Abstract: We define tests of boolean functions which distinguish between linear (or quadratic) polynomials, and functions which are very far, in an appropriate sense, from these polynomials. The tests have optimal or nearly optimal trade-offs between soundness and the number of queries.A central step in our analysis of quadraticity tests is the proof of aninverse theorem for the third Gowers uniformity norm of boolean functions.The last result implies that it ispossible to estimate efficiently the distance from the second-order Reed-Muller code on inputs lying far beyond its list-decoding radius.Our main technical tools are Fourier analysis on Z2n and methods from additive number theory. We observe that these methods can be used to give a tight analysis of the Abelian Homomorphism testing problemfor some families of groups, including powers of Zp.

Abstract: For the first time Boolean functions on 9 variables having nonlinearity 241 are discovered, that remained as an open question in literature for almost three decades. Such functions are found by heuristic search in the space of rotation symmetric Boolean functions (RSBFs). This shows that there exist Boolean functions on n (odd) variables having nonlinearity >2n-1-2n-1/2 if and only if n>7. Using similar search technique, balanced Boolean functions on 9, 10, and 11 variables are attained having autocorrelation spectra with maximum absolute value <2lceiln/2rceil. On odd number of variables, earlier such functions were known for 15, 21 variables; there was no evidence of such functions at all on even number of variables. In certain cases, our functions can be affinely transformed to obtain first-order resiliency or first-order propagation characteristics. Moreover, 10 variable functions having first-order resiliency and nonlinearity 492 are presented that had been posed as an open question at Crypto 2000. The functions reported in this paper are discovered using a suitably modified steepest descent based iterative heuristic search in the RSBF class along with proper affine transformations. It seems elusive to get a construction technique to match such functions

Abstract: In the past, complying with design rules was sufficient to ensure acceptable yields for a design. However, for sub-100-nm designs, this approach tends to create patterns that cannot be reliably printed for a given optical setup, thus leading to hot spots and systematic yield failures. Recent challenges faced by both the design and process communities call for a paradigm shift whereby circuits are constructed from a small set of lithography-friendly patterns that have previously been extensively characterized and ensured to print reliably. We describe the use of a regular design fabric for defining the underlying layout geometries of the circuit. While the direct application of this methodology to the current application-specific integrated circuit (ASIC) design flow would result in unnecessary area and performance penalties, we overcome these penalties via a unique design flow that ensures shape-level regularity by reducing the number of required logic functions as much as possible as part of the top-down design flow. We show that with a small set of Boolean functions and careful selection of lithography-friendly patterns, we not only mitigate but essentially eliminate such penalties. Additionally, we discuss the benefits of using extremely regular designs constructed from a limited set of lithography-friendly patterns not only to improve manufacturability but also to relax the pessimistic constraints defined by design rules. Specifically, we introduce the basis to exploit the regularity in the layout patterns by using "pushed-rules" for logic design, as is commonly done for static random access memory (SRAM). This in turn facilitates a common optical proximity correction (OPC) methodology for logic and SRAM. Moreover, by taking advantage of this newfound manufacturability and predictability of regular circuits, we show that the performance of logic built on regular fabrics can surpass that of seemingly more arbitrarily constructed logic.

Abstract: The use of error-correcting codes for tight control of the peak-to-mean envelope power ratio (PMEPR) in orthogonal frequency-division multiplexing (OFDM) transmission is considered in this correspondence. By generalizing a result by Paterson, it is shown that each q-phase (q is even) sequence of length 2m lies in a complementary set of size 2k+1, where k is a nonnegative integer that can be easily determined from the generalized Boolean function associated with the sequence. For small k this result provides a reasonably tight bound for the PMEPR of q-phase sequences of length 2 m. A new 2h-ary generalization of the classical Reed-Muller code is then used together with the result on complementary sets to derive flexible OFDM coding schemes with low PMEPR. These codes include the codes developed by Davis and Jedwab as a special case. In certain situations the codes in the present correspondence are similar to Paterson's code constructions and often outperform them

