Chih-Hung Chang | National University of Kaohsiung (original) (raw)

Papers by Chih-Hung Chang

In this paper, we study the quantitative behavior of one-dimensional linear cellular automata T f... more In this paper, we study the quantitative behavior of one-dimensional linear cellular automata T f [−r,r] , defined by local rule f (x −r , . . . , xr) = r i=−r λ i x i (mod m), acting on the space of all doubly infinite sequences with values in a finite ring Zm, m ≥ 2. Once generalize the formulas given by Ban et al. [J. Cellular Automata 6 (2011) 385-397] for measure-theoretic entropy and topological pressure of one-dimensional cellular automata, we calculate the measure entropy and the topological pressure of the linear cellular automata with respect to the Bernoulli measure on the set Z Z m . Also, it is shown that the uniform Bernoulli measure is the unique equilibrium measure for linear cellular automata. We compare values of topological entropy and topological directional entropy by using the formula obtained by Akın [J. Computation and Appl. Math. 225 (2) (2009) 459-466]. The topological directional entropy is interpreted by means of figures. As an application, we demonstrate that the Hausdorff of the limit set of a linear cellular automaton is the unique root of Bowen's equation. Some open problems remain to be of interest.

This paper investigates the kkk-mixing property of a multidimensional cellular automaton. Suppose... more This paper investigates the kkk-mixing property of a multidimensional cellular automaton. Suppose FFF is a cellular automaton with the local rule fff defined on a ddd-dimensional convex hull mathcalC\mathcal{C}mathcalC which is generated by an apex set CCC. Then FFF is kkk-mixing with respect to the uniform Bernoulli measure for all positive integer kkk if fff is a permutation at some apex in CCC. An algorithm called the \emph{Mixing Algorithm} is proposed to verify if a local rule fff is permutive at some apex in CCC. Moreover, the proposed conditions are optimal. An application of this investigation is to construct a multidimensional ergodic linear cellular automaton.

This investigation studies the ergodic properties of reversible linear cellular automata over Zm ... more This investigation studies the ergodic properties of reversible linear cellular automata over Zm for m ∈ N. We show that a reversible linear cellular automaton is either a Bernoulli automorphism or non-ergodic. This gives an affirmative answer to an open problem proposed in [Pivato, Ergodc theory of cellular automata, Encyclopedia of Complexity and Systems Science, 2009, pp. 2980-3015] for the case of reversible linear cellular automata.

Journal of Mathematics, 2013

ABSTRACT This paper studies the quantitative behavior of a class of one-dimensional cellular auto... more ABSTRACT This paper studies the quantitative behavior of a class of one-dimensional cellular automata, named weakly permutive cellular automata, acting on the space of all doubly infinite sequences with values in a finite ring , . We calculate the measure-theoretic entropy and the topological entropy of weakly permutive cellular automata with respect to any invariant measure on the space . As an application, it is shown that the uniform Bernoulli measure is the unique maximal measure for linear cellular automata among the Markov measures.

IEEE transactions on neural networks and learning systems, Jan 15, 2015

This paper focuses on the deep and shallow architecture of multilayer neural networks (MNNs). The... more This paper focuses on the deep and shallow architecture of multilayer neural networks (MNNs). The demonstration of whether or not an MNN can be replaced by another MNN with fewer layers is equivalent to studying the topological conjugacy of its hidden layers. This paper provides a systematic methodology to indicate when two hidden spaces are topologically conjugated. Furthermore, some criteria are presented for some specific cases.

