Computations on Nondeterministic Cellular Automata

The work is concerned with the trade-offs between the dimension and the time and space complexity of computations on nondeterministic cellular automata. It is proved, that 1). Every NCA \Cal A of dimension $r$, computing a predicate $P$ with time complexity T(n) and space complexity S(n) can be simulated by $r$-dimensional NCA with time and space complexity $O(T^{\frac{1}{r+1}} S^{\frac{r}{r+1}})$ and by $r+1$-dimensional NCA with time and space complexity $O(T^{1/2} +S)$. 2) For any predicate $P$ and integer $r>1$ if \Cal A is a fastest $r$-dimensional NCA computing $P$ with time complexity T(n) and space complexity S(n), then $T= O(S)$. 3). If $T_{r,P}$ is time complexity of a fastest $r$-dimensional NCA computing predicate $P$ then T_{r+1,P} &=O((T_{r,P})^{1-r/(r+1)^2}), T_{r-1,P} &=O((T_{r,P})^{1+2/r}). Similar problems for deterministic CA are discussed.Comment: 18 pages in AmsTex, 3 figures in PostScrip

A linear speed-up theorem for cellular automata

AbstractIbarra (1985) showed that, given a cellular automaton of range 1 recognizing some language in time n+1+R(n), we can obtain another CA of range 1 recognizing exactly the same language but in time n+1+R(n)/k (k⩾2 arbitrary). Their proof proceeds indirectly (through the simulation of CAs by a special kind of sequential machines, the STMs) and we think it misses that way some of the deep intuition of the problem. We, therefore, provide here a direct proof of this result (extended to the case of CAs of arbitrary range) involving the explicit construction of a CA working in time n+1+R(n)/k. This speeded-up automaton first gathers the cells of the line k by k in n+1 steps which then enables it to start computing by “leaps” of k steps, thus completing the R(n) remaining steps in time R(n)/k. The major problem arising from the obligation to pass from one phase to the other synchronously is solved using a synchronization process derived from the solutions of the well-known “firing-squad synchronization problem” (FSSP)

On generative morphological diversity of elementary cellular automata

Purpose: Studies in complexity of cellular automata do usually deal with measures taken on integral dynamics or statistical measures of space-time configurations. No one has tried to analyze a generative power of cellular-automaton machines. The purpose of this paper is to fill the gap and develop a basis for future studies in generative complexity of large-scale spatially extended systems. Design/methodology/approach: Let all but one cell be in alike state in initial configuration of a one-dimensional cellular automaton. A generative morphological diversity of the cellular automaton is a number of different three-by-three cell blocks occurred in the automaton's space-time configuration. Findings: The paper builds a hierarchy of generative diversity of one-dimensional cellular automata with binary cell-states and ternary neighborhoods, discusses necessary conditions for a cell-state transition rule to be on top of the hierarchy, and studies stability of the hierarchy to initial conditions. Research limitations/implications: The method developed will be used - in conjunction with other complexity measures - to built a complete complexity maps of one- and two-dimensional cellular automata, and to select and breed local transition functions with highest degree of generative morphological complexity. Originality/value: The hierarchy built presents the first ever approach to formally characterize generative potential of cellular automata. © Emerald Group Publishing Limited

On injective endomorphisms of symbolic schemes

Building on the seminal work of Gromov on endomorphisms of symbolic algebraic varieties [10], we introduce a notion of cellular automata over schemes which generalize affine algebraic cellular automata in [7]. We extend known results to this more general setting. We also establish several new ones regarding the closed image property, surjunctivity, reversibility, and invertibility for cellular automata over algebraic varieties with coefficients in an algebraically closed field. As a byproduct, we obtain a negative answer to a question raised in [7] on the existence of a bijective complex affine algebraic cellular automaton $\tau \colon A^{\mathbb Z} \to A^{\mathbb Z}$ whose inverse is not algebraic

Physical Universality, State-Dependent Dynamical Laws and Open-Ended Novelty

abstract: A major conceptual step forward in understanding the logical architecture of living systems was advanced by von Neumann with his universal constructor, a physical device capable of self-reproduction. A necessary condition for a universal constructor to exist is that the laws of physics permit physical universality, such that any transformation (consistent with the laws of physics and availability of resources) can be caused to occur. While physical universality has been demonstrated in simple cellular automata models, so far these have not displayed a requisite feature of life—namely open-ended evolution—the explanation of which was also a prime motivator in von Neumann’s formulation of a universal constructor. Current examples of physical universality rely on reversible dynamical laws, whereas it is well-known that living processes are dissipative. Here we show that physical universality and open-ended dynamics should both be possible in irreversible dynamical systems if one entertains the possibility of state-dependent laws. We demonstrate with simple toy models how the accessibility of state space can yield open-ended trajectories, defined as trajectories that do not repeat within the expected Poincaré recurrence time and are not reproducible by an isolated system. We discuss implications for physical universality, or an approximation to it, as a foundational framework for developing a physics for life