Abstract: We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty 'number on the forehead' model. Our method is based on the notion of voting polynomial degree of functions and extends the degree-discrepancy lemma in the recent work of Sherstov (2007). Using this we prove that depth three circuits consisting of a MAJORITY gate at the output, gates computing arbitrary symmetric function at the second layer and arbitrary gates of bounded fan-in at the base layer i.e. circuits of type MAJ o SYMM o ANYO(1) cannot simulate the circuit class AC0 in sub-exponential size. Further, even if the fan-in of the bottom ANY gates are increased to o(log log n), such circuits cannot simulate AC0 in quasi-polynomial size. This is in contrast to the classical result of Yao and Beigel-Tarui that shows that such circuits, having only MAJORITY gales, can simulate the class ACC0 in quasi-polynomial size when the bottom fan-in is increased to poly-logarithmic size. In the second part, we simplify the arguments in the breakthrough work of Bourgain (2005) for obtaining exponentially small upper bounds on the correlation between the boolean function MODq and functions represented bv polynomials of small degree over Zm, when m,q ges 2 are co-prime integers. Our calculation also shows similarity with techniques used to estimate discrepancy of functions in the multiparty communication setting. This results in a slight improvement of the estimates of Bourgain et al. (2005). It is known that such estimates imply that circuits of type MAJ o MODm o ANDisin log n cannot compute the MODq function in sub-exponential size. It remains a major open question to determine if such circuits can simulate ACC0 in polynomial size when the bottom fan-in is increased to poly-logarithmic size.

Abstract: Motivation: Probabilistic Boolean networks (PBNs) have been proposed to model genetic regulatory interactions. The steady-state probability distribution of a PBN gives important information about the captured genetic network. The computation of the steady-state probability distribution usually includes construction of the transition probability matrix and computation of the steady-state probability distribution. The size of the transition probability matrix is 2n-by-2n where n is the number of genes in the genetic network. Therefore, the computational costs of these two steps are very expensive and it is essential to develop a fast approximation method. Results: In this article, we propose an approximation method for computing the steady-state probability distribution of a PBN based on neglecting some Boolean networks (BNs) with very small probabilities during the construction of the transition probability matrix. An error analysis of this approximation method is given and theoretical result on the distribution of BNs in a PBN with at most two Boolean functions for one gene is also presented. These give a foundation and support for the approximation method. Numerical experiments based on a genetic network are given to demonstrate the efficiency of the proposed method. Contact: sqzhang@hkusua.hku.hk Supplementary information: Supplementary data are available at Bioinformatics online.

Abstract: Based on a method proposed by the first author, several classes of balanced Boolean functions with optimum algebraic immunity are constructed, and they have nonlinearities significantly larger than the previously best known nonlinearity of functions with optimal algebraic immunity. By choosing suitable parameters, the constructed n-variable functions have nonlinearity $${2^{n-1}-{n-1\choose\frac{n}{2}-1}+2{n-2\choose\frac{n}{2}-2}\Big/(n-2)}$$ for even $${n\geq 8\,{\rm and}\,2^{n-1}-{n-1\choose\frac{n-1}{2}}+\Delta(n)}$$ for odd n, where Δ(n) is a function increasing rapidly with n. The algebraic degrees of some constructed functions are also discussed.

Abstract: In this article we develop quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables. Our aim is to develop efficient algorithms: - whose sample complexity has no dependence on n, the dimension of the domain the Boolean functions are defined over; - with no access to any classical or quantum membership ("black-box") queries. Instead, our algorithms use only classical examples generated uniformly at random and fixed quantum superpositions of such classical examples; - which require only a few quantum examples but possibly many classical random examples (which are considered quite "cheap" relative to quantum examples). Our quantum algorithms are based on a subroutine FS which enables sampling according to the Fourier spectrum of f; the FS subroutine was used in earlier work of Bshouty and Jackson on quantum learning. Our results are as follows: - We give an algorithm for testing k-juntas to accuracy $\epsilon$ that uses $O(k/\epsilon)$ quantum examples. This improves on the number of examples used by the best known classical algorithm. - We establish the following lower bound: any FS-based k-junta testing algorithm requires $\Omega(\sqrt{k})$ queries. - We give an algorithm for learning $k$-juntas to accuracy $\epsilon$ that uses $O(\epsilon^{-1} k\log k)$ quantum examples and $O(2^k \log(1/\epsilon))$ random examples. We show that this learning algorithms is close to optimal by giving a related lower bound.