$Research paper thumbnail of Measures of the full Hausdorff dimension for a general Sierpi\'{n}ski carpet$

The measure of the full dimension for a general Sierpi\'{n}ski carpet is studied. In the firs... more The measure of the full dimension for a general Sierpi\'{n}ski carpet is studied. In the first part of this study, we give a criterion for the measure of the full Hausdorff dimension of a Sierpi\'{n}ski carpet. Meanwhile, it is the conditional equilibrium measure of zero potential with respect to some Gibbs measure nualpha\nu_{\alpha}nualpha of matrix-valued potential alphamathbfN\alpha\mathbf{N}alphamathbfN (defined later). On one hand, this investigation extends the result of [17] without condition \textbf{(H)}. On the other hand, it provides a checkable condition to ensure the existence and uniqueness of the measure of the full Hausdorff dimension for a general Sierpi\'{n}ski carpet. In the second part of this paper we give a criterion for the Markov projection measure and estimate its number of steps by means of the induced matrix-valued potential. The results enable us to answer some questions which arise from [1] and [4] on the projection measure and factors.

Journal of cellular automata

This paper studies cellular automata in two aspects: Ergodic and topological behavior. For ergodi... more This paper studies cellular automata in two aspects: Ergodic and topological behavior. For ergodic aspect, the formulae of measure-theoretic entropy, topological entropy and topological pressure are given in closed forms and Parry measure is demonstrated to be an equilibrium measure for some potential function. For topological aspect, an example is examined to show that the exhibition of snap-back repellers for a cellular automaton infers Li-Yorke chaos. In addition, bipermutive cellular automata are optimized for the exhibition of snap-back repellers in permutive cellular automata whenever two-sided shift space is considered.

Neural Networks, 2015

This paper investigates whether the output space of a multi-layer cellular neural network can be ... more This paper investigates whether the output space of a multi-layer cellular neural network can be realized via a single layer cellular neural network in the sense of the existence of finite-to-one map from one output space to the other. Whenever such realization exists, the phenomena exhibited in the output space of the revealed single layer cellular neural network is at most a constant multiple of the phenomena exhibited in the output space of the original multi-layer cellular neural network. Meanwhile, the computation complexity of a single layer system is much less than the complexity of a multi-layer system. Namely, one can trade the precision of the results for the execution time. We remark that a routine extension of the proposed methodology in this paper can be applied to the substitution of hidden spaces although the detailed illustration is omitted.

Advances in Pure Mathematics, 2013

This elucidation investigates the Hausdorff dimension of the output space of multi-layer neural n... more This elucidation investigates the Hausdorff dimension of the output space of multi-layer neural networks. When the factor map from the covering space of the output space to the output space has a synchronizing word, the Hausdorff dimension of the output space relates to its topological entropy. This clarifies the geometrical structure of the output space in more details.

Springer Proceedings in Mathematics & Statistics, 2013

2011 International Conference on Multimedia Computing and Systems, 2011

ABSTRACT This study investigates the complexity of the global set of output patterns for two-dime... more ABSTRACT This study investigates the complexity of the global set of output patterns for two-dimensional multi-layer cellular neural networks. Applying labeling to the output space produces a two-dimensional sofic shift space. The ordering matrices and symbolic transition matrices are introduced to study the spatial entropy of the output space.

Proceedings of the American Mathematical Society, 2011

Neural Networks, 2013

This manuscript considers the learning problem of multi-layer neural networks (MNNs) with an acti... more This manuscript considers the learning problem of multi-layer neural networks (MNNs) with an activation function which comes from cellular neural networks. A systematic investigation of the partition of the parameter space is provided. Furthermore, the recursive formula of the transition matrix of an MNN is obtained. By implementing the well-developed tools in the symbolic dynamical systems, the topological entropy of an MNN can be computed explicitly. A novel phenomenon, the asymmetry of a topological diagram that was seen in Ban, [J. Differential Equations 246, pp. 552-580, 2009], is revealed.

Journal of Differential Equations, 2012

Let Y ⊆ {−1, 1} Z∞×n be the mosaic solution space of an n-layer cellular neural network. We decou... more Let Y ⊆ {−1, 1} Z∞×n be the mosaic solution space of an n-layer cellular neural network. We decouple Y into n subspaces, say Y (n) , and give a necessary and sufficient condition for the existence of factor maps between them. In such a case, Y (i) is a sofic shift for 1 i n. This investigation is equivalent to study the existence of factor maps between two sofic shifts. Moreover, we investigate whether Y (i) and Y ( j) are topological conjugate, strongly shift equivalent, shift equivalent, or finitely equivalent via the well-developed theory in symbolic dynamical systems. This clarifies, in a multi-layer cellular neural network, each layer's structure. As an extension, we can decouple Y into arbitrary k-subspaces, where 2 k n, and demonstrates each subspace's structure.

