Showing papers on "Boolean function published in 2014"

Analysis of Boolean Functions

Abstract: Boolean functions are perhaps the most basic objects of study in theoretical computer science. They also arise in other areas of mathematics, including combinatorics, statistical physics, and mathematical social choice. The field of analysis of Boolean functions seeks to understand them via their Fourier transform and other analytic methods. This text gives a thorough overview of the field, beginning with the most basic definitions and proceeding to advanced topics such as hypercontractivity and isoperimetry. Each chapter includes a "highlight application" such as Arrow's theorem from economics, the Goldreich-Levin algorithm from cryptography/learning theory, Hstad's NP-hardness of approximation results, and "sharp threshold" theorems for random graph properties. The book includes roughly 450 exercises and can be used as the basis of a one-semester graduate course. It should appeal to advanced undergraduates, graduate students, and researchers in computer science theory and related mathematical fields.

Optimal Control of Boolean Control Networks

Abstract: In this paper, we address the optimal control problem for Boolean control networks (BCNs). We first consider the problem of finding the input sequences that minimize a given cost function over a finite time horizon. The problem solution is obtained by means of a recursive algorithm that represents the analogue for BCNs of the difference Riccati equation for linear systems. We prove that a significant number of optimal control problems for BCNs can be easily reframed into the present setup. In particular, the cost function can be adjusted so as to include penalties on the switchings, provided that we augment the size of the BCN state variable. In the second part of the paper, we address the infinite horizon optimal control problem and we provide necessary and sufficient conditions for the problem solvability. The solution is obtained as the limit of the solution over the finite horizon [0,T], and it is always achieved in a finite number of steps. Finally, the average cost problem over the infinite horizon, investigated in “Optimal control of logical control networks” (Y. Zhao , IEEE Trans. Autom. Control, vol 56, no. 8, pp. 1766-1776, Aug. 2011), is addressed by making use of the results obtained in the previous sections.

Majority-Inverter Graph: A Novel Data-Structure and Algorithms for Efficient Logic Optimization

Abstract: In this paper, we present Majority-Inverter Graph (MIG), a novel logic representation structure for efficient optimization of Boolean functions. An MIG is a directed acyclic graph consisting of three-input majority nodes and regular/complemented edges. We show that MIGs include any AND/OR/Inverter Graphs (AOIGs), containing also the well-known AIGs. In order to support the natural manipulation of MIGs, we introduce a new Boolean algebra, based exclusively on majority and inverter operations, with a complete axiomatic system. Theoretical results show that it is possible to explore the entire MIG representation space by using only five primitive transformation rules. Such feature opens up a great opportunity for logic optimization and synthesis. We showcase the MIG potential by proposing a delay-oriented optimization technique. Experimental results over MCNC benchmarks show that MIG optimization reduces the number of logic levels by 18%, on average, with respect to AIG optimization performed by ABC academic tool. Employed in a traditional optimization-mapping circuit synthesis flow, MIG optimization enables an average reduction of {22%, 14%, 11%} in the estimated {delay, area, power} metrics, before physical design, as compared to academic/commercial synthesis flows.

Reusing Building Blocks of Extracted Knowledge to Solve Complex, Large-Scale Boolean Problems

Abstract: Evolutionary computation techniques have had limited capabilities in solving large-scale problems due to the large search space demanding large memory and much longer training times. In the work presented here, a genetic programming like rich encoding scheme has been constructed to identify building blocks of knowledge in a learning classifier system. The fitter building blocks from the learning system trained against smaller problems have been utilized in a higher complexity problem in the domain to achieve scalable learning. The proposed system has been examined and evaluated on four different Boolean problem domains: 1) multiplexer, 2) majority-on, 3) carry, and 4) even-parity problems. The major contribution of this paper is to successfully extract useful building blocks from smaller problems and reuse them to learn more complex large-scale problems in the domain, e.g., 135-bit multiplexer problem, where the number of possible instances is 2 135 ≈ 4 × 10 40 , is solved by reusing the extracted knowledge from the learned lower level solutions in the domain. Autonomous scaling is, for the first time, shown to be possible in learning classifier systems. It improves effectiveness and reduces the number of training instances required in large problems, but requires more time due to its sequential build-up of knowledge.

