Abstract: Reversible logic is the basis for several emerging technologies such as quantum computing, optical computing, or DNA computing and has further applications in domains like low-power design and nanotechnologies. However, current methods for the synthesis of reversible logic are limited, i.e. they are applicable to relatively small functions only. In this paper, we propose a synthesis approach, that can cope with Boolean functions containing more than a hundred of variables. We present a technique to derive reversible circuits for a function given by a Binary Decision Diagram (BDD). The circuit is obtained using an algorithm with linear worst case behavior regarding run-time and space requirements. Furthermore, the size of the resulting circuit is bounded by the BDD size. This allows to transfer theoretical results known from BDDs to reversible circuits. Experiments show better results (with respect to the circuit cost) and a significantly better scalability in comparison to previous synthesis approaches.

Abstract: In this paper computation with memristors is studied in terms of how many memristors are needed to perform a given logic operation. It has been shown that memristors are naturally suited for performing implication logic (combination of implication and false operation) instead of Boolean logic. Also, it should be noted that a memristor can be used as both a logic gate and a latch (stateful logic). Being functionally complete, implication logic can be used to compute any Boolean function. However, by performing implication logic with stateful devices, storage of intermediate results requires additional memristors to keep data yet to be used from being written over. This paper describes an effective way to compute any Boolean function with a small number of memristors. Also, the length of the corresponding computing sequence is considered.

Abstract: This paper investigates the structure of Boolean networks via input-state structure. Using the algebraic form proposed by the author, the logic-based input-state dynamics of Boolean networks, called the Boolean control networks, is converted into an algebraic discrete-time dynamic system. Then the structure of cycles of Boolean control systems is obtained as compounded cycles. Using the obtained input-state description, the structure of Boolean networks is investigated, and their attractors are revealed as nested compounded cycles, called rolling gears. This structure explains why small cycles mainly decide the behaviors of cellular networks. Some illustrative examples are presented.

Abstract: Several successful approaches to software verification are based on the construction and analysis of an abstract reachability tree (ART). The ART represents unwindings of the control-flow graph of the program. Traditionally, a transition of the ART represents a single block of the program, and therefore, we call this approach single-block encoding (SBE). SBE may result in a huge number of program paths to be explored, which constitutes a fundamental source of inefficiency. We propose a generalization of the approach, in which transitions of the ART represent larger portions of the program; we call this approach large-block encoding (LBE). LBE may reduce the number of paths to be explored up to exponentially. Within this framework, we also investigate symbolic representations: for representing abstract states, in addition to conjunctions as used in SBE, we investigate the use of arbitrary Boolean formulas; for computing abstract-successor states, in addition to Cartesian predicate abstraction as used in SBE, we investigate the use of Boolean predicate abstraction. The new encoding leverages the efficiency of state-of-the-art SMT solvers, which can symbolically compute abstract large-block successors. Our experiments on benchmark C programs show that the large-block encoding outperforms the single-block encoding.

Abstract: The general adversary bound is a semi-definite program (SDP) that lower-bounds the quantum query complexity of a function. We turn this lower bound into an upper bound, by giving a quantum walk algorithm based on the dual SDP that has query complexity at most the general adversary bound, up to a logarithmic factor. In more detail, the proof has two steps, each based on "span programs," a certain linear-algebraic model of computation. First, we give an SDP that outputs for any boolean function a span program computing it that has optimal "witness size." The optimal witness size is shown to coincide with the general adversary lower bound. Second, we give a quantum algorithm for evaluating span programs with only a logarithmic query overhead on the witness size. The first result is motivated by a quantum algorithm for evaluating composed span programs. The algorithm is known to be optimal for evaluating a large class of formulas. The allowed gates include all constant-size functions for which there is an optimal span program. So far, good span programs have been found in an ad hoc manner, and the SDP automates this procedure. Surprisingly, the SDP's value equals the general adversary bound. A corollary is an optimal quantum algorithm for evaluating "balanced" formulas over any finite boolean gate set. The second result extends span programs' applicability beyond the formula evaluation problem. A strong universality result for span programs follows. A good quantum query algorithm for a problem implies a good span program, and vice versa. Although nearly tight, this equivalence is nontrivial. Span programs are a promising model for developing more quantum algorithms.

Abstract: There have existed several "card-based protocols" for secure computations of a Boolean function such as AND and XOR. The best result currently known is that AND and XOR can be securely computed using 8 cards and 10 cards, respectively. In this paper, we improve the result: we design a 6-card AND protocol and a 4-card XOR protocol. Thus, this paper succeeds in reducing the number of required cards for secure computations.

