On the dynamics and recursive properties of multidimensional symbolic systems

We study the (sub)dynamics of multidimensional shifts of finite type and sofic shifts, and the action of cellular automata on their limit sets. Such a subaction is always an effective dynamical system: i.e. it is isomorphic to a subshift over the Cantor set the complement of which can be written as the union of a recursive sequence of basic sets. Our main result is that, to varying degrees, this recursive-theoretic condition is also sufficient. We show that the class of expansive subactions of multidimensional sofic shifts is the same as the class of expansive effective systems, and that a general effective system can be realized, modulo a small extension, as the subaction of a shift of finite type or as the action of a cellular automaton on its limit set (after removing a dynamically trivial set). As applications, we characterize, in terms of their computational properties, the numbers which can occur as the entropy of cellular automata, and construct SFTs and CAs with various interesting properties.     

Sofic systems and graphs

A subclass of the class of the subshifts of finite-state symbolic shifts, which was introduced byB. Weiss under the name “sofic systems”, is characterized and studied by using graphs. It is proved that topologically transitive sofic systems are intrinsically ergodic.     

Is prediction possible? Chaotic behavior of Multiple Equilibria Regulation Model in cellular automata topology

In this article, we present the Multiple Equilibria Regulation (MER) Model in cellular automata topology. As argued in previous explorations of the model, for certain parameter values, the behavior of the system exhibits transient chaos (namely, the system is unpredictable but ends in a final steady state). In order to approach empirical reality, we introduce a cellular automata topology. Examining the outcome of the simulations leads us to conclude that for certain parameter values tested, the system yields chaotic behavior. Thus, cellular automata contribution has proven crucial, because the introduced topology converts the behavior of the system from transient chaos to “pure” chaos, i.e., the system is not only unpredictable on the long run but, in addition, it will never rest in a final steady state. According to these findings, authors argue the theoretical hypothesis that the urge for “prediction” in social sciences should be reconsidered in terms of “predictability horizon”. © 2004 Wiley Periodicals, Inc. Complexity 10: 23–36, 2004     

Prioritized Composition With Exclusion and Generation for the Interaction and Control of Discrete Event Systems

Interaction of multiple discrete event systems (DESs) represented as automata are carried out using composition operations. These operations on automata enforce concurrency, wherein an event exists in the composed automaton if it exists in the participating states of the interacting automata possessing the event in their event set. Heymann generalized this by introducing event priorities, wherein an event exists in the composed automaton if it exists in the participating state of the interacting automata having priority over the event. For two interacting automata P and Q, while prioritized composition can model the P, Q, AND, and OR boolean interactions, it cannot model boolean interactions which require exclusivity of participation, namely, “exclusive P”, “exclusive Q”, “exclusive P or exclusive Q”, “exclusive P and exclusive Q”. In order to also model these additional interactions we propose a generalization of prioritized composition by introducing an exclusivity set besides the existing priority sets. The resulting composition is called prioritized composition with exclusion. We also introduce prioritized composition with exclusion and generation that allows for all sixteen boolean modes of interaction possible when two automata interact. This is done by the further introduction of a nor-generative set. This event set together with the two priority sets and an exclusivity set makes it possible to model eight additional boolean interactions which do not require either of the interacting automata to participate for the event to be enabled in the composed automaton. The applicability of these interactions to decentralized supervisory decision fusion and in composing the rules based model of systems has been illustrated.     

An interconnection of local maps inducing onto global maps

“Local maps and the global maps induced by them” treated in this paper is a subject not only on cellular or tessellation automata but also on shift dynamical systems. We study an interconnection of local maps inducing onto global maps from a graph-theoretical viewpoint.     

Reducibility of covers of AFT shifts

In this paper we show that the reducibility structure of several covers of sofic shifts is a flow invariant. In addition, we prove that for an irreducible subshift of almost finite type the left Krieger cover and the past set cover are reducible. We provide an example which shows that there are non almost finite type shifts which have reducible left Krieger covers. As an application we show that the Matsumoto algebra of an irreducible, strictly sofic shift of almost finite type is not simple.     

