always at least as large as the additive adversary bound, which is known to characterize bounded-error quantum query complexity.

Abstract: Differentially 4-uniform permutations on $\gf_{2^{2k}}$ with high nonlinearity are often chosen as Substitution boxes in both block and stream ciphers. Recently, Qu et al. introduced a class of functions, which are called preferred functions, to construct a lot of infinite families of such permutations \cite{QTTL}. In this paper, we propose a particular type of Boolean functions to characterize the preferred functions. On the one hand, such Boolean functions can be determined by solving linear equations, and they give rise to a huge number of differentially 4-uniform permutations over $\gf_{2^{2k}}$. Hence they may provide more choices for the design of Substitution boxes. On the other hand, by investigating the number of these Boolean functions, we show that the number of CCZ-inequivalent differentially 4-uniform permutations over $\gf_{2^{2k}}$ grows exponentially when $k$ increases, which gives a positive answer to an open problem proposed in \cite{QTTL}.

Abstract: We consider the power of "linear reconstruction attacks" in statistical data privacy, showing that they can be applied to a much wider range of settings than previously understood. Linear attacks have been studied before [3, 6, 11, 1, 14] but have so far been applied only in settings with releases that are "obviously" linear.Consider a database curator who manages a database of sensitive information but wants to release statistics about how a sensitive attribute (say, disease) in the database relates to some nonsensitive attributes (e.g., postal code, age, gender, etc). This setting is widely considered in the literature, partly since it arises with medical data. Specifically, we show one can mount linear reconstruction attacks based on any release that gives:1. the fraction of records that satisfy a given non-degenerate boolean function. Such releases include contingency tables (previously studied by Kasiviswanathan et al. [11]) as well as more complex outputs like the error rate of classifiers such as decision trees;2. any one of a large class of M-estimators (that is, the output of empirical risk minimization algorithms), including the standard estimators for linear and logistic regression.We make two contributions: first, we show how these types of releases can be transformed into a linear format, making them amenable to existing polynomial-time reconstruction algorithms. This is already perhaps surprising, since many of the above releases (like M-estimators) are obtained by solving highly nonlinear formulations.Second, we show how to analyze the resulting attacks under various distributional assumptions on the data. Specifically, we consider a setting in which the same statistic (either 1 or 2 above) is released about how the sensitive attribute relates to all subsets of size k (out of a total of d) nonsensitive boolean attributes.

Abstract: The $\epsilon$-approximate degree of a Boolean function $f: \{-1, 1\}^n \to \{-1, 1\}$ is the minimum degree of a real polynomial that approximates $f$ to within $\epsilon$ in the $\ell_\infty$ norm. We prove several lower bounds on this important complexity measure by explicitly constructing solutions to the dual of an appropriate linear program. Our first result resolves the $\epsilon$-approximate degree of the two-level AND-OR tree for any constant $\epsilon > 0$. We show that this quantity is $\Theta(\sqrt{n})$, closing a line of incrementally larger lower bounds. The same lower bound was recently obtained independently by Sherstov using related techniques. Our second result gives an explicit dual polynomial that witnesses a tight lower bound for the approximate degree of any symmetric Boolean function, addressing a question of Spalek. Our final contribution is to reprove several Markov-type inequalities from approximation theory by constructing explicit dual solutions to natural linear programs. These inequalities underly the proofs of many of the best-known approximate degree lower bounds, and have important uses throughout theoretical computer science.

Abstract: How to compute a linear Boolean operator by a small circuit using only unbounded fanin addition gates? Because this question is about one of the simplest and most basic circuit models, it has been considered by many authors since the early 1950s. This has led to a variety of upper and lower bound arguments—ranging from algebraic (determinant and matrix rigidity), to combinatorial (Ramsey properties, coverings, and decompositions) to graph-theoretic (the superconcentrator method). We provide a thorough survey of the research in this direction, and prove some new results to fill out the picture. The focus is on the cases in which the addition operation is either the boolean OR or XOR, but the model in which arbitrary boolean functions are allowed as gates is considered as well.

Abstract: We prove that any total boolean function of rank $r$ can be computed by a deterministic communication protocol of complexity $O(\sqrt{r} \cdot \log(r))$. Equivalently, any graph whose adjacency matrix has rank $r$ has chromatic number at most $2^{O(\sqrt{r} \cdot \log(r))}$. This gives a nearly quadratic improvement in the dependence on the rank over previous results.

