Efficient universal pushdown cellular automata and their application to complexity

In order to obtain universal classical cellular automata an infinite space is required. Therefore, the number of required processors depends on the length of input data and, additionally, may increase during the computation. On the other hand, Turing machines are universal devices which have one processor only and additionally an infinite storage tape. Here an in some sense intermediate model is studied. The pushdown cellular automata are a stack augmented generalization of classical cellular automata. They form a massively parallel universal model where the number of processors is bounded by the length of input data. Effcient universal pushdown cellular automata and their efficiently verifiable encodings are proposed. They are applied to computational complexity, and tight time and stack-space hierarchies are shown. CR Subject Classification (1998): F.1, F.4.3, B.6.1, E.

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)

Parallel turing machines with one-head control units and cellular automata

Parallel Turing machines (PTM) can be viewed as a generalization of cellular automata (CA) where an additional measure called processor complexity can be defined which indicates the ``amount of parallelism\u27\u27 used. In this paper PTM are investigated with respect to their power as recognizers of formal languages. A combinatorial approach as well as diagonalization are used to obtain hierarchies of complexity classes for PTM and CA. In some cases it is possible to keep the space complexity of PTM fixed. Thus for the first time it is possible to find hierarchies of complexity classes (though not CA classes) which are completely contained in the class of languages recognizable by CA with space complexity n and in polynomial time. A possible collapse of the time hierarchy for these CA would therefore also imply some unexpected properties of PTM

On relations between arrays of processing elements of different dimensionality

We are examining the power of $d$-dimensional arrays of processing elements in view of a special kind of structural complexity. In particular simulation techniques are shown, which allow to reduce the dimension at an increased cost of time only. Conversely, it is not possible to regain the speed by increasing the dimension. Moreover, we demonstrate that increasing the computation time (just by a constant factor) can have a more favorable effect than increasing the dimension (arbitrari

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

MFCS\u2798 Satellite Workshop on Cellular Automata

For the 1998 conference on Mathematical Foundations of Computer Science (MFCS\u2798) four papers on Cellular Automata were accepted as regular MFCS\u2798 contributions. Furthermore an MFCS\u2798 satellite workshop on Cellular Automata was organized with ten additional talks. The embedding of the workshop into the conference with its participants coming from a broad spectrum of fields of work lead to interesting discussions and a fruitful exchange of ideas. The contributions which had been accepted for MFCS\u2798 itself may be found in the conference proceedings, edited by L. Brim, J. Gruska and J. Zlatuska, Springer LNCS 1450. All other (invited and regular) papers of the workshop are contained in this technical report. (One paper, for which no postscript file of the full paper is available, is only included in the printed version of the report). Contents: F. Blanchard, E. Formenti, P. Kurka: Cellular automata in the Cantor, Besicovitch and Weyl Spaces K. Kobayashi: On Time Optimal Solutions of the Two-Dimensional Firing Squad Synchronization Problem L. Margara: Topological Mixing and Denseness of Periodic Orbits for Linear Cellular Automata over Z_m B. Martin: A Geometrical Hierarchy of Graph via Cellular Automata K. Morita, K. Imai: Number-Conserving Reversible Cellular Automata and Their Computation-Universality C. Nichitiu, E. Remila: Simulations of graph automata K. Svozil: Is the world a machine? H. Umeo: Cellular Algorithms with 1-bit Inter-Cell Communications F. Reischle, Th. Worsch: Simulations between alternating CA, alternating TM and circuit families K. Sutner: Computation Theory of Cellular Automat

Artificial Evolution of Arbitrary Self-Replicating Cellular Automata

Since John von Neumann's seminal work on developing cellular automata models of self-replication, there have been numerous computational studies that have sought to create self-replicating structures or "machines". Cellular automata (CA) has been the most widely used method in these studies, with manual designs yielding a number of specific self-replicating structures. However, it has been found to be very difficult, in general, to design local state-transition rules that, when they operate concurrently in each cell of the cellular space, produce a desired global behavior such as self-replication. This has greatly limited the number of different self-replicating structures designed and studied to date. In this dissertation, I explore the feasibility of overcoming this difficulty by using genetic programming (GP) to evolve novel CA self-replication models. I first formulate an approach to representing structures and rules in cellular automata spaces that is amenable to manipulation by the genetic operations used in GP. Then, using this representation, I demonstrate that it is possible to create a "replicator factory" that provides an unprecedented ability to automatically generate a whole class of new self-replicating structures and that allows one to systematically investigate the properties of replicating structures as one varies the initial configuration, its size, shape, symmetry, and allowable states. This approach is then extended to incorporate multi-objective fitness criteria, resulting in production of diversified replicators. For example, this allows generation of target structures whose complexity greatly exceeds that of the seed structure itself. Finally, the extended multi-objective replicator factory is further generalized into a structure/rule co-evolution model, such that replicators with unspecified seed structures can also be concurrently evolved, resulting in different structure/rule combinations and having the capability of not only replicating but also carrying out a secondary pre-specified task with different strategies. I conclude that GP provides a powerful method for creating CA models of self-replication

Optimización e inferencia en procesos físico-químicos representados mediante autómatas celulares

Objetivos y método de estudio: Un autómata celular probabilístico Markoviano para capturar el fenómeno esencial de la ruptura y agregación en presencia de agitación externa es propuesto y su espacio de parámetros es explorado a fondo. El modelo propuesto es una extensión multidimensional de un modelo conceptual unidimensional, el cual logro reproducir algunas características básicas de la distribución del tamaño de partículas en la remoción de metales pesados en una prueba de jarras. Reportamos experimentos numéricos del espacio de parámetros para el autómata multidimensional para identificar los parámetros que mejor reproducen la distribución de tamaño de partículas observada, obteniéndose un mejor acuerd