Superposable Trellis Automata

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.

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.

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.

We study a simple triangular partitioned cellular automaton (TPCA), and clarify its complex behavior. It is a CA with triangular cells, each of which is divided into three parts. The next state of a cell is determined by the three adjacent parts of its neighbor cells. This framework makes it easy to design reversible triangular CAs. Among them, isotropic and eight-state (i.e., each part has only two states) TPCAs are called elementary TPCAs (ETPCAs). They are extremely simple, since each of their local transition functions is described by only four local rules. In this paper, we investigate a specific reversible ETPCA $$T_{0347}$$ , where 0347 is its identification number in the class of 256 ETPCAs. In spite of the simplicity of the local function and the constraint of reversibility, evolutions of configurations in $$T_{0347}$$ have very rich varieties. It is shown that a glider, which is a space-moving pattern, and glider guns exist in this cellular space We also show that the trajectory and the timing of a glider can be fully controlled by appropriately placing stable patterns called blocks. Furthermore, using gliders to represent signals, we can implement universal reversible logic gates in it. By this, computational universality of $$T_{0347}$$ is derived.

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.

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

В наших предыдущих работах мы продемонстрировали, что теория конечных подмножеств различных ассоциативных алгебр позволяет интерпретировать элементарную арифметику, в частности, она неразрешима. Например, это было показано для любых бесконечных абелевых групп. Возникает естественный вопрос: можно ли обобщить этот результат на более широкий класс алгебр, скажем, все коммутативные моноиды. В некоторых случаях нами ответ тоже получен ранее: для коммутативных моноидов с сокращением, имеющим элемент бесконечного порядка или произвольных абелевых групп. Сейчас же мы продемонстрируем, что не для всяких коммутативных моноидов это верно. Более того, мы дадим описание конструкции, которая позволяет строить такого рода системы из разного рода исходных алгебр. Вместе с тем, будут указаны и некоторые границы её применимости. In our previous works, we have proved for various associative algebras that the finite subsets theory allows to interpret elementary arithmetic, in particular, such theory is undecidable. For example, this is proved for all infinite Abelian groups. A natural question arises: can we generalize this result to a wider class of algebras, for example, all commutative monoids. In some cases, we also have proved analogous result: for commutative cancellative monoids with an element of infinite order, or arbitrary Abelian groups. In this paper we prove that this is not true for arbitrary commutative monoids. Moreover, we propose a method that allows to construct such algebras by various original algebras. Also, we have found a limitation of this method.

Divisibility and cellular automata

Picking binomial coefficients which cannot be divided by a given prime from Pascal's triangle, we find that they form a set with self-similarity. Essouabri studied on a class of meromorphic functions associated to the above set. These functions are related to fractal geometry and it is a problem whether such a function has a non-real pole on its axis of absolute convergence. Essouabri gave a proof of existence of such a non-real pole in the simplest case. The keys of his proof are Stein's and Wilson's estimates on how fast the points multiply in Pascal's triangle modulo a prime. This article will give an extension of Essouabri's result to some cases with certain ways to count the points in Pascal's triangle modulo a prime which are different from the traditional one.

Counting: Basic Subsets of a Set Pascal's Triangle Binomial Coefficient Identities Counting: Intermediate Finding a Polynomial The Upward-Extended Pascal's Triangle Recurrence Relations and Fibonacci Numbers Counting: Advanced Generating Functions and Making Change Integer Triangles Rook Paths and Queen Paths Discrete Probability Probability Spaces and Distributions Markov Chains Random Tournaments Number Theory Divisibility of Factorials and Binomial Coefficients Covering Systems Partitions of an Integer Information Theory What Is Surprise? A Coin-Tossing Game Shannon's Theorems Games A Little Graph Theory Background The Ramsey Game Tic-Tac-Toe and Animal Games Algorithms Counters Listing Permutations and Combinations Sudoku Solving and Polycube Packing Appendix A: Hints and Solutions to Exercises Appendix B: Notation Bibliography Index

I read with interest Thomas Byrne's article: "An Extension of Pascal's Triangle and Work concerning Binomial Expansion Coefficients" in the February 1990 issue. On page 28 Mr. Byrne states: "I was seeking a mechanism to generate multinomial expansion coefficients but this doesn't do it . . . Maybe a three dimensional extension will prove useful."

Mass spectrometry (MS) is widely used for isotopic studies of metabolism in which detailed information about biochemical processes is obtained from the analysis of isotope incorporation into metabolites. The biological value of such experiments is dependent on the accuracy of the isotopic measurements. Using MS, isotopologue distributions are measured from the quantitative analysis of isotopic clusters. These measurements are prone to various biases, which can occur during the experimental workflow and/or MS analysis. The lack of relevant standards limits investigations of the quality of the measured isotopologue distributions. To meet that need, we developed a complete theoretical and experimental framework for the biological production of metabolites with fully controlled and predictable labeling patterns. This strategy is valid for different isotopes and different types of metabolisms and organisms, and was applied to two model microorganisms, Pichia augusta and Escherichia coli, cultivated on (13)C-labeled methanol and acetate as sole carbon source, respectively. The isotopic composition of the substrates was designed to obtain samples in which the isotopologue distribution of all the metabolites should give the binomial coefficients found in Pascal's triangle. The strategy was validated on a liquid chromatography-tandem mass spectrometry (LC-MS/MS) platform by quantifying the complete isotopologue distributions of different intracellular metabolites, which were in close agreement with predictions. This strategy can be used to evaluate entire experimental workflows (from sampling to data processing) or different analytical platforms in the context of isotope labeling experiments.

The Pascal triangle is a geometric representation of binomial coefficients in triangular form. We utilize this formalism to deterministically arrange silver nanocylinders of different sizes (30, 60, and 90 nm) on a triangle and numerically study their near-field optical properties. We show that near-field intensities at specific points on this triangle depend on the wavelength and angle of incidence. From the wavelength-dependent studies at various junctions of nanocylinders, we obtain maximum near-field intensity at 350 and 380 nm. By varying the angle of incidence of the TM-polarized plane wave, we find systematic variation in the near-field intensity at different junctions of the geometry. Our study will lead to insights in designing controllable electromagnetic hot spots for chip-based plasmonic devices.

Generalized Pascal triangle for binomial coefficients of words