LAD models, trees and an analog of the fundamental theorem of arithmetic

International audienceMotivated by applications of Logical Analysis Data (LAD) in medical contexts, original discrete optimization problems are proposed. When a patient arrives with a presumption of a disease, he is submitted to a sequence of tests. From one patient to another, the tests allowing to detect the disease may vary. A subset of tests whose results detect the disease in a given part of the population is called a pattern, which has its own prevalence in the population. If there is only a limited number of tests that can be done, which ones must be selected in order to maximize the number of spotted patients ? Or, if each test has a cost, in which order the tests have to be done, in order to minimize the cost ? It is the kind of questions that are investigated in this paper. For various special cases, polynomial algorithms are proposed, especially when the hypergraph whose vertices are the tests and whose edges are the patterns is a tree graph. One of these questions involves a criterion which is not a number but a sequence of numbers. The objective is then to find the best sequence for the lexicographic order. To solve this question, a new product on finite sequences is defined, namely the maximum shuffle product, which maps two sequences to their shuffle that is maximal for the lexicographic order. Surprisingly, this product leads to a theorem similar to the fundamental theorem of arithmetic: every sequence can be written uniquely as the product of prime sequences, with the suitable definition of prime sequences

Boundary Algebra: A Simpler Approach to Boolean Algebra and the Sentential Connectives

Boundary algebra [BA] is a algebra of type , and a simplified notation for Spencer-Brown’s (1969) primary algebra. The syntax of the primary arithmetic [PA] consists of two atoms, () and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting, indifferently, the presence or absence of () into a PA formula yields a BA formula. The BA axioms are A1: ()()= (), and A2: “(()) [abbreviated ‘⊥’] may be written or erased at will,” implying (⊥)=(). The repeated application of A1 and A2 simplifies any PA formula to either () or ⊥. The basis for BA is B1: abc=bca (concatenation commutes & associates); B2, ⊥a=a (BA has a lower bound, ⊥); B3, (a)a=() (BA is a complemented lattice); and B4, (ba)a=(b)a (implies that BA is a distributive lattice). BA has two intended models: (1) the Boolean algebra 2 with base set B={(),⊥}, such that () ⇔ 1 [dually 0], (a) ⇔ a′, and ab ⇔ a∪b [a∩b]; and (2) sentential logic, such that () ⇔ true [false], (a) ⇔ ~a, and ab ⇔ a∨b [a∧b]. BA is a self-dual notation, facilitates a calculational style of proof, and simplifies clausal reasoning and Quine’s truth value analysis. BA resembles C.S. Peirce’s graphical logic, the symbolic logics of Leibniz and W.E. Johnson, the 2 notation of Byrne (1946), and the Boolean term schemata of Quine (1982).Boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; G. Spencer-Brown; C.S. Peirce; existential graphs

Boundary Algebra: A Simple Notation for Boolean Algebra and the Truth Functors

Boundary algebra [BA] is a simpler notation for Spencer-Brown’s (1969) primary algebra [pa], the Boolean algebra 2, and the truth functors. The primary arithmetic [PA] consists of the atoms ‘()’ and the blank page, concatenation, and enclosure between ‘(‘ and ‘)’, denoting the primitive notion of distinction. Inserting letters denoting the presence or absence of () into a PA formula yields a BA formula. The BA axioms are "()()=()" (A1), and "(()) [=?] may be written or erased at will” (A2). Repeated application of these axioms to a PA formula yields a member of B= {(),?} called its simplification. (a) has two intended interpretations: (a) ? a? (Boolean algebra 2), and (a) ? ~a (sentential logic). BA is self-dual: () ? 1 [dually 0] so that B is the carrier for 2, ab ? a?b [a?b], and (a)b [(a(b))] ? a=b, so that ?=() [()=?] follows trivially and B is a poset. The BA basis abc= bca (Dilworth 1938), a(ab)= a(b), and a()=() (Bricken 2002) facilitates clausal reasoning and proof by calculation. BA also simplifies normal forms and Quine’s (1982) truth value analysis. () ? true [false] yields boundary logic.G. Spencer Brown; boundary algebra; boundary logic; primary algebra; primary arithmetic; Boolean algebra; calculation proof; C.S. Peirce; existential graphs.

Recent advances in the theory and practice of logical analysis of data

Logical Analysis of Data (LAD) is a data analysis methodology introduced by Peter L. Hammer in 1986. LAD distinguishes itself from other classification and machine learning methods by the fact that it analyzes a significant subset of combinations of variables to describe the positive or negative nature of an observation and uses combinatorial techniques to extract models defined in terms of patterns. In recent years, the methodology has tremendously advanced through numerous theoretical developments and practical applications. In the present paper, we review the methodology and its recent advances, describe novel applications in engineering, finance, health care, and algorithmic techniques for some stochastic optimization problems, and provide a comparative description of LAD with well-known classification methods

The Second NASA Formal Methods Workshop 1992

The primary goal of the workshop was to bring together formal methods researchers and aerospace industry engineers to investigate new opportunities for applying formal methods to aerospace problems. The first part of the workshop was tutorial in nature. The second part of the workshop explored the potential of formal methods to address current aerospace design and verification problems. The third part of the workshop involved on-line demonstrations of state-of-the-art formal verification tools. Also, a detailed survey was filled in by the attendees; the results of the survey are compiled