Contextuality in Measurement-based Quantum Computation

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.

On controllability and stabilizability of probabilistic Boolean control networks

Abstract: The controllability of probabilistic Boolean control networks (PBCNs) is first considered. Using the input-state incidence matrices of all models, we propose a reachability matrix to characterize the joint reachability. Then we prove that the joint reachability and the controllability of PBCNs are equivalent, which leads to a necessary and sufficient condition of the controllability. Then, the result of controllability is used to investigate the stability of probabilistic Boolean networks (PBNs) and the stabilization of PBCNs. A necessary and sufficient condition for the stability of PBNs is obtained first. By introducing the control-fixed point of Boolean control networks (BCNs), the stability condition has finally been developed into a necessary and sufficient condition of the stabilization of PBCNs. Both necessary and sufficient conditions for controllability and stabilizability are based on reachability matrix, which are easily computable. Hence the two necessary and sufficient conditions are straightforward verifiable. Numerical examples are provided from case to case to demonstrate the corresponding theoretical results.

Noise Sensitivity of Boolean Functions and Percolation

Abstract: This is a graduate-level introduction to the theory of Boolean functions, an exciting area lying on the border of probability theory, discrete mathematics, analysis, and theoretical computer science Certain functions are highly sensitive to noise; this can be seen via Fourier analysis on the hypercube The key model analyzed in depth is critical percolation on the hexagonal lattice For this model, the critical exponents, previously determined using the now-famous Schramm-Loewner evolution, appear here in the study of sensitivity behavior Even for this relatively simple model, beyond the Fourier-analytic set-up, there are three crucially important but distinct approaches: hypercontractivity of operators, connections to randomized algorithms, and viewing the spectrum as a random Cantor set This book assumes a basic background in probability theory and integration theory Each chapter ends with exercises, some straightforward, some challenging

New Algorithms and Lower Bounds for Monotonicity Testing

Abstract: We consider the problem of testing whether an unknown Boolean function f : { -- 1, 1}n a#x21C6; { -- 1, 1} is monotone versus aepsi;-far from every monotone function. The two main results of this paper are a new lower bound and a new algorithm for this well-studied problem. Lower bound: We prove an a#x03A9;(n1/5) lower bound on the query complexity of any non-adaptive two-sided error algorithm for testing whether an unknown Boolean function f is monotone versus constant-far from monotone. This gives an exponential improvement on the previous lower bound of a#x03A9;(log n) due to Fischer et al.[1]. We show that the same lower bound holds for monotonicity testing of Boolean-valued functions over hypergrid domains {1,,m}n for all m age; 2. Upper bound: We present an O(n5/6) poly(1/aepsi;)-query algorithm that tests whether an unknown Boolean function f is monotone versus aepsi;-far from monotone. Our algorithm, which is non-adaptive and makes one-sided error, is a modified version of the algorithm of Chakrabarty and Seshadhri[2], which makes O(n7/8) poly(1/aepsi;) queries.

Rectangles Are Nonnegative Juntas.

Abstract: We develop a new method to prove communication lower bounds for composed functions of the form $f\circ g^n$, where $f$ is any boolean function on $n$ inputs and $g$ is a sufficiently “hard” two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of $f \circ g^n$ can be simulated by a nonnegative combination of juntas. This is a new formalization for the intuition that each low-communication randomized protocol can only “query” a few inputs of $f$ as encoded by the gadget $g$. Consequently, we characterize the communication complexity of $f\circ g^n$ in all known one-sided (i.e., not closed under complement) zero-communication models by a corresponding query complexity measure of $f$. These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy. As applications, we resolve several open problems from prior work. We show that $\mathsf{SBP}^{\sf cc}$ (a class characterized...

