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At least 109 records · Page 6Linked to original sources
arXiv2010

Tiling-Recognizable Two-Dimensional Languages: From Non-Determinism to Determinism through Unambiguity

Tiling recognizable two-dimensional languages, also known as REC, generalize recognizable string languages to two dimensions and share with them several theoretical properties. Nevertheless REC is not closed under complementation and the membership problem is NP-complete. This implies that this family REC is intrinsically non-deterministic. The natural and immediate definition of unambiguity corresponds to a family UREC of languages that is strictly contained in REC. On the other hand this definition of unambiguity leads to an undecidability result and therefore it cannot correspond to any deterministic notion. We introduce the notion of line-unambiguous tiling recognizable languages and prove that it corresponds or somehow naturally introduces different notions of determin- ism that define a hierarchy inside REC.

nlin.CG↗
arXiv2010

A Simulation of Oblivious Multi-Head One-Way Finite Automata by Real-Time Cellular Automata

In this paper, we present the simulation of a simple, yet significantly powerful, sequential model by cellular automata. The simulated model is called oblivious multi-head one-way finite automata and is characterized by having its heads moving only forward, on a trajectory that only depends on the length of the input. While the original finite automaton works in linear time, its corresponding cellular automaton performs the same task in real time, that is, exactly the length of the input. Although not truly a speed-up, the simulation may be interesting and reminds us of the open question about the equivalence of linear and real times on cellular automata.

nlin.CG↗
arXiv2010

A Categorical Outlook on Cellular Automata

In programming language semantics, it has proved to be fruitful to analyze context-dependent notions of computation, e.g., dataflow computation and attribute grammars, using comonads. We explore the viability and value of similar modeling of cellular automata. We identify local behaviors of cellular automata with coKleisli maps of the exponent comonad on the category of uniform spaces and uniformly continuous functions and exploit this equivalence to conclude some standard results about cellular automata as instances of basic category-theoretic generalities. In particular, we recover Ceccherini-Silberstein and Coornaert's version of the Curtis-Hedlund theorem.

nlin.CG↗
arXiv2010

Infinite Time Cellular Automata: A Real Computation Model

We define a new transfinite time model of computation, infinite time cellular automata. The model is shown to be as powerful than infinite time Turing machines, both on finite and infinite inputs; thus inheriting many of its properties. We then show how to simulate the canonical real computation model, BSS machines, with infinite time cellular automata in exactly ωsteps.

nlin.CG↗
arXiv2010

Computing (or not) Quasi-Periodicity Functions of Tilings

We know that tilesets that can tile the plane always admit a quasi-periodic tiling [4, 8], yet they hold many uncomputable properties [3, 11, 21, 25]. The quasi-periodicity function is one way to measure the regularity of a quasi-periodic tiling. We prove that the tilings by a tileset that admits only quasi-periodic tilings have a recursively (and uniformly) bounded quasi-periodicity function. This corrects an error from [6, theorem 9] which stated the contrary. Instead we construct a tileset for which any quasi-periodic tiling has a quasi-periodicity function that cannot be recursively bounded. We provide such a construction for 1-dimensional effective subshifts and obtain as a corollary the result for tilings of the plane via recent links between these objects [1, 10].

nlin.CG↗
arXiv2010

Time-Symmetric Cellular Automata

Together with the concept of reversibility, another relevant physical notion is time-symmetry, which expresses that there is no way of distinguishing between backward and forward time directions. This notion, found in physical theories, has been neglected in the area of discrete dynamical systems. Here we formalize it in the context of cellular automata and establish some basic facts and relations. We also state some open problems that may encourage further research on the topic.

cs.DM↗
arXiv2010

Construction of $μ$-Limit Sets

The $μ$-limit set of a cellular automaton is a subshift whose forbidden patterns are exactly those, whose probabilities tend to zero as time tends to in- finity. In this article, for a given subshift in a large class of subshifts, we propose the construction of a cellular automaton which realizes this subshift as $μ$-limit set where $μ$ is the uniform Bernoulli measure.

cs.DM↗
arXiv2010

The Block Neighborhood

We define the block neighborhood of a reversible CA, which is related both to its decomposition into a product of block permutations and to quantum computing. We give a purely combinatorial characterization of the block neighborhood, which helps in two ways. First, it makes the computation of the block neighborhood of a given CA relatively easy. Second, it allows us to derive upper bounds on the block neighborhood: for a single CA as function of the classical and inverse neighborhoods, and for the composition of several CAs. One consequence of that is a characterization of a class of "elementary" CAs that cannot be written as the composition of two simpler parts whose neighborhoods and inverse neighborhoods would be reduced by one half.