Abstract: This paper presents a unified and simple treatment of basic questions concerning two computational models: multiparty communication complexity and GF(2) polynomials. The key is the use of (known) norms on Boolean functions, which capture their approximability in each of these models. The main contributions are new XOR lemmas. We show that if a Boolean function has correlation at most epsi les 1/2 with any of these models, then the correlation of the parity of its values on m independent instances drops exponentially with m. More specifically: For GF(2) polynomials of degree d, the correlation drops to exp (-m/4d). No XOR lemma was known even for d = 2. For c-bit k-party protocols, the correlation drops to 2c ldrepsim/2 k . No XOR lemma was known for k ges 3 parties. Another contribution in this paper is a general derivation of direct product lemmas from XOR lemmas. In particular, assuming that f has correlation at most epsi les 1/2 with any of the above models, we obtain the following bounds on the probability of computing m independent instances of f correctly: For GF(2) polynomials of degree d we again obtain a bound of exp(-m/4d). For c-bit k-party protocols we obtain a bound of 2-Omega(m) in the special case when epsi les exp (-c ldr 2k). In this range of epsi, our bound improves on a direct product lemma for two-parties by Parnafes, Raz, and Wigderson (STOC '97). We also use the norms to give improved (or just simplified) lower bounds in these models. In particular we give a new proof that the Modm function on n bits, for odd m, has correlation at most exp(-n/4d) with degree-d GF(2) polynomials.

Abstract: The world is unpredictable, and acting intelligently requires anticipating possible consequences of actions that are taken. Assuming that the actions and the world are deterministic, planning can be represented in the classical propositional logic. Introducing nondeterminism (but not probabilities) or several initial states increases the complexity of the planning problem and requires the use of quantified Boolean formulae (QBF). The currently leading logic-based approaches to conditional planning use explicitly or implicitly a QBF with the prefix ∃∀∃. We present formalizations of the planning problem as QBF which have an asymptotically optimal linear size and the optimal number of quantifier alternations in the prefix: ∃∀ and ∀∃. This is in accordance with the fact that the planning problem (under the restriction to polynomial size plans) is on the second level of the polynomial hierarchy, not on the third.

Abstract: It was found recently that natural gene regulatory systems are governed by hierarchically canalyzing functions (HCFs), a special subclass of Boolean functions Here we study the HCF class in detail We present a new minimal logical expression for all HCFs Based on this formula, we calculate the cardinality of the HCF class Moreover, we define HCF subclasses and calculate their cardinality as well Using the well-known critical connectivity condition 2 K c p ( 1 − p ) = 1 , we discuss order–chaos transitions of Boolean networks (BNs) regulated by functions of given HCF subclasses Finally, analysing real gene regulatory rules we show that nearly all of the biologically relevant functions belong to the simplest HCF subclasses This restriction is important for reverse engineering of transcription regulatory networks and for ensemble approach studies in systems biology It is shown that Boolean networks with functions belonging to the biologically realized HCF subclasses show ordered behavior

Abstract: In this article we develop quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables. Our aim is to develop efficient algorithms: (1) whose sample complexity has no dependence on n, the dimension of the domain the Boolean functions are defined over; (2) with no access to any classical or quantum membership ("black-box") queries. Instead, our algorithms use only classical examples generated uniformly at random and fixed quantum superpositions of such classical examples; (3) which require only a few quantum examples but possibly many classical random examples (which are considered quite "cheap" relative to quantum examples). Our quantum algorithms are based on a subroutine FS which enables sampling according to the Fourier spectrum of f; the FS subroutine was used in earlier work of Bshouty and Jackson on quantum learning. Our results are as follows: (1) We give an algorithm for testing k-juntas to accuracy ? that uses O(k/?) quantum examples. This improves on the number of examples used by the best known classical algorithm. (2) We establish the following lower bound: any FS-based k-junta testing algorithm requires $$\Omega(\sqrt{k})$$ queries. (3) We give an algorithm for learning k-juntas to accuracy ? that uses O(??1 k log k) quantum examples and O(2 k log(1/?)) random examples. We show that this learning algorithm is close to optimal by giving a related lower bound.

