A universal non-conservative reversible elementary triangular partitioned cellular automaton that shows complex behavior

A reversible cellular automaton (RCA) is an abstract spatiotemporal model of a reversible world. Using the framework of an RCA, we study the problem of how we can elegantly compose reversible computers from simple reversible microscopic operations. The CA model used here is an elementary triangular partitioned CA (ETPCA), whose spatial configurations evolve according to an extremely simple local transition function. We focus on the particular reversible ETPCA No. 0347, where 0347 is an ID number in the class of 256 ETPCAs. Based on our past studies, we explain that reversible Turing machines (RTMs) can be constructed in a systematic and hierarchical manner in this cellular space. Though ETPCA 0347 is an artificial CA model, this method gives a new vista to find a pathway from a reversible microscopic law to reversible computers. In particular, we shall see that RTMs can be easily realized in a unique method by using a reversible logic element with memory (RLEM) in the intermediate step of the pathway.KeywordsReversible cellular automatonElementary triangular partitioned cellular automatonReversible Turing machineReversible logic element with memory

A complete classification of the computational complexity of the fixed-point existence problem for Boolean dynamical systems, i.e., finite discrete dynamical systems over the domain {0, 1}, is presented. For function classes $${\mathcal{F}}$$ and graph classes $${\mathcal{G}}$$ , an ( $${\mathcal{F}},{\mathcal{G}}$$ )-system is a Boolean dynamical system such that all local transition functions lie in $${\mathcal{F}}$$ and the underlying graph lies in $${\mathcal{G}}$$ . Let $${\mathcal{F}}$$ be a class of Boolean functions which is closed under composition and let $${\mathcal{G}}$$ be a class of graphs which is closed under taking minors. The following dichotomy theorems are shown: (1) If $${\mathcal{F}}$$ contains the self-dual functions and $${\mathcal{G}}$$ contains the planar graphs, then the fixed-point existence problem for ( $${\mathcal{F}},{\mathcal{G}}$$ )-systems with local transition function given by truth-tables is NP-complete; otherwise, it is decidable in polynomial time. (2) If $${\mathcal{F}}$$ contains the self-dual functions and $${\mathcal{G}}$$ contains the graphs having vertex covers of size one, then the fixed-point existence problem for ( $${\mathcal{F}},{\mathcal{G}}$$ )-systems with local transition function given by formulas or circuits is NP-complete; otherwise, it is decidable in polynomial time.

We consider the problem of inferring the local transition functions of discrete dynamical systems from observed behavior. Our focus is on synchronous systems whose local transition functions are threshold functions. We assume that the topology of the system is known and that the goal is to infer a threshold value for each node so that the system produces the observed behavior. We show that some of these inference problems are efficiently solvable while others are NP-complete, even when the underlying graph of the dynamical system is a simple path. We also identify a fixed parameter tractable problem in this context.

A property of superposability is defined here for Treillis Automata as an equivalent of the linearity property of Cellular Automata. We take advantage of the binary local transition function of TAs, which enables us to use an infix notation for it, to study this property in an algebraic framework. We show that a TA is superposable iff its local transition function operates on its set of states as a commutative monoid law and that the general case of commutative monoids is reducible to the case of Abelian groups in which the space-time diagrams obtained are isomorphic to a superposition of figures representing the triangle of binomial coefficients (Pascal's triangle) modulo integers.KeywordsCellular AutomatonInitial ConfigurationNeutral ElementBinomial CoefficientCommutative MonoidThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Inverse rules of ECA with rule number 150

A number-conserving cellular automaton (NCCA) is a cellular automaton whose states are integers and whose transition function keeps the sum of all cells constant throughout its evolution. It can be seen as a kind of modeling of the physical conservation laws of mass or energy. In this paper we show a construction method of radius 1/2 NCCAs. The local transition function is expressed via a single unary function which can be regarded as 'flows' of numbers. In spite of the strong constraint, we constructed radius 1/2 NCCAs that simulate any radius 1/2 cellular automata or any radius 1 NCCA. We also consider the state complexity of these non-splitting simulations (4n2 + 2n + 1 and 8n2 + 12n - 16, respectively). These results also imply existence of an intrinsically universal radius 1/2 NCCA.

