Fangyue Chen

Universal Perceptron and DNA-Like Learning Algorithm for Binary Neural Networks: LSBF and PBF Implementations

Implementing linearly nonseparable Boolean functions (non-LSBF) has been an important and yet challenging task due to the extremely high complexity of this kind of functions and the exponentially increasing percentage of the number of non-LSBF in the entire set of Boolean functions as the number of input variables increases. In this paper, an algorithm named DNA-like learning and decomposing algorithm (DNA-like LDA) is proposed, which is capable of effectively implementing non-LSBF. The novel algorithm first trains the DNA-like offset sequence and decomposes non-LSBF into logic XOR operations of a sequence of LSBF, and then determines the weight-threshold values of the multilayer perceptron (MLP) that perform both the decompositions of LSBF and the function mapping the hidden neurons to the output neuron. The algorithm is validated by two typical examples about the problem of approximating the circular region and the well-known n -bit parity Boolean function (PBF).

Chaos of elementary cellular automata rule 42 of Wolfram's class II.

In this paper, the dynamics of elementary cellular automata rule 42 is investigated in the bi-infinite symbolic sequence space. Rule 42, a member of Wolframs class II which was said to be simply as periodic before, actually defines a chaotic global attractor; that is, rule 42 is topologically mixing on its global attractor and possesses the positive topological entropy. Therefore, rule 42 is chaotic in the sense of both Li-Yorke and Devaney. Meanwhile, the characteristic function and the basin tree diagram of rule 42 are explored for some finite length of binary strings, which reveal its Bernoulli characteristics. The method presented in this work is also applicable to studying the dynamics of other rules, especially the 112 Bernoulli-shift rules of the elementary cellular automata.

Realization of Boolean Functions via CNN: Mathematical Theory, LSBF and Template Design

As a paradigm for nonlinear spatial-temporal processing, cellular nonlinear networks (CNN) are biologically inspired systems where computation emerges from a collection of simple locally coupled nonlinear cells. Our investigation is an exploration of an important and difficult aspect of implementing arbitrary Boolean functions by using CNN. A typical class of basic key Boolean functions is the class of linearly separable ones. In this paper, we focus on establishing a complete set of mathematical theories for the linearly separable Boolean functions (LSBF) that are identical to a class of uncoupled CNN. First, we obtain an essential relationship between the template and the offset levels as well as the basis of the binary input vector set in the uncoupled CNN. More precisely, we construct a neat binary input-output truth table and some interesting properties of the offset levels of the uncoupled CNN, and develop a practical design formula for the class of CNN template. Especially, we found a criterion for LSBF, which depends only on symbolic relations between a Boolean functions outputs. Furthermore, we develop a method for representing any linearly nonseparable Boolean function into a logic operation of a sequence of linearly separable ones for a small number of inputs

REALIZATION AND BIFURCATION OF BOOLEAN FUNCTIONS VIA CELLULAR NEURAL NETWORKS

In this work, we study the realization and bifurcation of Boolean functions of four variables via a Cellular Neural Network (CNN). We characterize the basic relations between the genes and the offsets of an uncoupled CNN as well as the basis of the binary input vectors set. Based on the analysis, we have rigorously proved that there are exactly 1882 linearly separable Boolean functions of four variables, and found an effective method for realizing all linearly separable Boolean functions via an uncoupled CNN. Consequently, any kind of linearly separable Boolean function can be implemented by an uncoupled CNN, and all CNN genes that are associated with these Boolean functions, called the CNN gene bank of four variables, can be easily determined. Through this work, we will show that the standard CNN invented by Chua and Yang in 1988 indeed is very essential not only in terms of engineering applications but also in the sense of fundamental mathematics.

