Abstract: A Boolean network is a finite dynamical system, whose variables take values from a binary set. The value update rule for each variable is a Boolean function, depending on a selected subset of variables. Boolean networks have been widely used in modeling gene regulatory networks. We focus in this paper on a special class of Boolean networks, termed as conjunctive Boolean networks. A Boolean network is conjunctive if the associated value update rule is comprised of only AND operations. It is known that any trajectory of a finite dynamical system will enter a periodic orbit. We characterize in this paper all periodic orbits of a conjunctive Boolean network whose underlying graph is strongly connected. In particular, we establish a bijection between the set of periodic orbits and the set of binary necklaces of a certain length. We further investigate the stability of a periodic orbit. Specifically, we perturb a state in the periodic orbit by changing the value of a single entry of the state. The trajectory, with the perturbed state being the initial condition, will enter another (possibly the same) periodic orbit in finite time steps. We then provide a complete characterization of all such transitions from one periodic orbit to another. In particular, we construct a digraph, with the vertices being the periodic orbits, and the (directed) edges representing the transitions among the orbits. We call such a digraph the stability structure of the conjunctive Boolean network.

Abstract: A Majority-Inverter Graph (MIG) is a homogeneous logic network, where each node represents the majority function. Recently, a logic optimization package based on the MIG datastructure, with 3-input majority node (M3) has been proposed [2],[30]. It is demonstrated to have efficient area-delay-power results compared to state-of-the-art logic optimization packages. In this paper, the Boolean algebraic transformations based on majority logic, i.e., majority Boolean algebra is studied. In the first part of this paper, we summarize a range of identities for majority Boolean algebra with their corresponding proofs. In the second part, we venture towards heterogeneous logic network and provide reversible logic mapping of majority nodes.

Abstract: The sensitivity of a Boolean function f is the maximum, over all inputs x, of the number of sensitive coordinates of x (namely the number of Hamming neighbors of x with different f-value). The well-known sensitivity conjecture of Nisan (see also Nisan and Szegedy) states that every sensitivity-s Boolean function can be computed by a polynomial over the reals of degree poly(s). The best known upper bounds on degree, however, are exponential rather than polynomial in s. Our main result is an approximate version of the conjecture: every Boolean function with sensitivity s can be e-approximated (in e2) by a polynomial whose degree is s · polylog(1/e). This is the first improvement on the folklore bound of s/e. We prove this via a new "switching lemma for low-sensitivity functions" which establishes that a random restriction of a low-sensitivity function is very likely to have low decision tree depth. This is analogous to the well-known switching lemma for AC0 circuits. Our proof analyzes the combinatorial structure of the graph Gf of sensitive edges of a Boolean function f. Understanding the structure of this graph is of independent interest as a means of understanding Boolean functions. We propose several new complexity measures for Boolean functions based on this graph, including tree sensitivity and component dimension, which may be viewed as relaxations of worst-case sensitivity, and we introduce some new techniques, such as proper walks and shifting, to analyze these measures. We use these notions to show that the graph of a function of full degree must be sufficiently complex, and that random restrictions of low-sensitivity functions are unlikely to lead to such complex graphs. We postulate a robust analogue of the sensitivity conjecture: if most inputs to a Boolean function f have low sensitivity, then most of the Fourier mass of f is concentrated on small subsets. We prove a lower bound on tree sensitivity in terms of decision tree depth, and show that a polynomial strengthening of this lower bound implies the robust conjecture. We feel that studying the graph Gf is interesting in its own right, and we hope that some of the notions and techniques we introduce in this work will be of use in its further study.

Abstract: As they are conventionally formulated, Boolean games assume that players make their choices in ignorance of the choices being made by other players – they are games of simultaneous moves. For many settings, this is clearly unrealistic. In this paper, we show how Boolean games can be enriched by dependency graphs which explicitly represent the informational dependencies between variables in a game. More precisely, dependency graphs play two roles. First, when we say that variable x depends on variable y, then we mean that when a strategy assigns a value to variable x, it can be informed by the value that has been assigned to y. Second, and as a consequence of the first property, they capture a richer and more plausible model of concurrency than the simultaneous-action model implicit in conventional Boolean games. Dependency graphs implicitly define a partial ordering of the run-time events in a game: if x is dependent on y, then the assignment of a value to y must precede the assignment of a value to x; if x and y are independent, however, then we can say nothing about the ordering of assignments to these variables—the assignments may occur concurrently. We refer to Boolean games with dependency graphs as partial-order Boolean games. After motivating and presenting the partial-order Boolean games model, we explore its properties. We show that while some problems associated with our new games have the same complexity as in conventional Boolean games, for others the complexity blows up dramatically. We also show that the concurrency in partial-order Boolean games can be modelled using a closure-operator semantics, and conclude by considering the relationship of our model to Independence-Friendly (IF) logic.