Synthesis of one-dimensional binary scope-2 flexible cellular systems from initial final configuration pairs

In this paper, we formalize precisely the sense in which the application of a cellular automaton to partial configurations is a natural extension of its local transition function through the categorical notion of Kan extension. In fact, the two possible ways to do such an extension and the ingredients involved in their definition are related through Kan extensions in many ways. These relations provide additional links between computer science and category theory, and also give a new point of view on the famous Curtis–Hedlund theorem of cellular automata from the extended topological point of view provided by category theory. These links also allow to relatively easily generalize concepts pioneered by cellular automata to arbitrary kinds of possibly evolving spaces. No prior knowledge of category theory is assumed for the most part.

This paper presents a groove reconstruction method based on the local rule of Cellular Automata (CA). A local rule for groove reconstruction is proposed to design and optimize the groove of milling insert. The insert is firstly divided into a large number of cells that typically form a regular grid over the domain. The domain consists of regular square cells with discrete variables and the discrete CA model is built. As the cells in the CA domain only interact with their neighboring cells when performing local computations, the governing equation for the whole domain is not necessary. Then the states of cells are modified according to the local rule applied. The state of the entire system is updated based on the state of the cell and its neighboring cells. Collectively, these cells’ states define the state of the entire domain, and the groove can be reconstructed according to the state of the domain. The reconstructed groove is tested by a FEM simulation. The simulation results show that the reconstructed groove has a satisfied performance on the stress field.

Open asynchronous dynamic cellular learning automata and its application to allocation hub location problem

Numerical simulation of the debris-flow process is commonly based on the shallow water equations. However, as a two-phase anisotropic mixture, debris flows display complex rheological behavior, making it difficult to model or to simulate these using standard approaches. In this paper, an improved cellular automaton (CA) model is developed for simulating the extent of debris-flow run-out. The CA model consists of three essential components: cellular space, lattice relation, and transition function. A two-dimensional rectangular cellular space is generated from mesh grid in the digital terrain model data, and the Moore neighborhood type is selected as the lattice relation. We also use a transition function based on a Monte Carlo iteration algorithm to automatically search the flow direction and flow routine. Specifically, this new transition function combines the topography function and persistence function (due to the flow inertia), and is advanced in its ability to avoid certain illogical lateral spreading due to abrupt changes in topography. In addition, in contrast to previous studies, in the present work, we regressed the persistence function from a well-documented flume experiment, rather than using a manipulated constant value as described in earlier empirical studies. Our results show that the debris-flow persistence function is closely related to the channel slope. It approximates the law of cosines at a steep slope and Gamma law at a gentle slope. To illustrate the performance of the improved CA model, we selected the 2010 Yohutagawa debris-flow event in Japan as a case study. Our results show that the simulated deposition perimeter pattern and run-out distance are in high accordance with the data from in situ investigation.

CATS – A process-based model for turbulent turbidite systems at the reservoir scale

There have been many attempts to understand complexity and to represent it in terms of computable quantities. To date, however, these attempts have had little success. Although we find complexity in a broad range of scientific domains, precise definitions escape our grasp (Bak, 1996; Morin, 2001; Prigogine & Stengers, 1984). One of the key models in complexity science is the Cellular Automaton (CA), a class of system in which small changes in the initial conditions or in local rules can provoke unpredictable behavior (Wolfram, 1984; Wolfram, 2002; Langton, 1986; 1990). The key issue, here as in other kinds of complex system, is to discover the rules governing the emergence of complex phenomena. If such rules were known we could use them to model and predict the behavior of complex physical and biological systems. Taking it for granted that complex behavior is the result of interactions among multiple components of a larger system; we can ask a number of fundamental questions.

Emergence of self-replicating structures in a cellular automata space

Cellular Automata (CA) is capable of (re)producing complex spatial-temporal dynamic patterns by using a set of local evolution rules involving the states of immediate neighbouring cells. Classical Cellular Automata consist of a regular lattice of cells, which lead to a geometric staircase representation at the boundaries. This paper developed unstructured cellular automata (UCA) with both unstructured triangular and polygon elements. In order to test the capabilities of the unstructured cellular automata, a Grass-vegetarian model and a macro-plankton model were developed based on UCA. The grass-vegetarian model combined UCA with optimization techniques. The macro-plankton model was set up by defining separate rules for every individual. The unstructured grid varied with growth and mortality of plants. The competition among plants evolved according to the UCA approach.