cs.DM↗
arXiv2011

Algebraic approach to directed stochastic avalanches

A two-dimensional directed stochastic sandpile model is studied analytically with the use of directed Abelian algebras recently introduced by Alcaraz and V. Rittenberg [Phys. Rev. E {\bf 78}, 041126 (2008)]. Exact expressions for the probabilities of all possible toppling events which follow the transfer of arbitrary number of particles to a site in the stationary configuration are derived. A description of the virtual-time evolution of directed avalanches on two dimensional lattices is suggested. Due to intractability of the general problem, the algebraic approach is applied only to the solution of the special cases of directed deterministic avalanches and trivial stochastic avalanches describing simple random walks of two particles. The study of these cases has clarified the role of each particular kind of toppling in the process of avalanche growth. In the general case of the quadratic directed algebra we have determined exactly the maximum possible values of: (1) the current of particles at any given moment of virtual time and (2) the occupation number (`height') of each site at any moment of time.

nlin.CG↗
arXiv2011

Solving the Satisfiability Problem Through Boolean Networks

In this paper we present a new approach to solve the satisfiability problem (SAT), based on boolean networks (BN). We define a mapping between a SAT instance and a BN, and we solve SAT problem by simulating the BN dynamics. We prove that BN fixed points correspond to the SAT solutions. The mapping presented allows to develop a new class of algorithms to solve SAT. Moreover, this new approach suggests new ways to combine symbolic and connectionist computation and provides a general framework for local search algorithms.

cs.AI↗
arXiv2011

Schema Redescription in Cellular Automata: Revisiting Emergence in Complex Systems

We present a method to eliminate redundancy in the transition tables of Boolean automata: schema redescription with two symbols. One symbol is used to capture redundancy of individual input variables, and another to capture permutability in sets of input variables: fully characterizing the canalization present in Boolean functions. Two-symbol schemata explain aspects of the behaviour of automata networks that the characterization of their emergent patterns does not capture. We use our method to compare two well-known cellular automata for the density classification task: the human engineered CA GKL, and another obtained via genetic programming (GP). We show that despite having very different collective behaviour, these rules are very similar. Indeed, GKL is a special case of GP. Therefore, we demonstrate that it is more feasible to compare cellular automata via schema redescriptions of their rules, than by looking at their emergent behaviour, leading us to question the tendency in complexity research to pay much more attention to emergent patterns than to local interactions.

nlin.CG↗
arXiv2011

Cellular Automata and Discrete Geometry

In this paper, we look at the possibility to implement the algorithm to construct a discrete line devised by the first author in cellular automata. It turns out that such an implementation is feasible.

cs.CG↗
arXiv2011

Guaranteeing Convergence of Iterative Skewed Voting Algorithms for Image Segmentation

In this paper we provide rigorous proof for the convergence of an iterative voting-based image segmentation algorithm called Active Masks. Active Masks (AM) was proposed to solve the challenging task of delineating punctate patterns of cells from fluorescence microscope images. Each iteration of AM consists of a linear convolution composed with a nonlinear thresholding; what makes this process special in our case is the presence of additive terms whose role is to "skew" the voting when prior information is available. In real-world implementation, the AM algorithm always converges to a fixed point. We study the behavior of AM rigorously and present a proof of this convergence. The key idea is to formulate AM as a generalized (parallel) majority cellular automaton, adapting proof techniques from discrete dynamical systems.

math.FA↗
arXiv2011

Phase transition behavior in a cellular automaton model with different initial configurations

We investigate the dynamical transition from free-flow to jammed traffic, which is related to the divergence of the relaxation time and susceptibility of the energy dissipation rate $E_d$, in the Nagel-Schreckenberg (NS) model with two different initial configurations. Different initial configurations give rise to distinct phase transition. We argue that the phase transition of the deterministic NS model with megajam and random initial configuration is first- and second-order phase transition, respectively. The energy dissipation rate $E_d$ and relaxation time follow power-law behavior in some cases. The associated dynamic exponents have also been presented.

nlin.CG↗
arXiv2011

A Deterministic Approach to the Synchronization of Cellular Automata

In this work we introduce a deterministic scheme of synchronization of linear and nonlinear cellular automata (CA) with complex behavior, connected through a master-slave coupling. By using a definition of Boolean derivative, we use the linear approximation of the automata to determine a function of coupling that promotes synchronization without perturbing all the sites of the slave system.

nlin.CD↗
arXiv2011

A Carbon-Cycle Based Stochastic Cellular Automata Climate Model

In this article a stochastic cellular automata model is examined, which has been developed to study a "small" world, where local changes may noticeably alter global characteristics. This is applied to a climate model, where global temperature is determined by an interplay between atmospheric carbon dioxide and carbon stored by plant life. The latter can be relased by forest fires, giving rise to significant changes of global conditions within short time.

nlin.CG↗
arXiv2011

Can anything from Noether's theorem be salvaged for discrete dynamical systems?

The dynamics of a physical system is linked to its phase-space geometry by Noether's theorem, which holds under standard hypotheses including continuity. Does an analogous theorem hold for discrete systems? As a testbed, we take the Ising spin model with both ferromagnetic and antiferromagnetic bonds. We show that---and why---energy not only acts as a generator of the dynamics for this family of systems, but is also conserved when the dynamics is time-invariant.

nlin.CG↗