Which Boolean Functions Maximize Mutual Information on Noisy Inputs

Abstract: We pose a simply stated conjecture regarding the maximum mutual information a Boolean function can reveal about noisy inputs. Specifically, let X n be independent identically distributed Bernoulli(1/2), and let Yn be the result of passing X n through a memoryless binary symmetric channel with crossover probability α. For any Boolean function b : {0, 1} n → {0, 1}, we conjecture that I(b(X n ); Y n ) ≤ 1 - H(α). While the conjecture remains open, we provide substantial evidence supporting its validity. Connections are also made to discrete isoperimetric inequalities.

Boolean function monotonicity testing requires (almost) $n^{1/2}$ non-adaptive queries

Abstract: We prove a lower bound of $\Omega(n^{1/2 - c})$, for all $c>0$, on the query complexity of (two-sided error) non-adaptive algorithms for testing whether an $n$-variable Boolean function is monotone versus constant-far from monotone. This improves a $\tilde{\Omega}(n^{1/5})$ lower bound for the same problem that was recently given in [CST14] and is very close to $\Omega(n^{1/2})$, which we conjecture is the optimal lower bound for this model.

BDD based synthesis of Boolean functions using memristors

Abstract: Very recently a new passive circuit element called memristor has been extensively investigated by researchers, which can be used for a variety of applications. This two-terminal device having few nanometer dimensions has been experimentally shown to possess both memory and resistor properties. This has also received great attention due to the fact that these devices can very easily be integrated on CMOS subsystems. Most of the logic design works in this context are based on material implication operation which can be very efficiently implemented using memristors. In this paper we propose an efficient realization of 2-to-1 multiplexer using memristors, and hence present a synthesis methodology that represents a given Boolean function as a Reduced Ordered Binary Decision Diagram (ROBDD) and then maps the same to memristor implementation.

Construction of rotation symmetric Boolean functions with optimal algebraic immunity and high nonlinearity

Abstract: Recent research shows that the class of rotation symmetric Boolean functions is potentially rich in functions of cryptographic significance. In this paper, based on the knowledge of compositions of an integer, we present two new kinds of construction of rotation symmetric Boolean functions having optimal algebraic immunity on either odd variables or even variables. Our new functions are of much better nonlinearity than all the existing theoretical constructions of rotation symmetric Boolean functions with optimal algebraic immunity. Further, the algebraic degree of our rotation symmetric Boolean functions are also high enough.

Candidate weak pseudorandom functions in AC0 ○ MOD2

Abstract: Pseudorandom functions (PRFs) play a fundamental role in symmetric-key cryptography. However, they are inherently complex and cannot be implemented in the class AC0 (MOD2). Weak pseudorandom functions (weak PRFs) do not suffer from this complexity limitation, yet they suffice for many cryptographic applications. We study the minimal complexity requirements for constructing weak PRFs. To this end We conjecture that the function family FA(x) = g(Ax), where A is a random square GF(2) matrix and g is a carefully chosen function of constant depth, is a weak PRF. In support of our conjecture, we show that functions in this family are inapproximable by GF(2) polynomials of low degree and do not correlate with any fixed Boolean function family of subexponential size. We study the class AC0 ○ MOD2 that captures the complexity of our construction. We conjecture that all functions in this class have a Fourier coefficient of magnitude exp(- poly log n) and prove this conjecture in the case when the MOD2 function is typical. We investigate the relation between the hardness of learning noisy parities and the existence of weak PRFs in AC0 ○ MOD2. We argue that such a complexity-driven approach can play a role in bridging the gap between the theory and practice of cryptography.