Sublinear-Time Language Recognition and Decision by One-Dimensional Cellular Automata

After an apparent hiatus of roughly 30 years, we revisit a seemingly neglected subject in the theory of (one-dimensional) cellular automata: sublinear-time computation. The model considered is that of ACAs, which are language acceptors whose acceptance condition depends on the states of all cells in the automaton. We prove a time hierarchy theorem for sublinear-time ACA classes, analyze their intersection with the regular languages, and, finally, establish strict inclusions in the parallel computation classes $\mathsf{SC}$ and (uniform) $\mathsf{AC}$. As an addendum, we introduce and investigate the concept of a decider ACA (DACA) as a candidate for a decider counterpart to (acceptor) ACAs. We show the class of languages decidable in constant time by DACAs equals the locally testable languages, and we also determine $\Omega(\sqrt{n})$ as the (tight) time complexity threshold for DACAs up to which no advantage compared to constant time is possible.Comment: 16 pages, 2 figures, to appear at DLT 202

An Algorithmic Framework for Robot Navigation in Unknown Terrains.

The problem of navigating a robot body through a terrain whose model is a priori known is well-solved problem in many cases. Comparatively, a lesser number of research results have been reported about the navigation problem in unknown terrains i.e., the terrains whose model are not a priori known. The focus of our work is to obtain an algorithmic framework that yields algorithms to solve certain navigational problems in unknown terrains. We consider a finite-sized two-dimensional terrain populated by a finite set of obstacles $O$ = \{O\sb1,O\sb2,\...,O\sb{n}\} where O\sb{i} is a simple polygon with a finite number of vertices. Consider a circular body R, of diameter $\delta\geq$ O, capable of translational and rotational motions. R houses a computational device with storage capability. Additionally, R is equipped with a sensor system capable of detecting all visible vertices and edges. We consider two generic problems of navigation in unknown terrains: the Visit Problem, VP, and the Terrain model acquisition Problem, TP. In the visit problem, R is required to visit a sequence of destination points d\sb1,d\sb2,\...,d\sb{M} in the specified order. In the terrain model acquisition problem, R is required to acquire the model of the terrain so that it can navigate to any destination without using sensors and by using only the path planning algorithms of known terrains. We present a unified algorithmic framework that yields correct algorithms to solve both VP and TP. In this framework, R \u27simulates\u27 a graph exploration algorithm on an incrementally-constructible graph structure, called the navigation course, that satisfies the properties of finiteness, connectivity, terrain-visibility and local-constructibility. Additionally, we incorporate the incidental learning feature in our solution to VP so as to enhance the performance. We consider solutions to VP and TP using navigation courses based two geometric structures, namely the visibility graph and the Voronoi diagram. In all the cases, we analyze the performance of the algorithms for VP and TP in terms of the number of scan operations, the distance traversed and the computational complexity

Enhancing the diversity of self-replicating structures using active self-adapting mechanisms

Numerous varieties of life forms have filled the earth throughout evolution. Evolution consists of two processes: self-replication and interaction with the physical environment and other living things around it. Initiated by von Neumann et al. studies on self-replication in cellular automata have attracted much attention, which aim to explore the logical mechanism underlying the replication of living things. In nature, competition is a common and spontaneous resource to drive self-replications, whereas most cellular-automaton-based models merely focus on some self-protection mechanisms that may deprive the rights of other artificial life (loops) to live. Especially, Huang et al. designed a self-adaptive, self-replicating model using a greedy selection mechanism, which can increase the ability of loops to survive through an occasionally abandoning part of their own structural information, for the sake of adapting to the restricted environment. Though this passive adaptation can improve diversity, it is always limited by the loop’s original structure and is unable to evolve or mutate new genes in a way that is consistent with the adaptive evolution of natural life. Furthermore, it is essential to implement more complex self-adaptive evolutionary mechanisms not at the cost of increasing the complexity of cellular automata. To this end, this article proposes new self-adaptive mechanisms, which can change the information of structural genes and actively adapt to the environment when the arm of a self-replicating loop encounters obstacles, thereby increasing the chance of replication. Meanwhile, our mechanisms can also actively add a proper orientation to the current construction arm for the sake of breaking through the deadlock situation. Our new mechanisms enable active self-adaptations in comparison with the passive mechanism in the work of Huang et al. which is achieved by including a few rules without increasing the number of cell states as compared to the latter. Experiments demonstrate that this active self-adaptability can bring more diversity than the previous mechanism, whereby it may facilitate the emergence of various levels in self-replicating structures