On the complexity of enumerating possible dynamics of sparsely connected Boolean network automata with simple update rules

We study asynchronous cellular automata (ACA) induced by symmetric Boolean functions [1]. These systems can be considered as sequential dynamical systems (SDS) over words, a class of dynamical systems that consists of (a) a finite, labeled graph Y with vertex set {v1,…,vn} and where each vertex vi has a state xvi in a finite field K, (b) a sequence of functions (Fvi,Y)i, and (c) a word w = (w1,…,wk), where each wi is a vertex in Y. The function Fvi,Y updates the state of vertex vi as a function of the state of vi and its Y-neighbors and maps all other vertex states identically. The SDS is the composed map [Formula: see text]. In the particular case of ACA, the graph is the circle graph on n vertices (Y = Circ n), and all the maps Fvi are induced by a common Boolean function. Our main result is the identification of all w-independent ACA, that is, all ACA with periodic points that are independent of the word (update schedule) w. In general, for each w-independent SDS, there is a finite group whose structure contains information about for example SDS with specific phase space properties. We classify and enumerate the set of periodic points for all w-independent ACA, and we also compute their associated groups in the case of Y = Circ 4. Finally, we analyze invertible ACA and offer an interpretation of S35 as the group of an SDS over the three-dimensional cube with local functions induced by nor3 + nand3.

Sequential dynamical systems over words

The conceptions of $\chi$-value and K-rotation symmetric Boolean functions are introduced by Cusick. K-rotation symmetric Boolean functions are a special rotation symmetric functions, which are invariant under the $k-th$ power of $\rho$.In this paper, we discuss cubic 2-value 2-rotation symmetric Boolean function with $2n$ variables, which denoted by $F^{2n}(x^{2n})$. We give the recursive formula of weight of $F^{2n}(x^{2n})$, and prove that the weight of $F^{2n}(x^{2n})$ is the same as its nonlinearity.

Two important classes of symmetric Boolean functions are the equal-weight Boolean functions and the elementary (or homogeneous) symmetric Boolean functions. In this paper we studied the equal-weight symmetric Boolean functions. First the Walsh spectra of the equal-weight symmetric Boolean functions are given. Second the sufficient and necessary condition on correlation-immunity of the equal-weight symmetric Boolean function is derived and other cryptology properties such as the nonlinearity, balance and propagation criterion are taken into account. In particular, the nonlinearity of the equal-weight symmetric Boolean functions with n (n ≥ 10) variables is determined by their Hamming weight. Considering these properties will be helpful in further investigations of symmetric Boolean functions.

Two Boolean functions are affine equivalent if one can be obtained from the other by applying an affine transformation to the input variables. For a long time, there have been efforts to investigate the affine equivalence of Boolean functions. Due to the complexity of the general problem, only affine equivalence under certain groups of permutations is usually considered. Boolean functions which are invariant under the action of cyclic rotation of the input variables are known as rotation symmetric (RS) Boolean functions. Due to their speed of computation and the prospect of being good cryptographic Boolean functions, this class of Boolean functions has received a lot of attention from cryptographic researchers. In this paper, we study affine equivalence for the simplest rotation symmetric Boolean functions, called MRS functions, which are generated by the cyclic permutations of a single monomial. Using Pólya’s enumeration theorem, we compute the number of equivalence classes, under certain large groups of permutations, for these MRS functions in any number

We consider average-case complexity of computing monotone Boolean functions by straight-line programs with a conditional stop over the basis of all Boolean functions of at most two variables. For the set of all n -ary monotone Boolean functions new Shannon-type upper and lower bounds for the average-case complexity as n → ∞ are established.

We study the realization of monotone Boolean functions by networks. Our main result is a precise version of the following statement: the complexity of realizing a monotone Boolean function ofn arguments is less by the factor (2/πn)1/2, whereπ is the circular ratio, than the complexity of realizing an arbitrary Boolean function ofn arguments. The proof combines known results concerning monotone Boolean functions with new methods relating the computing abilities of networks and machines.