Multi-level approximate logic synthesis under general error constraints

Abstract: We address the problem of multi-level approximate logic synthesis. Our strategy assumes existence of an optimized exact Boolean network, which is critical in practice since arithmetic blocks are rarely synthesized from 2-level representation automatically. The goal is to produce minimum cost circuits whose logic function deviates in a controlled manner from the exact function with deviations quantified by the magnitude and frequency of errors. We rely on network simplifications allowed by external don't cares (EXDCs). We formulate the error-magnitude constrained problem by using Boolean relations to capture the allowed error behavior in a more general manner compared to incompletely specified functions. Our key contribution is in finding sets of external don't cares that maximally approach the Boolean relation. The algorithm starts with an EXDC set that is overly relaxed and iteratively, and in a greedy fashion, identifies a feasible EXDC set by solving a series of conventional EXDC-based network optimizations. The algorithm then ensures compliance to error frequency constraints by recovering the correct outputs on the sought number of error-producing inputs while aiming to minimize the network cost increase. We applied the algorithm to several well-known adder and multiplier designs of varying bit-width. Even for small error magnitudes, the algorithm produces networks with gate count reduced by 30–50%, when the error frequency constraint is loose. This is up to 20% fewer gates than a naive EXDC-based approach.

On Quadratization of Pseudo-Boolean Functions

Abstract: We survey current term-wise techniques for quadratizing high-degree pseudo-Boolean functions and introduce a new one, which allows multiple splits of terms. We also introduce the first aggregative approach, which splits a collection of terms based on their common parts.

Generalized Maiorana–McFarland Construction of Resilient Boolean Functions With High Nonlinearity and Good Algebraic Properties

Abstract: A new framework concerning the construction of small-order resilient Boolean functions whose nonlinearity is strictly greater than 2 n-1 - 2 [n/2] is given. First, a generalized Maiorana-McFarland construction technique is described, which extends the current approaches by combining the usage of affine and nonlinear functions in a controllable manner. It is shown that for any given m, this technique can be used to construct a large class of n-variable (n both even and odd) m-resilient degree-optimized Boolean functions with currently best known nonlinearity. This class may also provide functions with excellent algebraic properties, measured through the resistance to (fast) algebraic attacks, if the number of n/2-variable affine subfunctions used in the construction is relatively low. Due to a potentially low hardware implementation cost, along with overall good cryptographic properties, this class of functions is an attractive candidate for the use in certain stream cipher schemes.

Approximation algorithms for stochastic boolean function evaluation and stochastic submodular set cover

Abstract: We present approximation algorithms for two problems: Stochastic Boolean Function Evaluation (SBFE) and Stochastic Submodular Set Cover (SSSC) Our results for SBFE problems are obtained by reducing them to SSSC problems through the construction of appropriate utility functionsWe give a new algorithm for the SSSC problem that we call Adaptive Dual Greedy We use this algorithm to obtain a 3-approximation algorithm solving the SBFE problem for linear threshold formulas We also get a 3-approximation algorithm for the closely related Stochastic Min-Knapsack problem, and a 2-approximation for a natural special case of that problem In addition, we prove a new approximation bound for a previous algorithm for the SSSC problem, Adaptive GreedyWe consider an approach to approximating SBFE problems using existing techniques, which we call the Q-value approach This approach easily yields a new result for evaluation of CDNF formulas, and we apply variants of it to simultaneous evaluation problems and a ranking problem However, we show that the Q-value approach provably cannot be used to obtain a sublinear approximation factor for the SBFE problem for linear threshold formulas or read-once DNF