Abstract: We show that any $k$-wise independent probability distribution on $\{0,1\}^n$ $O(m^{2.2}$ $2^{-\sqrt{k}/10})$-fools any boolean function computable by an $m$-clause disjunctive normal form (DNF) (or conjunctive normal form (CNF)) formula on $n$ variables. Thus, for each constant $e>0$, there is a constant $c>0$ such that any boolean function computable by an $m$-clause DNF (or CNF) formula is $m^{-e}$-fooled by any $c\log^2m$-wise probability distribution. This resolves up to an $O(\log m)$ factor the depth-2 circuit case of a conjecture due to Linial and Nisan [Combinatorica, 10 (1990), pp. 349-365]. The result is equivalent to a new characterization of DNF (or CNF) formulas by low degree polynomials. It implies a similar statement for probability distributions with the small bias property. Using known explicit constructions of small probability spaces having the limited independence property or the small bias property, we directly obtain a large class of explicit pseudorandom generators of $O(\log^2m\log n)$-seed length for $m$-clause DNF (or CNF) formulas on $n$ variables, improving previously known seed lengths.

Abstract: Symmetric Boolean functions with even variables 2k and maximum algebraic immunity AI(f) = k have been constructed in Braeken's thesis (2006). In this paper, we show more constructions of such Boolean functions including the generalization of a result and prove a conjecture raised in Braeken's thesis (2006).

Abstract: Determining the equivalence of reversible circuits designed to meet a common specification is considered. The circuits' primary inputs and outputs must be in pure logic states but the circuits may include elementary quantum gates in addition to reversible logic gates. The specification can include don't-cares arising from constant inputs, garbage outputs, and total or partial don't-cares in the underlying target function. The paper explores well-known techniques from irreversible equivalence checking and how they can be applied in the domain of reversible circuits. Two approaches are considered. The first employs decision diagram techniques and the second uses Boolean satisfiability. Experimental results show that for both methods, circuits with up to 27,000 gates, as well as adders with more than 100 inputs and outputs, are handled in under three minutes with reasonable memory requirements.

Abstract: The use of firewalls to enforce access control policies can result in extremely complex networks. Each individual firewall may have hundreds or thousands of rules, and when combined in a network, they may result in unexpected combined behavior. To mitigate this problem, there has been recent interest in the use of model checking techniques for analyzing the behavior of firewall policy configurations, and reporting anomalies. Existing techniques for firewall policy analysis are based on decision diagrams, most normally reduced ordered Binary Decision Diagrams (BDDs). BDDs are a rich data structure, supporting more logical operations than just solving boolean formulae. Typically, search algorithms for boolean satisfiability (so-called SAT-solvers) outperform BDDs. In this paper, we show that the extra structure provided by BDDs is not necessary for firewall policy analysis, and that SAT solvers are sufficient. This argument is supported both by theoretical analysis and by experimental data.

Abstract: It is well known that the control/intervention of some genes in a genetic regulatory network is useful for avoiding undesirable states associated with some diseases like cancer. For this purpose, both optimal finite-horizon control and infinite-horizon control policies have been proposed. Boolean networks (BNs) and its extension probabilistic Boolean networks (PBNs) as useful and effective tools for modelling gene regulatory systems have received much attention in the biophysics community. The control problem for these models has been studied widely. The optimal control problem in a PBN can be formulated as a probabilistic dynamic programming problem. In the previous studies, the optimal control problems did not take into account the hard constraints, i.e. to include an upper bound for the number of controls that can be applied to the captured PBN. This is important as more treatments may bring more side effects and the patients may not bear too many treatments. A formulation for the optimal finite-horizon control problem with hard constraints introduced by the authors. This model is state independent and the objective function is only dependent on the distance between the desirable states and the terminal states. An approximation method is also given to reduce the computational cost in solving the problem. Experimental results are given to demonstrate the efficiency of our proposed formulations and methods.

Abstract: We give a simple, more efficient and uniform proof of the hard-core lemma, a fundamental result in complexity theory with applications in machine learning and cryptography. Our result follows from the connection between boosting algorithms and hard-core set constructions discovered by Klivans and Servedio [11]. Informally stated, our result is the following: suppose we fix a family of boolean functions. Assume there is an efficient algorithm which for every input length and every smooth distribution (i.e. one that doesn't assign too much weight to any single input) over the inputs produces a circuit such that the circuit computes the boolean function noticeably better than random. Then, there is an efficient algorithm which for every input length produces a circuit that computes the function correctly on almost all inputs.Our algorithm significantly simplifies previous proofs of the uniform and the non-uniform hard-core lemma, while matching or improving the previously best known parameters. The algorithm uses a generalized multiplicative update rule combined with a natural notion of approximate Bregman projection. Bregman projections are widely used in convex optimization and machine learning. We present an algorithm which efficiently approximates the Bregman projection onto the set of high density measures when the Kullback-Leibler divergence is used as a distance function. Our algorithm has a logarithmic runtime over any domain from which we can efficiently sample. High density measures correspond to smooth distributions which arise naturally, for instance, in the context of online learning. Hence, our technique may be of independent interest.