Let $T_t$ denote the $t$-threshold function on the $n$-cube: $T_t(x) = 1$ if $|\{i : x_i=1\}| \geq t$, and $0$ otherwise. Define the distance between Boolean functions $g$ and $h$, $d(g,h)$, to be the number of points on which $g$ and $h$ disagree. We consider the following extremal problem: Over a monotone Boolean function $g$ on the $n$-cube with $s$ zeros, what is the maximum of $d(g,T_t)$? We show that the following monotone function $p_s$ maximizes the distance: For $x \in \{0,1\}^n$, $p_s(x)=0$ if and only if $N(x) < s$, where $N(x)$ is the integer whose $n$-bit binary representation is $x$. Our result generalizes the previous work for the case $t=\lceil n/2 \rceil$ and $s=2^{n-1}$ by Blum, Burch, and Langford [BBL98-FOCS98], who considered the problem to analyze the behavior of a learning algorithm for monotone Boolean functions, and the previous work for the same $t$ and $s$ by Amano and Maruoka [AM02-ALT02].

The Aleksander model of neural networks replaces the connection weights of conventional models by logic devices (or Boolean functions). Learning is achieved by adjusting the Boolean functions stepwise via a 'training-with-noise' algorithm. The authors present a theory of the statistical dynamical properties of the randomly connected model and demonstrate that, in the limit of large but dilute connectivity c of the nodes, the storage capacity for associative memory is of the order (2/c2)2c, which corresponds, roughly speaking, to an average of one nearest-neighbouring pattern stored at site distances 2 on each node. Two parameters are introduced into the learning algorithm: qr and qc being respectively the probabilities to register a correct bit and erase an incorrect one. The effects of varying qr, qc and the training noise level on the storage capacity (after very long training) are discussed. In the limit of low training noise level, the training algorithm is equivalent to the so-called 'proximity rules'. Study of its retrieval properties shows that the model can be described as 'short ranged', whereas the Hopfield model is 'long ranged'. The advantages and disadvantages of introducing the intermediate u state into the system are also discussed.

The Stochastic Boolean Function Evaluation problem for symmetric Boolean functions

In this work, we consider a new type of Fourier-like representation of Boolean function f\colon\{+1,-1\}^n\to\{+1,-1\}: f(x) = \cos(\pi\sum_{S\subseteq[n]}\phi_S \prod_{i\in S} x_i). This representation, which we call the periodic Fourier representation, of Boolean function is closely related to a certain type of multipartite Bell inequalities and non-adaptive measurement-based quantum computation with linear side-processing NMQCp. The minimum number of non-zero coefficients in the above representation, which we call the periodic Fourier sparsity, is equal to the required number of qubits for the exact computation of f by \NMQCp. Periodic Fourier representations are not unique, and can be directly obtained both from the Fourier representation and the F_2-polynomial representation. In this work, we first show that Boolean functions related to ZZZZ-polynomial have small periodic Fourier sparsities. Second, we show that the periodic Fourier sparsity is at least 2^{\deg_{\mathbb{F}_2}(f)}-1, which means that NMQCp efficiently computes a Boolean function $f$ if and only if F_2-degree of f is small. Furthermore, we show that any symmetric Boolean function, e.g., AND_n, Mod^3_n, Maj_n, etc, can be exactly computed by depth-2 NMQCp using a polynomial number of qubits, that implies exponential gaps between NMQCp and depth-2 NMQCp.