Mining Circuit Lower Bound Proofs for Meta-algorithms

Abstract: We show that circuit lower bound proofs based on the method of random restrictions yield non-trivial compression algorithms for easy Boolean functions from the corresponding circuit classes. The compression problem is defined as follows: given the truth table of an n-variate Boolean function f computable by some unknown small circuit from a known class of circuits, find in deterministic time poly(2^n) a circuit C (no restriction on the type of C) computing f so that the size of C is less than the trivial circuit size 2^n/n. We get non-trivial compression for functions computable by AC0 circuits, (de Morgan) formulas, and (read-once) branching programs of the size for which the lower bounds for the corresponding circuit class are known. These compression algorithms rely on the structural characterizations of "easy" functions, which are useful both for proving circuit lower bounds and for designing "meta-algorithms" (such as Circuit-SAT). For (de Morgan) formulas, such structural characterization is provided by the "shrinkage under random restrictions" results cite {Sub61, Has98-shrinkage}, strengthened to the "high-probability" version by {San10, IMZ12, KR13}. We give a new, simple proof of the "high-probability" version of the shrinkage result for (de Morgan) formulas, with improved parameters. We use this shrinkage result to get both compression and #SAT algorithms for (de Morgan) formulas of size about n^2. We also use this shrinkage result to get an alternative proof of the recent result by Komargodski and Raz {KR13} of the average-case lower bound against small (de Morgan) formulas. Finally, we show that the existence of any non-trivial compression algorithm for a circuit class mathcal {C}subseteqP/poly would imply the circuit lower bound NEXPnotsubseteq mathcal {C}. This complements Williams's result {Wil10} that any non-trivial Circuit-SAT algorithm for a circuit class mathcal {C} would imply a super polynomial lower bound against mathcal {C} for a language in NEXP also proves such an implication in NEXP.

Topological structure and the disturbance decoupling problem of singular Boolean networks

Abstract: The general singular Boolean networks are proposed in this study, motivated by the algebraic form of dynamic-algebraic Boolean networks via the semi-tensor product of matrices. First, one of the most important problems for this kind of networks, solvability problem, is discussed. Then, in order to calculate the fixed points and cycles, the transition matrix of a singular Boolean network is defined, which contains all the state transferring information. At last, the general singular Boolean control networks are considered with their solvability and the disturbance decoupling problem is presented and solved by a constant control. Illustrative examples are given to show the feasibility of the results.

Fast Algorithm of Attribute Reduction Based on the Complementation of Boolean Function

Abstract: In this chapter we propose a new method of solving the attribute reduction problem. Our method is different to the classical approach using the so-called discernibility function and its CNF into DNF transformation. We have proved that the problem is equivalent to very efficient unate complementation algorithm. That is why we propose new algorithm based on recursive execution of the procedure, which at every step of recursion selects the splitting variable and then calculates the cofactors with respect to the selected variables (Shannon expansion procedure). The recursion continues until at each leaf of the recursion tree the easily computable rules for complement process can be applied. The recursion process creates a binary tree so that the final result is obtained merging the results in the subtrees. The final matrix represents all the minimal reducts of a decision table or all the minimal dependence sets of input variables, respectively. According to the results of computer tests, better results can be achieved by application of our method in combination with the classical method.

S-box, SET, Match: A Toolbox for S-box Analysis

Abstract: Boolean functions and substitution boxes (S-boxes) represent the only nonlinear part in many algorithms and therefore play the crucial role in their security. Despite the fact that some algorithms today reuse theoretically secure and carefully constructed S-boxes, there is a clear need for a tool that can analyze security properties of S-boxes and hence the corresponding primitives. This need is especially evident in the scenarios where the goal is to create new S-boxes. Even in the cases when some common properties of S-boxes are known, we believe it is prudent to exhaustively investigate all possible sets of cryptographic properties. In this paper we present a tool for the evaluation of Boolean functions and S-boxes suitable for cryptography.

Threshold Boolean form for joint probabilistic constraints with random technology matrix

Abstract: We develop a new modeling and solution method for stochastic programming problems that include a joint probabilistic constraint in which the multirow random technology matrix is discretely distributed. We binarize the probability distribution of the random variables in such a way that we can extract a threshold partially defined Boolean function (pdBf) representing the probabilistic constraint. We then construct a tight threshold Boolean minorant for the pdBf. Any separating structure of the tight threshold Boolean minorant defines sufficient conditions for the satisfaction of the probabilistic constraint and takes the form of a system of linear constraints. We use the separating structure to derive three new deterministic formulations for the studied stochastic problem, and we derive a set of strengthening valid inequalities. A crucial feature of the new integer formulations is that the number of integer variables does not depend on the number of scenarios used to represent uncertainty. The computational study, based on instances of the stochastic capital rationing problem, shows that the mixed-integer linear programming formulations are orders of magnitude faster to solve than the mixed-integer nonlinear programming formulation. The method integrating the valid inequalities in a branch-and-bound algorithm has the best performance.