Abstract: We give a new model of learning motivated by smoothed analysis (Spielman and Teng, 2001). In this model, we analyze two new algorithms, for PAC-learning DNFs and agnostically learning decision trees, from random examples drawn from a constant-bounded product distributions. These two problems had previously been solved using membership queries (Jackson, 1995; Gopalan et al, 2005). Our analysis demonstrates that the "heavy" Fourier coefficients of a DNF suffice to recover the DNF. We also show that a structural property of the Fourier spectrum of any boolean function over "typical" product distributions. In a second model, we consider a simple new distribution over the boolean hypercube, one which is symmetric but is not the uniform distribution, from which we can learn O(log n)-depth decision trees in polynomial time.

Abstract: During the IC design process, functional specifications are often modified late in the design cycle, after placement and routing are completed. However, designers are left either to manually process such modifications by hand or to restart the design process from scratch — a very costly option. In order to address this issue, we present DeltaSyn, a method for generating a highly optimized logic difference between a modified high-level specification and an implemented design. DeltaSyn has the ability to locate boundaries in implemented logic within which changes can be confined. Delta-Syn demarcates the boundary in two phases. The first phase employs fast functional and structural analysis techniques to identify equivalent signals forming the input-side boundary of the changes. The second phase locates the output-side boundary of the changes through a novel dynamic algorithm that detects matching logic downstream from the changes required by the ECO. Experiments on industrial designs show that together these techniques successfully implement ECOs while preserving an average of 97% of the existing logic. Unlike previous approaches, the use of bit-parallel logic simulation and fast SAT solvers enables high performance and scalability. DeltaSyn can process and verify a typical ECO for a design of around 10K gates in about 200 seconds or less.

Abstract: In a recent paper, the authors introduced a method for constructing new quadratic APN functions from known ones. Applying this method, they obtained the function x3 + tr n (x9) which is APN over F 2 n for any positive integer n. The present paper is a continuation of this work. We give sufficient conditions on linear functions L 1 and L 2 from F 2 n to itself such that the function L 1 (x3) + L 2 (x9) is APN over F 2 n . We show that this can lead to many new cases of APN functions. In particular, we get two families of APN functions x3 + a−1 tr3 n (a3x9 + a6x18) and x3 + a−1 tr3 n (a6x18 + a12x36) over F 2 n for any n divisible by 3 and a Є F∗ 2 n . We prove that for n=9, these families are pairwise different and differ from all previously known families of APN functions, up to the most general equivalence notion, the CCZ-equivalence. We also investigate further sufficient conditions under which the conditions on the linear functions L 1 and L 2 are satisfied.

Abstract: With cryptographic investigations, the design of Boolean functions is a wide area. The Boolean functions play important role in the construction of a symmetric cryptosystem. In this paper the modified hill climbing method is considered. Using hill climbing techniques, the method allows modifying bent functions used to design balanced, highly non-linear Boolean functions with high algebraic degree and low autocorrelation. The experimental results of constructing the cryptographically strong Boolean functions are presented.

Abstract: In this paper, a technique on constructing nonlinear resilient Boolean functions is described. By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number of variables can be constructed. It is shown that given any $m$, one can construct infinitely many $n$-variable ($n$ even), $m$-resilient functions with nonlinearity $>2^{n-1}-2^{n/2}$. A large class of highly nonlinear resilient functions which were not known are obtained. Then one method to optimize the degree of the constructed functions is proposed. Last, an improved version of the main construction is given.

Abstract: Boolean relations are an important tool in system synthesis and verification to characterize solutions to a set of Boolean constraints. For physical realization as hardware, a deterministic function often has to be extracted from a relation. Prior methods however are unlikely to handle large problem instances. From the scalability standpoint this paper demonstrates how interpolation can be exploited to extend deter-minization capacity. A comparative study is performed on several proposed computation techniques. Experimental results show that Boolean relations with thousands of variables can be effectively determinized and the extracted functional implementations are of reasonable quality.

Abstract: Linear functions are a source of weakness in a cryptographic algorithm because such functions are easy to invert and have very predictable outputs; hence the essential cryptographic property of randomness is made more difficult to achieve if any part of the algorithm is linear, or close to linear. Thus it is desirable for the functions used in the algorithm to have high nonlinearity. Bent Boolean functions, which are only defined for an even number n of variables, are functions which have the maximum Hamming distance from the set of all linear functions, and so have the maximum possible nonlinearity. This chapter gives a full discussion of all aspects of the theory of bent functions, including equivalent definitions of bent, ways to construct bent functions, properties of bent functions and counts of bent functions as n varies. The chapter concludes with discussions of the related partially bent functions and semi-bent functions.