Realization of Boolean functions via CNN with von Neumann neighborhoods

Recently, an effective method for realizing linearly separable Boolean functions via Cellular Neural Networks (CNN), called the threshold bifurcation method, was introduced, with a CNN gene bank of four variables established [Chen & Chen, 2005]. Based on this success, the present paper is to further explore the realization of all linearly separable Boolean functions of five variables via CNN with von Neumann neighborhoods. This paper provides: (i) important and essential relations among the genes (or templates) and the offsets of an uncoupled CNN as well as the basis of the binary input vectors set, (ii) a neat truth table of uncoupled CNN with five input variables, (iii) 94572 linearly separable Boolean functions (LSBF) in the family of 225 = 4.294967296 × 109 Boolean functions of five variables, realizable by a single CNN, and (iv) all 94572 CNN linearly separable Boolean genes (LSBG), which can be determined to form the CNN gene bank of five variables.

Realization of Boolean functions and gene bank of cellular neural networks

A paradigm for nonlinear spatial-temporal processing, cellular neural networks (CNN), was created by inspiration from the cellular automata and neural networks. This article is an exploration of the important aspect of realizing Boolean functions by using standard CNN. A neat CNN truth table of n binary variables and an essential formula of an uncoupled CNN are discovered, and an effective method of realizing all linearly separable Boolean functions (LSBF) via CNN is proposed. Borrowed from biological concepts and terms, the parameter group in a CNN is a metaphor for gene which completely determines the dynamical properties of the CNN. The CNN gene bank, which consists of the family of all linearly separable Boolean genes (LSBG) that are associated with all the LSBF, can be easily determined and progressively established. An interesting phenomenon is that the number of LSBG with the von Neumann neighborhood is 94572, which is close to the number of genes existing in the human genome.

SOME NONROBUST BERNOULLI-SHIFT RULES

In this paper, it is shown that elementary cellular automata rule 172, as a member of the Chuas robust period-1 rules and the Wolfram class I, is also a nonrobust Bernoulli-shift rule. This rule actually exhibits complex Bernoulli-shift dynamics in the bi-infinite binary sequence space. More precisely, in this paper, it is rigorously proved that rule 172 is topologically mixing and has positive topological entropy on a subsystem. Hence, rule 172 is chaotic in the sense of both Li–Yorke and Devaney. The method developed in this paper is also applicable to checking the subshifts contained in other robust period-1 rules, for example, rules 168 and 40, which also represent nonrobust Bernoulli-shift dynamics.

Implementation of Arbitrary Boolean Functions via CNN

As a paradigm for nonlinear spatial-temporal processing, cellular nonlinear networks (CNN) are biologically inspired systems where computation emerges from a collection of simple locally coupled nonlinear cells. Our investigation is an exploration of implementing arbitrary Boolean functions by using CNN. A class of basic key Boolean functions is the class of linearly separable ones, which is identical to the class of uncoupled CNN with binary inputs and binary outputs. In our recent studies, we not only construct a neat binary input-output truth table and some interesting properties of the offset levels of uncoupled CNN, but also develop a practical design formula for the uncoupled CNN template. Especially, we obtain a criterion for LSBF (abbreviation of linearly separable Boolean function), which depends only on symbolic relations between a Boolean functions outputs. Furthermore, we show that any linearly non-separable Boolean function can be decomposed as a logic operation of a series of linearly separable ones and can be implemented on CNN-UM

Chaos emerged on the ‘edge of chaos’

Rule 110 is a complex cellular automaton (CA) in Wolframs system of identification, capable of supporting universal computation. It has been suggested that a universal CA should be on the ‘edge of chaos’, which means that the dynamical behaviour of such a system is neither simple nor chaotic. There is no doubt that the dynamical property of Rule 110 is extremely complex and still not well understood. This paper proves the existence of subsystems on which this rule is chaotic in the sense of Devaney.

DNA-like learning algorithm of CNN template implementing Boolean functions

Inspired by the concept of DNA sequence in biological systems, we developed a novel learning algorithm named DNA-like learning, which is enable to quickly train the CNN template (or named CNN gene) implementing linearly separable Boolean function (LSBF). This algorithm has many advantages including in particular faster running speed and better robustness, and without the need to consider its convergence property. For example, the “AND” and “OR” operations only needs 6 iterations and computations by using the algorithm, compared to the error-correction algorithm which needs 20 operations for the same task, and for judging and implementing a 9-bit linearly separable Boolean function can be finished within only one second on a program based on the new algorithm.