Friedgut-Kalai-Naor Theorem for Slices of the Boolean Cube.

Abstract: The Friedgut–Kalai–Naor theorem states that if a Boolean function f : t0, 1u N t0, 1u is close (in L-distance) to an affine function `px1, . . . , xnq “ c0 ` ř i cixi, then f is close to a Boolean affine function (which necessarily depends on at most one coordinate). We prove a similar theorem for functions defined over `rns k “ tpx1, . . . , xnq P t0, 1u : ř i xi “ ku.

Bounding the Sensitivity of Polynomial Threshold Functions

Abstract: We give the first nontrivial upper bounds on the average sensitivity and noise sensitivity of polynomial threshold functions. More specifically, for a Boolean function f on n variables equal to the sign of a real, multivariate polynomial of total degree d, we prove • The average sensitivity of f is at most O(n 1 1=(4d+6) ). (We also give a combinatorial

Average Sensitivity and Noise Sensitivity of Polynomial Threshold Functions

Abstract: We give the first nontrivial upper bounds on the Boolean average sensitivity and noise sensitivity of degree-$d$ polynomial threshold functions (PTFs). Our bound on the Boolean average sensitivity of PTFs represents the first progress toward the resolution of a conjecture of Gotsman and Linial [Combinatorica, 14 (1994), pp. 35--50], which states that the symmetric function slicing the middle $d$ layers of the Boolean hypercube has the highest average sensitivity of all degree-$d$ PTFs. Via the $L_1$ polynomial regression algorithm of Kalai et al. [SIAM J. Comput., 37 (2008), pp. 1777--1805], our bound on Boolean noise sensitivity yields the first polynomial-time agnostic learning algorithm for the broad class of constant-degree PTFs under the uniform distribution. To obtain our bound on the Boolean average sensitivity of PTFs, we generalize the “critical-index” machinery of [R. Servedio, Comput. Complexity, 16 (2007), pp. 180--209] (which in that work applies to halfspaces, i.e., degree-1 PTFs) to general...

Stochastic Multiple-Valued Gene Networks

Abstract: Among various approaches to modeling gene regulatory networks (GRNs), Boolean networks (BNs) and its probabilistic extension, probabilistic Boolean networks (PBNs), have been studied to gain insights into the dynamics of GRNs. To further exploit the simplicity of logical models, a multiple-valued network employs gene states that are not limited to binary values, thus providing a finer granularity in the modeling of GRNs. In this paper, stochastic multiple-valued networks (SMNs) are proposed for modeling the effects of noise and gene perturbation in a GRN. An SMN enables an accurate and efficient simulation of a probabilistic multiple-valued network (as an extension of a PBN). In a k-level SMN of n genes, it requires a complexity of O(nLkn) to compute the state transition matrix, where L is a factor related to the minimum sequence length in the SMN for achieving a desired accuracy. The use of randomly permuted stochastic sequences further increases computational efficiency and allows for a tunable tradeoff between accuracy and efficiency. The analysis of a p53-Mdm2 network and a WNT5A network shows that the proposed SMN approach is efficient in evaluating the network dynamics and steady state distribution of gene networks under random gene perturbation.

Synchronization of coupled large-scale Boolean networks.

Abstract: This paper investigates the complete synchronization and partial synchronization of two large-scale Boolean networks. First, the aggregation algorithm towards large-scale Boolean network is reviewed. Second, the aggregation algorithm is applied to study the complete synchronization and partial synchronization of large-scale Boolean networks. Finally, an illustrative example is presented to show the efficiency of the proposed results.