Abstract: Universal perceptron (UP), a generalization of Rosenblatt's perceptron, is considered in this paper, which is capable of implementing all Boolean functions (BFs). In the classification of BFs, there are: 1) linearly separable Boolean function (LSBF) class, 2) parity Boolean function (PBF) class, and 3) non-LSBF and non-PBF class. To implement these functions, UP takes different kinds of simple topological structures in which each contains at most one hidden layer along with the smallest possible number of hidden neurons. Inspired by the concept of DNA sequences in biological systems, a novel learning algorithm named DNA-like learning is developed, which is able to quickly train a network with any prescribed BF. The focus is on performing LSBF and PBF by a single-layer perceptron (SLP) with the new algorithm. Two criteria for LSBF and PBF are proposed, respectively, and a new measure for a BF, named nonlinearly separable degree (NLSD), is introduced. In the sense of this measure, the PBF is the most complex one. The new algorithm has many advantages including, in particular, fast running speed, good robustness, and no need of considering the convergence property. For example, the number of iterations and computations in implementing the basic 2-bit logic operations such as and, or, and xor by using the new algorithm is far smaller than the ones needed by using other existing algorithms such as error-correction (EC) and backpropagation (BP) algorithms. Moreover, the synaptic weights and threshold values derived from UP can be directly used in designing of the template of cellular neural networks (CNNs), which has been considered as a new spatial-temporal sensory computing paradigm.

Abstract: A Boolean relation can specify some types of flexibility of a combinational circuit that cannot be expressed with don't cares. Several problems in logic synthesis, such as Boolean decomposition or multilevel minimization, can be modeled with Boolean relations. However, solving Boolean relations is a computationally expensive task. This paper presents a novel recursive algorithm for solving Boolean relations. The algorithm has several features: efficiency, wide exploration of solutions, and customizable cost function. The experimental results show the applicability of the method in logic minimization problems and tangible improvements with regard to previous heuristic approaches.

Abstract: An approximate representation for the state space of a context-sensitive probabilistic Boolean network has previously been proposed and utilized to devise therapeutic intervention strategies. Whereas the full state of a context-sensitive probabilistic Boolean network is specified by an ordered pair composed of a network context and a gene-activity profile, this approximate representation collapses the state space onto the gene-activity profiles alone. This reduction yields an approximate transition probability matrix, absent of context, for the Markov chain associated with the context-sensitive probabilistic Boolean network. As with many approximation methods, a price must be paid for using a reduced model representation, namely, some loss of optimality relative to using the full state space. This paper examines the effects on intervention performance caused by the reduction with respect to various values of the model parameters. This task is performed using a new derivation for the transition probability matrix of the context-sensitive probabilistic Boolean network. This expression of transition probability distributions is in concert with the original definition of context-sensitive probabilistic Boolean network. The performance of optimal and approximate therapeutic strategies is compared for both synthetic networks and a real case study. It is observed that the approximate representation describes the dynamics of the context-sensitive probabilistic Boolean network through the instantaneously random probabilistic Boolean network with similar parameters.

Abstract: We give a new version of the adversary method for proving lower bounds on quantum query algorithms. The new method is based on analyzing the eigenspace structure of the problem at hand. We use it to prove a new and optimal strong direct product theorem for 2-sided error quantum algorithms computing k independent instances of a symmetric Boolean function: if the algorithm uses significantly less than k times the number of queries needed for one instance of the function, then its success probability is exponentially small in k. We also use the polynomial method to prove a direct product theorem for 1-sided error algorithms for k threshold functions with a stronger bound on the success probability. Finally, we present a quantum algorithm for evaluating solutions to systems of linear inequalities, and use our direct product theorems to show that the time-space tradeoff of this algorithm is close to optimal.

Abstract: In a seminal paper from 1985, Sistla and Clarke showed that satisfiability for Linear Temporal Logic (LTL) is either NP-complete or PSPACE-complete, depending on the set of temporal operators used. If, in contrast, the set of propositional operators is restricted, the complexity may decrease. This paper undertakes a systematic study of satisfiability for LTL formulae over restricted sets of propositional and temporal operators. Since every propositional operator corresponds to a Boolean function, there exist infinitely many propositional operators. In order to systematically cover all possible sets of them, we use Post's lattice. With its help, we determine the computational complexity of LTL satisfiability for all combinations of temporal operators and all but two classes of propositional functions. Each of these infinitely many problems is shown to be either PSPACE-complete, NP-complete, or in P.