Abstract: In this note, it is proved that for each odd positive integer n there are exactly two n-variable symmetric Boolean functions with maximum algebraic immunity.

Abstract: Probabilistic Boolean networks (PBNs) have been recently introduced as a paradigm for modeling genetic regulatory networks. One of the objectives of PBN modeling is to use the network for the design and analysis of intervention strategies aimed at moving the network out of undesirable states, such as those associated with disease, and into desirable ones. To date, a number of intervention strategies have been proposed in the context of PBNs. However, all these techniques assume perfect knowledge of the transition probability matrix of the PBN. Such an assumption cannot be satisfied in practice since the presence of noise and the availability of limited number of samples will prevent the transition probabilities from being accurately determined. Moreover, even if the exact transition probabilities could be estimated from the data, mismatch between the PBN model and the actual genetic regulatory network will invariably be present. Thus, it is important to study the effect of modeling errors on the final outcome of an intervention strategy and one of the goals of this paper is to do precisely that when the uncertainties are in the entries of the transition probability matrix. In addition, the paper develops a robust intervention strategy that is obtained by minimizing the worst-case cost over the uncertainty set.

Abstract: The inference of gene regulatory networks is a key issue for genomic signal processing. This paper addresses the inference of probabilistic Boolean networks (PBNs) from observed temporal sequences of network states. Since a PBN is composed of a finite number of Boolean networks, a basic observation is that the characteristics of a single Boolean network without perturbation may be determined by its pairwise transitions. Because the network function is fixed and there are no perturbations, a given state will always be followed by a unique state at the succeeding time point. Thus, a transition counting matrix compiled over a data sequence will be sparse and contain only one entry per line. If the network also has perturbations, with small perturbation probability, then the transition counting matrix would have some insignificant nonzero entries replacing some (or all) of the zeros. If a data sequence is sufficiently long to adequately populate the matrix, then determination of the functions and inputs underlying the model is straightforward. The difficulty comes when the transition counting matrix consists of data derived from more than one Boolean network. We address the PBN inference procedure in several steps: (1) separate the data sequence into "pure" subsequences corresponding to constituent Boolean networks; (2) given a subsequence, infer a Boolean network; and (3) infer the probabilities of perturbation, the probability of there being a switch between constituent Boolean networks, and the selection probabilities governing which network is to be selected given a switch. Capturing the full dynamic behavior of probabilistic Boolean networks, be they binary or multivalued, will require the use of temporal data, and a great deal of it. This should not be surprising given the complexity of the model and the number of parameters, both transitional and static, that must be estimated. In addition to providing an inference algorithm, this paper demonstrates that the data requirement is much smaller if one does not wish to infer the switching, perturbation, and selection probabilities, and that constituent-network connectivity can be discovered with decent accuracy for relatively small time-course sequences.