Abstract: Papert Strauss (Proc. London Math. Soc. 18(3), 217–230, 1968) used Pontryagin duality to prove that a compact Hausdorff topological Boolean algebra is a powerset algebra. We give a more elementary proof of this result that relies on a version of Bogolyubov’s lemma.

Abstract: VBF is a collection of C++ classes designed for analyzing vector Boolean functions (functions that map a Boolean vector to another Boolean vector) from a cryptographic perspective. This implementation uses the NTL library from Victor Shoup, adding new modules that call NTL functions and complement the existing ones, making it better suited to cryptography. The class representing a vector Boolean function can be initialized by several alternative types of data structures such as Truth Table, Trace Representation, and Algebraic Normal Form (ANF), among others. The most relevant cryptographic criteria for both block and stream ciphers as well as for hash functions can be evaluated with VBF: it obtains the nonlinearity, linearity distance, algebraic degree, linear structures, and frequency distribution of the absolute values of the Walsh Spectrum or the Autocorrelation Spectrum, among others. In addition, operations such as equality testing, composition, inversion, sum, direct sum, bricklayering (parallel application of vector Boolean functions as employed in Rijndael cipher), and adding coordinate functions of two vector Boolean functions are presented. Finally, three real applications of the library are described: the first one analyzes the KASUMI block cipher, the second one analyzes the Mini-AES cipher, and the third one finds Boolean functions with very high nonlinearity, a key property for robustness against linear attacks.

Abstract: Given a 0-dimensional polynomial system in a polynomial ring over F_2 having only F_2-rational solutions, we optimize the Border Basis Algorithm (BBA) for solving this system by introducing a Boolean BBA. This algorithm is further improved by optimizing the linear algebra steps. We discuss ways to combine it with SAT solvers, optimized methods for performing the combinatorial steps involved in the algorithm, and various approaches to implement the linear algebra steps. Based on our C++ implementation, we provide some timings to compare sparse and dense representations of the coefficient matrices and to Groebner basis methods.

Abstract: The r-th order nonlinearity of a Boolean function is the minimum number of elements that have to be changed in its truth table to arrive at a Boolean function of degree at most r. It is shown that the (suitably normalised) r-th order nonlinearity of a random Boolean function converges strongly for all r ź 1. This extends results by Rodier for r = 1 and by Dib for r = 2. The methods in the present paper are mostly of elementary combinatorial nature and also lead to simpler proofs in the cases that r = 1 or 2.

Abstract: bound (if it exists) of every subset of I with power at most a belongs to I. A Boolean algebra that is N0-complete is also called a-complete. A field of sets B is a Boolean algebra where the operations +, *, and - are respectively the operations of set-union, set-intersection, and complementation with respect to the unit element of B. A field of sets B is a-complete if the union of any subset of B with power at most a belongs to B. By a theorem of Stone [7 ], everv Boolean algebra is isomorphic to a field of sets. On the other hand, not every a-complete Boolean algebra is isomorphic to some a-complete field of sets; a necessary and sufficient condition for such a representation is that every prinicipal ideal of the algebra be contained in an a-complete maximal ideal (cf. [5]). In 1947, Loomis [3] proved that every a-complete Boolean algebra A is isomorphic to a u-complete field of sets B modulo a a-complete ideal of B. The question was raised as to

Abstract: In this article, the controllability of Boolean networks via input controls under Harvey's update scheme is investigated. First, the model of Boolean control networks under Harvey's stochastic update is proposed, by means of semi-tensor product approach, which is converted into discrete-time linear representation. And, a general formula of control-depending network transition matrix is provided. Second, based on discrete-time dynamics, controllability of the proposed model is analytically discussed by revealing the necessary and sufficient conditions of the reachable sets, respectively, for three kinds of controls, i.e., free Boolean control sequence, input control networks, and close-loop control. Examples are showed to demonstrate the effectiveness and feasibility of the proposed scheme.

Abstract: In this paper, we show that checking some properties of Boolean functions which are given by the lists of monomials in their polynomial representations can be implemented in polynomial time. Multi-linear polynomials over GF(2) are often a convenient way to represent Boolean functions. There is a single polynomial for each Boolean function, and the length of the polynomial (i.e. the number of its monomials) which represents a function of n variables can be far less than 2n. Therefore, in some cases, polynomials are a compressed description of Boolean functions. Besides, polynomial representations of Boolean functions have applications in circuit lower bounds, computational learning, error-correcting codes, cryptography. We construct polynomial-time algorithms for checking some properties of Boolean functions which are given by the lists of monomials in their polynomial representations. The considered properties are self-anti-duality (evenness), self-duality, periodicity, 1-invariance (Mobius transform invariance, coincidence). Note that checking each of these properties directly by definition gives, in the general case, only exponential-time algorithms. The approach to construct our algorithms is the following. Firstly, we prove that if a Boolean function has a certain property then its polynomial has a special structure. And secondly, we check the property by its characterization, cutting negative cases by proven facts. More precisely, we analyze an exponential time algorithm and prove that if the number of steps of the algorithm exceeds a bound, which is polynomial in the input size, and may be computed in advance, then the input will be necessarily rejected.

