Finite Set Of Patterns Research Articles

Canonical projection tilings defined by patterns

We give a necessary and sufficient condition on a $d$-dimensional affine subspace of $\mathbb{R}^n$ to be characterized by a finite set of patterns which are forbidden to appear in its digitization. This can also be stated in terms of local rules for canonical projection tilings, or subshift of finite type. This provides a link between algebraic properties of affine subspaces and combinatorics of their digitizations. The condition relies on the notion of {\em coincidence} and can be effectively checked. As a corollary, we get that only algebraic subspaces can be characterized by patterns.

On the upper and lower quantization coefficient for probability measures on multiscale Moran sets

Given a finite set of patterns, we consider the Moran sets determined by using each of these patterns with a prescribed frequency. For certain infinite product measures μ on such Moran sets, we determine the exact values of the quantization dimensions Dr(μ). We give various sufficient conditions for the Dr(μ)-dimensional upper quantization coefficient and the lower one to be positive and finite. We also construct an example to illustrate our main result.

Geometrical Kernel Machine for Prediction and Novelty Detection of Disruptive Events in TOKAMAK Machines

This paper presents a recently introduced Kernel Machine, called Geometrical Kernel Machine, used to predict disruptive events in nuclear fusion reactors. The algorithm proposed to construct the Kernel Machine is able to automatically determine both the number of neurons and the synaptic weights of a Multilayer Perceptron neural network with a single hidden layer. It has been demonstrated that the resulting network is able to classify any finite set of patterns defined in a real domain. The prediction problem has been here modeled as a two classes classification problem. The geometrical interpretation of the network equations allows us both to develop the disruption predictor and to manage the so called ageing of the kernel machine. In fact, using the same kernel machine, a novelty detection system has been integrated in the predictor, increasing the overall system performance, and the reliability of the predictor.

Complexity of Clausal Constraints Over Chains

We investigate the complexity of the satisfiability problem of constraints over finite totally ordered domains. In our context, a clausal constraint is a disjunction of inequalities of the form x≥d and x≤d. We classify the complexity of constraints based on clausal patterns. A pattern abstracts away from variables and contains only information about the domain elements and the type of inequalities occurring in a constraint. Every finite set of patterns gives rise to a (clausal) constraint satisfaction problem in which all constraints in instances must have an allowed pattern. We prove that every such problem is either polynomially decidable or NP-complete, and give a polynomial-time algorithm for recognizing the tractable cases. Some of these tractable cases are new and have not been previously identified in the literature.

Higher-order pattern complement and the strict λ-calculus

We address the problem of complementing higher-order patterns without repetitions of existential variables. Differently from the first-order case, the complement of a pattern cannot, in general, be described by a pattern, or even by a finite set of patterns. We therefore generalize the simply-typed λ-calculus to include an internal notion of strict function so that we can directly express that a term must depend on a given variable. We show that, in this more expressive calculus, finite sets of patterns without repeated variables are closed under complement and intersection. Our principal application is the transformational approach to negation in higher-order logic programs.

High-order extensions of the Double Chain Markov Model

The Double Chain Markov Model is a fully Markovian model for the representation of time-series in random environments. In this article, we show that it can handle transitions of high-order between both a set of observations and a set of hidden states. In order to reduce the number of parameters, each transition matrix can be replaced by a Mixture Transition Distribution model. We provide a complete derivation of the algorithms needed to compute the model. Three applications, the analysis of a sequence of DNA, the song of the wood pewee, and the behavior of young monkeys show that this model is of great interest for the representation of data that can be decomposed into a finite set of patterns.

The Occurrence of Sequence Patterns in Repeated Dependent Experiments

A sequence of Markov dependent experiments is performed, each one of them producing a letter from a given alphabet. We study the following question: How many trials does it take to observe a member of a specified finite set of patterns of letters for the first time?

String overlaps, pattern matching, and nontransitive games

This paper studies several topics concerning the way strings can overlap. The key notion of the correlation of two strings is introduced, which is a representation of how the second string can overlap into the first. This notion is then used to state and prove a formula for the generating function that enumerates the q-ary strings of length n which contain none of a given finite set of patterns. Various generalizations of this basic result are also discussed. This formula is next used to study a wide variety of seemingly unrelated problems. The first application is to the nontransitive dominance relations arising out of a probabilistic coin-tossing game. Another application shows that no algorithm can check for the presence of a given pattern in a text without examining essentially all characters of the text in the worst case. Finally, a class of polynomials arising in connection with the main result are shown to be irreducible.