We describe a global optimization method called mean field annealing (MFA) and its application to two basic problems in seismic data processing: Seismic deconvolution and surface related multiple attenuation. MFA replaces the stochastic nature of the simulated annealing method with a set of deterministic update rules that act on the average value of the variables rather than on the variables themselves, based on the mean field approximation. As the temperature is lowered, the MFA rules update the variables in terms of their values at a previous temperature. By minimizing averages, it is possible to converge to an equilibrium state considerably faster than a standard simulated annealing method. The update rules are dependent on the form of the cost function and are obtained easily when the cost function resembles the energy function of a Hopfield network. The mapping of a problem onto a Hopfield network is not a precondition for using MFA, but it makes analytic calculations simpler. The seismic deconvolution problem can be mapped onto a Hopfield network by parameterizing the source and the reflectivity in terms of binary neurons. In this context, the solution of the problem is obtained when the neurons of the network reach their stable states. By minimizing the cost function of the network with MFA and using an appropriate cooling schedule, it is possible to escape local minima. A similar idea can also be applied to design an operator that attenuates surface related multiple reflections from plane‐wave transformed seismograms assuming a 1-D earth. The cost function for the multiple elimination problem is based on the criterion of minimum energy of the multiple suppressed data.

A new search method for combinatorial optimization problem inspired by the spin glass system

In this paper, we prove two general theorems on monotone Boolean functions which are useful for constructing an learning algorithm for monotone Boolean functions under the uniform distribution. A monotone Boolean function is called fair if it takes the value 1 on exactly half of its inputs. The first result proved in this paper is that the single variable function f(x) = x i has the minimum correlation with the majority function among all fair monotone functions (Theorem 1). This proves the conjecture by Blum, Burch and Langford (FOCS '98) and improves the performance guarantee of the best known learning algorithm for monotone Boolean functions under the uniform distribution proposed by them. Our second result is on the relationship between the influences and the average sensitivity of a monotone Boolean function. The influence of variable x i on f is defined as the probability that f(x) differs from f(x○+ e i ) where x is chosen uniformly from {0, 1} n and x ○+ e i means x with its i-th bit flipped. The average sensitivity of f is defined as the sum of the influences over all variables x i . We prove that a somewhat unintuitive result which says if the influence of every variable on a monotone Boolean function is small, i.e., O(1/n c ) for some constant c > 0, then the average sensitivity of the function must be large, i.e., Ω(log n) (Theorem 11). We also discuss how to apply this result to construct a new learning algorithm for monotone Boolean functions.

This thesis proposes a new parallel computing network--a binary relation inference network. The network topology is set up to match conventional optimization procedures. The network transforms some optimization problems (shortest path, transitive closure, assignment, etc.) into network convergence problems and divides the work load evenly among all processors. Link weights are known. It can operate in synchronous discrete-time, asynchronous discrete-time, and continuous-time domains. Inference network algorithms are derived directly from problem formulation, and are effective for a large range of problem sizes. The inference network consists of interconnected components; each component is composed of a unit, a number of sites attached to the unit, and links from the unit to sites on other units. Each unit is a simple computational engine. The topology of the inference network matches naturally to many optimization problems, since its deduction sites produce candidates to compete for the optimum and its decision-making units select the optimum from the site values. Either directly or through a transformation mapping to a systolic structure, discrete inference network algorithms can be implemented on a commercially available parallel processing facility, such as the Connection Machine. Analog inference network can be built using integrated circuits which solve problems efficiently in the continuous-time domain. In this thesis, mathematical analysis and numerical simulation were conducted for the synchronous, asynchronous, and analog inference network. Various applications has been discussed. The inference network has shown to solve the all-pair shortest path problem efficiently in various updating schemes. The inference network algorithm for the transitive closure problem parts derived straightforwardly from the formulation of the problem and the topology of the inference network. Efficient continuous-time solution was obtained from a non-synchronized logical circuit. The convergence of the network for the shortest path or transitive closure problems was shown to be independent of the problem size. It was demonstrated that the inference network can solve the assignment problem in a way similar to the Hopfield net's solution to the traveling salesman problem. However, the former was shown to converge for all test cases with problem size up to 128. (Abstract shortened by UMI.)