Petr Kurka

Languages, equicontinuity and attractors in cellular automata

We consider three related classifications of cellular automata: the first is based on the complexity of languages generated by clopen partitions of the state space, i.e. on the complexity of the factor subshifts; the second is based on the concept of equicontinuity and it is a modification of the classification introduced by Gilman [ 9 ]. The third one is based on the concept of attractors and it refines the classification introduced by Hurley [ 16 ]. We show relations between these classifications and give examples of cellular automata in the intersection classes. In particular, we show that every positively expansive cellular automaton is conjugate to a one-sided subshift of finite type and that every topologically transitive cellular automaton is sensitive to initial conditions. We also construct a cellular automaton with minimal quasi-attractor, whose basin has measure zero, answering a question raised in Hurley [ 16 ].

Topological and measure-theoretic properties of one-dimensional cellular automata

Abstract This is a survey of known results and open questions about the topological and ergodic dynamics of one-dimensional cellular automata.

On topological dynamics of Turing machines

Abstract We associate to a Turing machine two dynamical systems which we call Turing machine with moving tape (TMT) and Turing machine with moving head (TMH). TMT are equivalent to generalized shifts of Moore (1990) and they include two-sided full shifts. TMH are shift-commuting maps of two-sided sofic systems. In both classes we characterize systems with the shadowing property, show that a bijective expansive TMT is conjugate to a subshift of finite type and that topological entropy of every TMH is zero. We conjecture that every TMT has a periodic point.

Topological Dynamics of Cellular Automata

This is an overview of some classical and recent results in topological dynamics of cellular automata on the space of twosided symbolic sequences. The concepts studied include surjectivity, transitivity, equicontinuity, closingness, openness, expansivity, attractors and the shadowing property.

Cellular Automata Dynamical Systems

We present recent studies on cellular automata (CAs) viewed as discrete dynamical systems. In the first part, we illustrate the relations between two important notions: subshift attractors and signal subshifts, measure attractors and particle weight functions. The second part of the chapter considers some operations on the space of one-dimensional CA configurations, namely, shifting and lifting, showing that they conserve many dynamical properties while reducing complexity. The final part reports recent investigations on two-dimensional CA. In particular, we report a construction (slicing construction) that allows us to see a two-dimensional CA as a one-dimensional one and to lift some one-dimensional results to the two-dimensional case.

Zero-Dimensional Dynamical Systems, Formal Languages, and Universality

Abstract. We measure the complexity of dynamical systems on zero-dimensional compact metric spaces by the complexity of formal languages, which these systems generate on clopen partitions of the state space. We show that in the classes of recursive, context-sensitive, context-free, regular, etc., languages there exist universal dynamical systems which yield, by factor maps, all dynamical systems of the class. Universal systems are not unique, but in every class there exists a smallest universal system.

Simplicity Criteria for Dynamical Systems

We formulate two simplicity criteria for dynamical systems based on the concepts of finite automata and regular languages. Finite automata are regarded as dynamical systems on discontinuum and their factors yield the first simplicity class. A finite cover of a topological space is almost disjoint, if it consists of closed sets which have the same dimension as the space, and meet in sets whose dimension is smaller. A dynamical system is regular, if it yields a regular language when observed through any finite almost disjoint cover. Next we formulate two topological simplicity criteria. A dynamical system has finite attractors, if the Ω-limit of its state space is finite. A dynamical system has chaotic limits, if every point is included in a set whose Ω-limit is either a finite orbit or a chaotic subsystem. We show the relations between these criteria and classify according to them several classes of zero-dimensional and one-dimensional dynamical systems.

The Unary Arithmetical Algorithm in Bimodular Number Systems

We analyze the performance of the unary arithmetical algorithm which computes a Moebius transformation in bimodular number systems which extend the binary signed system. We give statistical evidence that in some of these systems, the algorithm has linear average time complexity.

Fast Arithmetical Algorithms in Möbius Number Systems

We analyze the time complexity of exact real arithmetical algorithms in Möbius number systems. Using the methods of Ergodic theory, we associate to any Möbius number system its transaction quotient T ≥ 1 and show that the norm of the state matrix after n transactions is of the order Tn. We argue that the Bimodular Möbius number system introduced in Kůrka [10] has transaction quotient less than 1.2, so that it computes the arithmetical operations faster than any standard positional system.

The Exact Real Arithmetical Algorithm in Binary Continued Fractions

The exact real binary arithmetical algorithm is an on-line algorithm which computes the sum, product or ratio of two real numbers to arbitrary precision. The algorithm works in general Moebius number systems which represent real numbers by infinite products of Moebius transformations. We consider a number system of binary continued fractions in which this algorithm is computed faster than in the binary signed system. Moreover, the number system of binary continued fractions circumvents the problem of nonredundancy and slow convergence of continued fractions.