Local variational principle concerning entropy of a sofic group action

Recently Lewis Bowen introduced a notion of entropy for measure-preserving actions of countable sofic groups admitting a generating measurable partition with finite entropy; and then David Kerr and Hanfeng Li developed an operator-algebraic approach to actions of countable sofic groups not only on a standard probability space but also on a compact metric space, and established the global variational principle concerning measure-theoretic and topological entropy in this sofic context. By localizing these two kinds of entropy, in this paper we prove a local version of the global variational principle for any finite open cover of the space, and show that these local measure-theoretic and topological entropies coincide with their classical counterparts when the acting group is an infinite amenable group.     

Auto-tuning unit norm frames

Finite unit norm tight frames provide Parseval-like decompositions of vectors in terms of redundant components of equal weight. They are known to be robust against additive noise and erasures, and as such, have great potential as encoding schemes. Unfortunately, up to this point, these frames have proven notoriously difficult to construct. Indeed, though the set of all unit norm tight frames, modulo rotations, is known to contain manifolds of nontrivial dimension, we have but a small finite number of known constructions of such frames. In this paper, we present a new iterative algorithm—gradient descent of the frame potential—for increasing the degree of tightness of any finite unit norm frame. The algorithm itself is easy to implement, and it preserves certain group structures present in the initial frame. In the special case where the number of frame elements is relatively prime to the dimension of the underlying space, we show that this algorithm converges to a unit norm tight frame at a linear rate, provided the initial unit norm frame is already sufficiently close to being tight. By slightly modifying this approach, we get a similar, but weaker, result in the non-relatively-prime case, providing an explicit answer to the Paulsen problem: “How close is a frame which is almost tight and almost unit norm to some unit norm tight frame?”     

Deterministic automata simulation, universality and minimality

Finite automata have been recently used as alternative, discrete models in theoretical physics, especially in problems related to the dichotomy between endophysical/intrinsic and exophysical/ extrinsic perception (see, for instance [3, 6, 18–21]). These studies deal with Moore experiments; the main result states that it is impossible to determine the initial state of an automaton, and, consequently, a discrete model of Heisenberg uncertainty has been suggested. For this aim the classical theory of finite automata — which considers automata with initial states — is not adequate, and a new approach is necessary. A study of finite deterministic automata without initial states is exactly the aim of this paper. We will define and investigate the complexity of various types of simulations between automata. Minimal automata will be constructed and proven to be unique up to an isomorphism. We will build our results on an extension of Myhill-Nerode technique; all constructions will make use of “automata responses” to simple experiments only, i.e., no information about the internal machinery will be considered available.     

On the structure of a sofic shift space

The structure of a sofic shift space is investigated, and Krieger's embedding theorem and Boyle's factor theorem are generalized to a large class of sofic shifts.

     

Set of periods of a subshift

In this article, subsets of \({\mathbb {N}}\) that can arise as sets of periods of the following subshifts are characterized: (i) subshifts of finite type, (ii) transitive subshifts of finite type, (iii) sofic shifts, (iv) transitive sofic shifts, and (v) arbitrary subshifts.     

Symmetry principles via interactions and symmetry-violations

In this work we would like to continue our ideas concerning the principle mentioned at the end of [Appl. Math. Comput. 113 (2000) 289]. From now on we will call this principle the principle of symmetry of states. We recall that:Principle of symmetry of states. Any system whose states have the same symmetry can take part in “interaction-chain”, until the symmetry of the states will be maximal and the system will be stable. In this case, the system has a minimal number of different states. Each “intermediate system” can be viewed like a homomorphic image of the anterior.We recall the fact that we have used the automata theory for modelling the systems. The “homomorphic image” words mentioned in the statement of our principle refer to automata-homomorphism. We mention also that we have applied this principle concerning radioactivity, making a conjecture, that we have considered each radioactive nucleus like an automaton and for each radioactive chain (viewed as an interaction-chain) applied our principle.Our purpose in this work is to try to find other examples of interactions which are in concordance with our principle (are all the interactions in concordance with this principle ?).     

A non-interior-point smoothing method for variational inequality problem

In this paper, we focus on the variational inequality problem. Based on the Fischer-Burmeister function with smoothing parameters, the variational inequality problem can be reformulated as a system of parameterized smooth equations, a non-interior-point smoothing method is presented for solving the problem. The proposed algorithm not only has no restriction on the initial point, but also has global convergence and local quadratic convergence, moreover, the local quadratic convergence is established without a strict complementarity condition. Preliminary numerical results show that the algorithm is promising.     