Abstract: We continue the recent line of work on the connection between semidefinite programming-based approximation algorithms and the Unique Games Conjecture. Given any-boolean 2-CSP (or more generally, any nonnegative objective function on two boolean variables), we show how to reduce the search for a good inapproximability result to a certain numeric minimization problem. The key objects in our analysis are the vector triples arising when doing clause-by-clause analysis of algorithms based on semidefinite programming. Given a weighted set of such triples of a certain restricted type, which are "hard" to round in a certain sense, we obtain a Unique Games-based inapproximability matching this "hardness" of rounding the set of vector triples. Conversely, any instance together with an SDP solution can be viewed as a set of vector triples, and we show that we can always find an assignment to the instance which is at least as good as the "hardness" of rounding the corresponding set of vector triples. We conjecture that the restricted type required for the hardness result is in fact no restriction, which would imply that these upper and lower bounds match exactly. This conjecture is supported by all existing results for specific 2-CSPs. As an application, we show that Max 2-AND is hard to approximate within 0.87435. This improves upon the best previous hardness of alphaGW + epsi ap 0.87856, and comes very close to matching the approximation ratio of the best algorithm known, 0.87401. It also establishes that balanced instances of Max 2-AND, i.e., instances in which each variable occurs positively and negatively equally often, are not the hardest to approximate, as these can be approximated within a factor alphaGW.

Abstract: A new technique for Boolean random masking of the logic and operation in terms of nand logic gates is proposed and applied for masking the integer addition. The new technique can be used for masking arbitrary cryptographic functions and is more efficient than previously known techniques, recently applied to the Advanced Encryption Standard (AES). New techniques for the conversions from Boolean to arithmetic random masking and vice versa are also developed. They are hardware oriented and do not require additional random bits. Unlike the previous, software-oriented techniques showing a substantial difference in the complexity of the two conversions, they have a comparable complexity being about the same as that of one integer addition only. All the techniques proposed are in theory secure against the first-order differential power analysis on the logic gate level. They can be applied in hardware implementations of various cryptographic functions, including AES, (keyed) SHA-1, IDEA, and RC6

Abstract: In recent work binary decision diagrams (BDDs) were introduced as a technique for postoptimality analysis for integer programming. In this paper we show that much smaller BDDs can be used for the same analysis by employing cost bounding techniques in their construction.

Abstract: In this paper we present a theoretical construction of Rotation Symmetric Boolean Functions (RSBFs) on odd number of variables with maximum possible algebraic immunity (AI) and further these functions are not symmetric. Our RSBFs are of better nonlinearity than the existing theoretical constructions with maximum possible AI. To get very good nonlinearity, which is important for practical cryptographic design, we generalize our construction to a construction cum search technique in the RSBF class. We find 7, 9, 11 variable RSBFs with maximum possible AI having nonlinearities 56, 240, 984 respectively with very small amount of search after our basic construction.

Abstract: In this paper we propose a novel and efficient method for majority gate-based design. The basic Boolean primitive in quantum cellular automata (QCA) is the majority gate. Method for reducing the number of majority gates required for computing Boolean functions is developed to facilitate the conversion of sum of products (SOP) expression into QCA majority logic. This method is based on genetic algorithm and can reduce the hardware requirements for a QCA design. We will show that the proposed approach is very efficient in deriving the simplified majority expression in QCA design.

Abstract: Recently, 9-variable Boolean functions having nonlinearity 241, which is strictly greater than the bent concatenation bound of 240, have been discovered in the class of Rotation Symmetric Boolean Functions (RSBFs) by Kavut, Maitra and Yucel. In this paper, we present several 9-variable Boolean functions having nonlinearity of 242, which we obtain by suitably generalizing the classes of RSBFs and Dihedral Symmetric Boolean Functions (DSBFs). These functions do not have any zero in the Walsh spectrum values, hence they cannot be made balanced easily. This result also shows that the covering radius of the first order Reed-Muller code R(1, 9) is at least 242.

Abstract: Probabilistic Boolean networks (PBNs) have been recently introduced as a paradigm for modeling genetic regulatory networks. One of the objectives of PBN modeling is to use the network for the design and analysis of intervention strategies aimed at moving the network out of undesirable states, such as those associated with disease, and into desirable ones. To date, a number of intervention strategies have been proposed in the context of Probabilistic Boolean networks. However, all these techniques assume perfect knowledge of the transition probability matrix of the PBN. Such an assumption cannot be satisfied in practice since the presence of noise and the availability of limited number of samples will prevent the transition probabilities from being accurately determined. Moreover, even if the exact transition probabilities could be estimated from the data, mismatch between the PBN model and the actual genetic regulatory network will invariably be present. In this paper, we develop a robust intervention strategy that is obtained by minimizing the worst-case cost over the uncertainties in the entries of the transition probability matrix.

