Complexity of Two-Dimensional Patterns

In dynamical systems such as cellular automata and iterated maps, it is often useful to look at a language or set of symbol sequences produced by the system. There are well-established classification schemes, such as the Chomsky hierarchy, with which we can measure the complexity of these sets of sequences, and thus the complexity of the systems which produce them. In this paper, we look at the first few levels of a hierarchy of complexity for two-or-more-dimensional patterns. We show that several definitions of ``regular language'' or ``local rule'' that are equivalent in d=1 lead to distinct classes in d >= 2. We explore the closure properties and computational complexity of these classes, including undecidability and L-, NL- and NP-completeness results. We apply these classes to cellular automata, in particular to their sets of fixed and periodic points, finite-time images, and limit sets. We show that it is undecidable whether a CA in d >= 2 has a periodic point of a given period, and that certain ``local lattice languages'' are not finite-time images or limit sets of any CA. We also show that the entropy of a d-dimensional CA's finite-time image cannot decrease faster than t^{-d} unless it maps every initial condition to a single homogeneous state.Comment: To appear in J. Stat. Phy

Living Melodies - Coevolution Of Sonic Communication

A man address the audience during a program at the Cesar Chavez School in Fort Worth, Texas

Universal Computation in Simple One-Dimensional Cellular Automata

The existence of computation-universal one-dimensional cellular automata with seven states per cell for a transition function depending on the cell itself and its nearest neighbors (r= 1), and four states per cell for r= 2 (when next-nearest neighbors also are included), is shown. It is also demonstrated that a Turing machine with m tape symbols and n internal states can be simulated by a cellular automaton of range r= 1 with m+ n+ 2 states per cell

Coevolving Pursuit-Evasion Strategies in Open and Confined Regions

We have studied pursuit-evasion games where the players can move in the plane or in a square region. Confining the pursuer and evader to a restricted region, or making them different, makes the problem more complex, and excludes simple solutions such as running straight towards infinity as fast as possible. In particular we study the case when the pursuer is made faster but less maneuverable. A steady improvement in performance measured against a fixed set of strategies is often found. We study the how the behavior changes with the parameters in the problem, such as the degree of asymmetry, and investigate the occurrence of unpredictable (protean) evasion behavior. 1 Introduction Coevolution of strategies has been studied extensively for discrete games, both for two-person games such as the Prisoner's Dilemma [Lindgren, 1991, Lindgren & Nordahl, 1994a, Lindgren & Nordahl, 1994b], and multi-person games (e.g., [Akiyama & Kaneko, 1995]). Continuous strategies for differential games [Isa..

Predicting Lattice Gases is P-complete

. We show that predicting the HPP or FHP III lattice gas for finite time is equivalent to calculating the output of an arbitrary Boolean circuit, and is therefore P-complete: that is, it is just as hard as any other problem solvable by a serial computer in polynomial time. It is widely believed in computer science that there are inherently sequential problems, for which parallel processing gives no significant speedup. Unless this is false, it is impossible even with highly parallel processing to predict lattice gases much faster than by explicit simulation. More precisely, we cannot predict t time-steps of a lattice gas in parallel computation time O(log k t) for any k, or O(t ff ) for ff ! 1=2, unless the class P is equal to the class NC or SP respectively. 1 Introduction Given the initial conditions of a d-dimensional cellular automaton, suppose we want to know the state at a site t time-steps in the future. We can do this in O(t d+1 ) steps on a serial computer, or O(t) st..