Fuzzy clustering using multiple Gaussian kernels with optimized-parameters

In this paper, we propose a new kernel-based fuzzy clustering algorithm which tries to find the best clustering results using optimal parameters of each kernel in each cluster. It is known that data with nonlinear relationships can be separated using one of the kernel-based fuzzy clustering methods. Two common fuzzy clustering approaches are: clustering with a single kernel and clustering with multiple kernels. While clustering with a single kernel doesn’t work well with “multiple-density” clusters, multiple kernel-based fuzzy clustering tries to find an optimal linear weighted combination of kernels with initial fixed (not necessarily the best) parameters. Our algorithm is an extension of the single kernel-based fuzzy c-means and the multiple kernel-based fuzzy clustering algorithms. In this algorithm, there is no need to give “good” parameters of each kernel and no need to give an initial “good” number of kernels. Every cluster will be characterized by a Gaussian kernel with optimal parameters. In order to show its effective clustering performance, we have compared it to other similar clustering algorithms using different databases and different clustering validity measures.     

Entropy theory for sofic groupoids I: The foundations

This is the first part in a series in which sofic entropy theory is generalized to class-bijective extensions of sofic groupoids. Here we define topological and measure entropy and prove invariance. We also establish the variational principle, compute the entropy of Bernoulli shift actions and answer a question of Benjy Weiss pertaining to the isomorphism problem for non-free Bernoulli shifts. The proofs are independent of previous literature.     

Optimization in “self‐modeling” complex adaptive systems

When a dynamical system with multiple point attractors is released from an arbitrary initial condition, it will relax into a configuration that locally resolves the constraints or opposing forces between interdependent state variables. However, when there are many conflicting interdependencies between variables, finding a configuration that globally optimizes these constraints by this method is unlikely or may take many attempts. Here, we show that a simple distributed mechanism can incrementally alter a dynamical system such that it finds lower energy configurations, more reliably and more quickly. Specifically, when Hebbian learning is applied to the connections of a simple dynamical system undergoing repeated relaxation, the system will develop an associative memory that amplifies a subset of its own attractor states. This modifies the dynamics of the system such that its ability to find configurations that minimize total system energy, and globally resolve conflicts between interdependent variables, is enhanced. Moreover, we show that the system is not merely “recalling” low energy states that have been previously visited but “predicting” their location by generalizing over local attractor states that have already been visited. This “self‐modeling” framework, i.e., a system that augments its behavior with an associative memory of its own attractors, helps us better understand the conditions under which a simple locally mediated mechanism of self‐organization can promote significantly enhanced global resolution of conflicts between the components of a complex adaptive system. We illustrate this process in random and modular network constraint problems equivalent to graph coloring and distributed task allocation problems. © 2010 Wiley Periodicals, Inc. Complexity 16: 17–26, 2011     

具有有穷个模态辞的模态系统

<正> 在真值逻辑系统中如果加入“可能”“必然”等模熊概念,所得的逻辑系统叫做模态系统(modal system).如果该真值系就为伟统的二值系统,特名曰传统模态系统(下文的讨论不限于传统模态系统).纯由命题变元以及“～”(非)“◇”(可能)“口”(必然)三运算而组成的命题叫做模态辞(modality).若只经奇数次～运算的名曰负模态辞,经偶数次(包括0次)～运算的名曰正模态辞.     

Sofic systems

A symbolic flow is called a sofic system if it is a homomorphic image (factor) of a subshift of finite type. We show that every sofic system can be realized as a finite-to-one factor of a subshift of finite type with the same entropy. From this it follows that sofic systems share many properties with subshifts of finite type. We concentrate especially on the properties of TPPD (transitive with periodic points dense) sofic systems.     

具有输出字符功能的模糊自动机的最小化问题

通过文献[8]中两类具有输出字符功能的Fuzzy自动机和Fuzzy有限状态自动机的强等价性,等价性和弱等价性的条件,在以往仅仅给出的Fuzzy有限状态自动机的最小化问题基础上,讨论了具有更广泛意义的具有输出字符功能的Fuzzy自动机的最小化问题,以及其最小化自动机与Fuzzy有限状态自动机的最小化自动机在不同条件下的关系。