Journal of Differential Equations, 2009

This study investigates the complexity of the global set of output patterns for one-dimensional m... more This study investigates the complexity of the global set of output patterns for one-dimensional multi-layer cellular neural networks with input. Applying labeling to the output space produces a sofic shift space. Two invariants, namely spatial entropy and dynamical zeta function, can be exactly computed by studying the induced sofic shift space. This study gives sofic shift a realization through a realistic model. Furthermore, a new phenomenon, the broken of symmetry of entropy, is discovered in multi-layer cellular neural networks with input.

International Journal of Bifurcation and Chaos, 2009

ABSTRACT This work investigates the monotonicity of topological entropy for one-dimensional multi... more ABSTRACT This work investigates the monotonicity of topological entropy for one-dimensional multilayer cellular neural networks. The interacting radius and number of layers are treated as parameters. Fix either one of them; the set of topological entropies grows as a strictly nested sequence with respect to one another. Apart from the comparison of the set of topological entropies, maximal and minimal templates are indicators of a dynamical system. Our results demonstrate that maximal and minimal templates of larger interacting radius (respectively number of layers) dominate those of smaller one. To be precise, the strict monotonicity of topological entropy is demonstrated through the comparison of the maximal and minimal templates as the parameters are varied.

Entropy, 2009

This elucidation studies ergodicity and equilibrium measures for additive cellular automata with ... more This elucidation studies ergodicity and equilibrium measures for additive cellular automata with prime states. Additive cellular automata are ergodic with respect to Bernoulli measure unless it is either an identity map or constant. The formulae of measure-theoretic and topological entropies can be expressed in closed forms and the topological pressure is demonstrated explicitly for potential functions that depend on finitely many coordinates. According to these results, Parry measure is inferred to be an equilibrium measure.

Boundary Value Problems, 2013

This study considers the dynamics of cellular neural network-based inhomogeneous lattice dynamica... more This study considers the dynamics of cellular neural network-based inhomogeneous lattice dynamical systems (CNN-based ILDS). The influence of three kinds of boundary conditions, say, the periodic, Dirichlet, and Neumann boundary conditions, is elucidated. We reveal that the complete stability of CNN-based ILDS and, under some prescriptions, the topological entropies of CNN-based ILDS with/without the boundary condition are identical. MSC: 37B10

Applied Mathematics Letters, 2013

ABSTRACT The present investigation elucidates how the number of layers/variance of templates infl... more ABSTRACT The present investigation elucidates how the number of layers/variance of templates influences the phenomena of multi-layer cellular neural networks (MCNNs). This study relates to learning problems for MCNNs. We show that the greater the number of templates that MCNNs adopt, the richer the phenomena that are derived, while equivalently, such neural networks are more efficient as regards the learning aspect. Additionally, the MCNNs with more layers exhibit more phenomena than the ones with fewer layers. A novel phenomenon is seen in the study of the effect of the number of layers with respect to fixed templates.

Applied Mathematics and Computation, 2013

ABSTRACT This investigation considers the complexity of output spaces of multi-layer cellular neu... more ABSTRACT This investigation considers the complexity of output spaces of multi-layer cellular neural networks. Let B be a set of admissible local output patterns coupled with input and let (B) over tilde be the set of admissible output patterns extracting from B. Since topological entropy is an indicator for investigating the complexity of spaces, we study the topological entropy of output spaces Y-U and Y which are induced by B and (B) over tilde, respectively. A system has a diamond if h(Y-U)not equal h(Y). Necessary and sufficient conditions for the existence of diamond are demonstrated separately. Furthermore, numerical experiments exhibit some novel phenomena.