Abstract: We show that any k-wise independent probability measure on {0, 1}n can O(m2ldr2ldr2-radick/10)-fool any boolean function computable by an rn-clauses DNF (or CNF) formula on n variables. Thus, for each constant c > 0. there is a constant e > 0 such that any boolean function computable by an m-clauses DNF (or CNF) formula can be in m-e-fooled by any clog in-wise probability measure. This resolves, asymptotically and up to a logm factor, the depth-2 circuits case of a conjecture due to Linial and Nisan (1990). The result is equivalent to a new characterization of DNF (or CNF) formulas by low degree polynomials. It implies a similar statement for probability measures 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 PRG's ofO(log2 m log n)-seed length for m-clauses DNF (or CNF) formulas on n variables, improving previously known seed lengths.

Abstract: The research in the field of reversible logic is motivated by its application in low-power design, optical computing and quantum computing. Hence synthesis of reversible logic has become a very important research area in the last years. In this paper exact algorithms for the synthesis of generalized Toffoli networks are considered. We present an improvement of an existing synthesis approach that is based on Boolean Satisfiability. Furthermore, the principle limits of the original and the improved approach are shown. Then, we propose a new method using problem specific knowledge during the synthesis process to overcome these limits. Experimental results demonstrate improvements of the overall synthesis time up to four orders of magnitude.

Abstract: Boolean games are a logical setting for representing static games in a succinct way, taking advantage of the expressive power and conciseness of propositional logic. A Boolean game consists of a set of players, each of them controls a set of propositional variables and has a specific goal expressed by a propositional formula. There is a lot of graphical structures hidden in a Boolean game: the satisfaction of each player's goal depends on players whose actions have an influence on these goals. Even if these dependencies are not specific to Boolean games, in this particular setting they give a way of finding simple characterizations of Nash equilibria and computing them.

Abstract: Dassault Aviation have developed a reliability workbench based on the high-level formal description language AltaRica. The workbench includes a compiler of AltaRica models into fault trees. The fault trees generated for the largest industrial systems involve up to a thousand basic events and several dozen thousand gates; moreover, they are non-coherent. The assessment of such large formulae is challenging, even for binary decision diagrams, the state-of-the-art data structure to encode and to manipulate Boolean functions. This article describes the various heuristics and strategies that were used to make the assessment tractable.

Abstract: We develop a new technique of proving lower bounds for the randomized communication complexity of boolean functions in the multiparty 'number on the forehead' model Our method is based on the notion of voting polynomial degree of functions and extends the degree-discrepancy lemma in the recent work of Sherstov (2007) Using this we prove that depth three circuits consisting of a MAJORITY gate at the output, gates computing arbitrary symmetric function at the second layer and arbitrary gates of bounded fan-in at the base layer ie circuits of type MAJ o SYMM o ANYO(1) cannot simulate the circuit class AC0 in sub-exponential size Further, even if the fan-in of the bottom ANY gates are increased to o(log log n), such circuits cannot simulate AC0 in quasi-polynomial size This is in contrast to the classical result of Yao and Beigel-Tarui that shows that such circuits, having only MAJORITY gales, can simulate the class ACC0 in quasi-polynomial size when the bottom fan-in is increased to poly-logarithmic size In the second part, we simplify the arguments in the breakthrough work of Bourgain (2005) for obtaining exponentially small upper bounds on the correlation between the boolean function MODq and functions represented bv polynomials of small degree over Zm, when m,q ges 2 are co-prime integers Our calculation also shows similarity with techniques used to estimate discrepancy of functions in the multiparty communication setting This results in a slight improvement of the estimates of Bourgain et al (2005) It is known that such estimates imply that circuits of type MAJ o MODm o ANDisin log n cannot compute the MODq function in sub-exponential size It remains a major open question to determine if such circuits can simulate ACC0 in polynomial size when the bottom fan-in is increased to poly-logarithmic size