Abstract: Some techniques using linear algebra was introduced by Faugere in F4 to speed up the reduction process during Grobner basis computations. These techniques can also be used in fast implementations of F5 and some other signature-based Grobner basis algorithms. When these techniques are applied, a very important step is constructing matrices from critical pairs and existing polynomials by the Symbolic Preprocessing function (given in F4). Since multiplications of monomials and polynomials are involved in the Symbolic Preprocessing function, this step can be very costly when the number of involved polynomials/monomials is huge. In this paper, multiplications of monomials and polynomials for a Boolean polynomial ring are investigated and a specific method of implementing the Symbolic Preprocessing function over Boolean polynomial rings is reported. Many examples have been tested by using this method, and the experimental data shows that the new method is very efficient.

Abstract: Algebra deals with more than computations such as addition or exponentiation; it also studies relations. Calculus touches on this a bit with locating extreme values and determining where functions increase and decrease; and in elementary algebra you occasionally “solve” inequalities involving the order relations of < or ≤ , but this almost seems like an intrusion foreign to the main focus, which is making algebraic calculations. Relational ideas have become more important with the advent of computer science and the rise of discrete mathematics, however. Many contemporary mathematical applications involve binary or n-ary relations in addition to computations. We began discussing this topic in the last chapter when we introduced equivalence relations. In this chapter we will explore other kinds of relations (these will all be binary relations here), particularly ones that impose an order of one sort or another on a set. This will lead us to investigate certain order-structures (posets, lattices) and to introduce an abstract type of algebra known as Boolean Algebra. Our exploration of these ideas will nicely tie together some earlier ideas in logic and set theory as well as lead us into areas that are of crucial importance to computer science.

Abstract: The mathematical basics of the non-classical approach to the logical synthesis of k-valued digital structures based on the replacement of the classic mathematical apparatus of logic synthesis (Boolean algebra) to the proposed mathematical apparatus — linear algebra are considered. The logic synthesis process of two valued and multi-valued digital structures in linear algebra including the formation of bases of a linear space and original representation of the implemented logical function are discussed. Mathematical advantages of the proposed approach, which could be the basis for designing of high-speed digital logic structures for various applications are considered.

Abstract: Lattice induced threshold function is a Boolean function determined by a particular linear combination of lattice elements. We prove that every isotone Boolean function is a lattice induced threshold function and vice versa. We give the generalization of this result to Boolean functions on a k-element set. 2010 Mathematics Subject Classification: 06E30

Abstract: For a Boolean network we consider asynchronous updates and define the exclusive asynchronous basin of attraction for any steady state or cyclic attractor. An algorithm based on commutative algebra is presented to compute the exclusive basin. Finally its use for targeting desirable attractors by selective intervention on network nodes is illustrated with two examples, one cell signalling network and one sensor network measuring human mobility.

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Abstract: Linear algebra techniques are used throughout the area of financial engineering, and in particular in the analysis of options and other financial derivatives. These techniques are used for example to calculate the value of large portfolios, or to quickly price derivative instruments. This chapter contains an overview of LA algorithms and their implementation in C++.

Abstract: Abstract A polarized polynomial form (PPF) (modulo k) is a modulo k sum of products of variables x1, . . . , xn or their Post negations, where the number of negations of each variable is determined by the polarization vector of the PPF. The length of a PPF is the number of its pairwise distinct summands. The length of a function f(x1, . . . , xn)of k-valued logic in the class of PPFs is the minimum length among all PPFs realizing the function. The paper presents a sequence of symmetric functions fn(x1, . . . , xn)of three-valued logic such that the length of each function fn in the class of PPFs is not less than ⌊3n+1/4⌋, where ⌊a⌋ denotes the greatest integer less or equal to the number a. The complexity of a system of PPFs sharing the same polarization vector is the number of pairwise distinct summands entering into all of these PPFs. The complexity L k PPF (F) $L_k^{{\\rm{PPF}}}(F)$ of a system F ={f1,..., fm} of functions of k-valued logic depending on variables x1,..., xn in the class of PPFs is the minimum complexity among all systems of PPFs {p1,...,pm}such that all PPFs p1,...,pm share the same polarization vector and the PPF pj realizes the function fj, j = 1,...,m. Let L k PPF (m,n) = max F L 2 PPF (F) $L_k^{{\\rm{PPF}}}(m,n)\\, = \\,\\mathop {\\max }\\limits_F L_2^{{\\rm{PPF}}}(F)$ , where F runs through all systems consisting of m functions of k-valued logic depending on variables x1,..., xn. For prime values of k it is easy to derive the estimate L k PPF (m,n) ≤ k n $L_k^{{\\rm{PPF}}}(m,n)\\, \\le \\,{k^n}$. In this paper it is shown that L k PPF (m,n) = 2 n $L_k^{{\\rm{PPF}}}(m,n)\\, = \\,{2^n}$ and L k PPF (m,n) = 3 n $L_k^{{\\rm{PPF}}}(m,n)\\, = \\,{3^n}$ for all m ≥ 2, n= 1, 2, . . . Moreover, it is demonstrated that the estimates remain valid when consideration is restricted to systems of symmetric functions only.