Abstract: Tensors are becoming increasingly common in data mining, and consequently, tensor factorizations are becoming more important tools for data miners. When the data is binary, it is natural to ask if we can factorize it into binary factors while simultaneously making sure that the reconstructed tensor is still binary. Such factorizations, called Boolean tensor factorizations, can provide improved interpretability and find Boolean structure that is hard to express using normal factorizations. Unfortunately the algorithms for computing Boolean tensor factorizations do not usually scale well. In this paper we present a novel algorithm for finding Boolean CP and Tucker decompositions of large and sparse binary tensors. In our experimental evaluation we show that our algorithm can handle large tensors and accurately reconstructs the latent Boolean structure.

Abstract: We highlight the challenge of proving correlation bounds between boolean functions and real-valued polynomials, where any non-boolean output counts against correlation.We prove that real-valued polynomials of degree 1 2 lg2 lg2n have correlation with parity at most zero. Such a result is false for modular and threshold polynomials. Its proof is based on a variant of an anti-concentration result by Costello et al. [2006].

Abstract: A Boolean function f: {0,1} n ? {0,1} is said to be noise sensitive if inserting a small random error in its argument makes the value of the function almost unpredictable. Benjamini, Kalai and Schramm [3] showed that if the sum of squares of inuences of f is close to zero then f must be noise sensitive. We show a quantitative version of this result which does not depend on n, and prove that it is tight for certain parameters. Our results hold also for a general product measure µ p on the discrete cube, as long as log1/p?logn. We note that in [3], a quantitative relation between the sum of squares of the inuences and the noise sensitivity was also shown, but only when the sum of squares is bounded by n ?c for a constant c. Our results require a generalization of a lemma of Talagrand on the Fourier coefficients of monotone Boolean functions. In order to achieve it, we present a considerably shorter proof of Talagrand's lemma, which easily generalizes in various directions, including non-monotone functions.

Abstract: The synthesis of Boolean functions, as they are found in many quantum algorithms, is usually conducted in two steps. First, the function is realized in terms of a reversible circuit followed by a mapping into a corresponding quantum realization. During this process, the number of lines and the quantum costs of the resulting circuits have mainly been considered as optimization objectives thus far. However, beyond that also the depth of a quantum circuit is vital. Although first synthesis approaches that consider depth have recently been introduced, the majority of design methods did not consider this metric. In this paper, we introduce an optimization approach aiming for the reduction of depth in the process of mapping a reversible circuit into a quantum circuit. For this purpose, we present an improved (local) mapping of single gates as well as a (global) optimization scheme considering the whole circuit. In both cases, we incorporate the idea of exploiting additional circuit lines which are used in order to split a chain of serial gates. Our optimization techniques enable a concurrent application of gates which significantly reduces the depth of the circuit. Experiments show that reductions of approx. 40% on average can be achieved when following this scheme.

Abstract: We consider collocated wireless sensor networks, where each node's transmissions can be heard by every other node. Each node has a Boolean measurement and the goal of the network is to compute a given Boolean function of these measurements. We first consider the worst case setting and study optimal block computation strategies for computing symmetric Boolean functions. We study three classes of functions: threshold functions, delta functions and interval functions. We provide optimal strategies for the first two classes, and a scaling law order-optimal strategy with optimal preconstant for interval functions. We extend the results to the case of integer measurements and certain integer-valued functions. Next, we address the problem of minimizing the expected total number of bits that are transmitted when node measurements are random and drawn from independent Bernoulli distributions. In the case of computing a single instance of a Boolean threshold function, the problem reduces to one of determining the optimal order in which the nodes should transmit. We show that the optimal order of transmissions depends in an extremely simple way on the values of previously transmitted bits and the ordering of the marginal probabilities of the Boolean variables according to the k-th least likely rule: At any transmission, the node that transmits is the one that has the k-th least likely value of its Boolean variable, where k reduces by one whenever a node transmits a one. Initially the value of k is (n +1 - Threshold). Interestingly, the order of transmissions does not depend on the exact values of the probabilities of the Boolean variables. In the case of identically distributed measurements, we further show that the average-case complexity of block computation of a Boolean threshold function is O(θ), where θ is the threshold. We further show how to generalize to a pulse model of communication. One can also consider the related problem of approximate computation given a fixed number of bits. For the special case of the parity function, we show that the greedy strategy is optimal.

Abstract: Qubit (quantum) structures of data and computational processes for significantly improving performance when solving problems of discrete optimization and fault-tolerant design are proposed. We describe a hardware-focused models for parallel (one cycle) calculating the power set (the set of all subsets) on the universe of n primitives for solving coverage problems, minimization of Boolean functions, data compression, analysis and synthesis of digital systems through the implementation of the processor structure in the form of the Hasse diagram. A prototype of quantum device, implemented by programmable logic, is described.