Abstract: We characterize up to isomorphism the Boolean algebra (BA, for short) of regular piecewise testable languages and show the decidability of classes of regular languages related to this characterization. This BA turns out isomorphic to several other natural BAs of regular languages, in particular to the BA of regular aperiodic languages.

Abstract: In general, two overlapping Boolean algebras always extend to a common Boolean algebra, but three may not. We prove a new sufficient condition for $n$ overlapping Boolean algebras to have a common extension. Combining this with the set-theoretic technique of long $\omega_1$-approximation sequences (also known as Davies sequences), we obtain a flexible method of constructing (in ZFC) arbitrarily large Boolean algebras as direct limits of countable Boolean algebras. Along the way, we develop some category theory regarding (co)limits of faces of commutative $n$-cubes and an $n$-ary version of the Interpolation Theorem of propositional logic. Our most elaborate application of the above machinery is a Boolean algebra of size $\aleph_n$ with the $n$-ary FN but not the $(n+1)$-ary FN where the $n$-ary FN is a higher-arity variant of the Freese-Nation property. The Stone dual of the $n$-ary FN, $n$-open generation, generalizes Shchepin's concept of openly generated compact spaces. We also show that, given a Boolean algebra $A$ of size $\aleph_\alpha$ and letting $d=min(\alpha+2,\omega)$, we have $A$ projective iff it has every $(

Abstract: Генерический подход к алгоритмическим проблемам предложен А. Мясниковым, И. Каповичем, П. Шуппом и В. Шпильрайном в 2003 г. В рамках этого подхода рассматривается поведение алгоритмов на множествах почти всех входов. В данной работе изучается генерическая сложность проблемы общезначимости (тождественной истинности) булевых формул. Доказывается, что эта проблема неразрешима за полиномиальное время на любом полиномиальном строго генерическом множестве формул при условии её трудноразрешимости в худшем случае.

Abstract: The implicit function theorem (IFT) of logical algebraic equations (LAEs) and a class of more generalized singular Boolean networks (SBNs) are considered. Using matrix semi-tensor product (STP), three equivalent expressions of LAEs and their relationship are presented. LAEs have equivalent explicit formula, if and only if data submatrix has distinct rows. An algorithm is provided to find all the explicit formulas of LAEs. Combined with the derived results about minimum independent variables, a class of SBNs is normalizable. Furthermore, the SBNs with probability is investigated and some results such as the reachability and attractors are obtained.

Abstract: I show that any elementary cellular automata – a class of 1-dimensional, 2-state cellular automata – can be deconstructed into a set of two Boolean operators; I also present a conjecture concerning the computational completeness of a rule and its relationship to complete Boolean operators.

Abstract: In this paper, we propose a Boolean matrix approach to encode Dung’s acceptability semantics. Each semantics is encoded into one or more Boolean constraint models, which can be solved by Boolean constraint solvers. In addition, based on our Boolean matrix representations, we also propose a bit-vector-based approach to compute the grounded semantics, and the experimental results show that this approach can achieve a good performance.

Abstract: The problem of realization of Boolean functions by initial Boolean automata with two constant states and n inputs is considered. An initial Boolean automaton with two constant states and n inputs is an initial automaton with output such that in all states the output functions are n-ary constant Boolean functions 0 or 1. The maximum cardinality of set of n-ary Boolean functions, where n > 1, realized by an initial Boolean automaton with two constant states and n inputs is obtained.

Abstract: Decomposition (or factorization) of Boolean multi-relation data, i.e. data in the form of Boolean matrices, together with relation between them, containing zeros and ones only, received a considerable attention in data mining research. The main aim is to find new variables—factors—hidden in data that explain data. The main advantage of Boolean data is interpretability. In this paper we argue that considering only Boolean data can be limiting. Especially the relation between input matrices is not necessarily of a Boolean nature. Usually this relation represents linkages to some degree, e.g. how much a user likes or dislikes a movie. Using Boolean method for such data—data must be somehow binarized first—leads to a loss of information. First, we reformulate decomposition problem for multi-relational data with ordinal relations. Then we propose a new algorithm for such data along with an experimental evaluation.