Space Programs Summary No. 37-51, Volume 2 for the Period March 1 to April 30, 1968. the Deep Space Network

Deep Space Network - mission support, advanced engineering, development and implementation, and operations and facilitie

Quantum computation beyond the circuit model

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2008.Includes bibliographical references (p. 133-144).The quantum circuit model is the most widely used model of quantum computation. It provides both a framework for formulating quantum algorithms and an architecture for the physical construction of quantum computers. However, several other models of quantum computation exist which provide useful alternative frameworks for both discovering new quantum algorithms and devising new physical implementations of quantum computers. In this thesis, I first present necessary background material for a general physics audience and discuss existing models of quantum computation. Then, I present three new results relating to various models of quantum computation: a scheme for improving the intrinsic fault tolerance of adiabatic quantum computers using quantum error detecting codes, a proof that a certain problem of estimating Jones polynomials is complete for the one clean qubit complexity class, and a generalization of perturbative gadgets which allows k-body interactions to be directly simulated using 2-body interactions. Lastly, I discuss general principles regarding quantum computation that I learned in the course of my research, and using these principles I propose directions for future research.by Stephen Paul Jordan.Ph.D

Robust GNSS Carrier Phase-based Position and Attitude Estimation Theory and Applications

Mención Internacional en el título de doctorNavigation information is an essential element for the functioning of robotic platforms and intelligent transportation systems. Among the existing technologies, Global Navigation Satellite Systems (GNSS) have established as the cornerstone for outdoor navigation, allowing for all-weather, all-time positioning and timing at a worldwide scale. GNSS is the generic term for referring to a constellation of satellites which transmit radio signals used primarily for ranging information. Therefore, the successful operation and deployment of prospective autonomous systems is subject to our capabilities to support GNSS in the provision of robust and precise navigational estimates. GNSS signals enable two types of ranging observations: –code pseudorange, which is a measure of the time difference between the signal’s emission and reception at the satellite and receiver, respectively, scaled by the speed of light; –carrier phase pseudorange, which measures the beat of the carrier signal and the number of accumulated full carrier cycles. While code pseudoranges provides an unambiguous measure of the distance between satellites and receiver, with a dm-level precision when disregarding atmospheric delays and clock offsets, carrier phase measurements present a much higher precision, at the cost of being ambiguous by an unknown number of integer cycles, commonly denoted as ambiguities. Thus, the maximum potential of GNSS, in terms of navigational precision, can be reach by the use of carrier phase observations which, in turn, lead to complicated estimation problems. This thesis deals with the estimation theory behind the provision of carrier phase-based precise navigation for vehicles traversing scenarios with harsh signal propagation conditions. Contributions to such a broad topic are made in three directions. First, the ultimate positioning performance is addressed, by proposing lower bounds on the signal processing realized at the receiver level and for the mixed real- and integer-valued problem related to carrier phase-based positioning. Second, multi-antenna configurations are considered for the computation of a vehicle’s orientation, introducing a new model for the joint position and attitude estimation problems and proposing new deterministic and recursive estimators based on Lie Theory. Finally, the framework of robust statistics is explored to propose new solutions to code- and carrier phase-based navigation, able to deal with outlying impulsive noises.La información de navegación es un elemental fundamental para el funcionamiento de sistemas de transporte inteligentes y plataformas robóticas. Entre las tecnologías existentes, los Sistemas Globales de Navegación por Satélite (GNSS) se han consolidado como la piedra angular para la navegación en exteriores, dando acceso a localización y sincronización temporal a una escala global, irrespectivamente de la condición meteorológica. GNSS es el término genérico que define una constelación de satélites que transmiten señales de radio, usadas primordinalmente para proporcionar información de distancia. Por lo tanto, la operatibilidad y funcionamiento de los futuros sistemas autónomos pende de nuestra capacidad para explotar GNSS y estimar soluciones de navegación robustas y precisas. Las señales GNSS permiten dos tipos de observaciones de alcance: –pseudorangos de código, que miden el tiempo transcurrido entre la emisión de las señales en los satélites y su acquisición en la tierra por parte de un receptor; –pseudorangos de fase de portadora, que miden la fase de la onda sinusoide que portan dichas señales y el número acumulado de ciclos completos. Los pseudorangos de código proporcionan una medida inequívoca de la distancia entre los satélites y el receptor, con una precisión de decímetros cuando no se tienen en cuenta los retrasos atmosféricos y los desfases del reloj. En contraposición, las observaciones de la portadora son super precisas, alcanzando el milímetro de exactidud, a expensas de ser ambiguas por un número entero y desconocido de ciclos. Por ende, el alcanzar la máxima precisión con GNSS queda condicionado al uso de las medidas de fase de la portadora, lo cual implica unos problemas de estimación de elevada complejidad. Esta tesis versa sobre la teoría de estimación relacionada con la provisión de navegación precisa basada en la fase de la portadora, especialmente para vehículos que transitan escenarios donde las señales no se propagan fácilmente, como es el caso de las ciudades. Para ello, primero se aborda la máxima efectividad del problema de localización, proponiendo cotas inferiores para el procesamiento de la señal en el receptor y para el problema de estimación mixto (es decir, cuando las incógnitas pertenecen al espacio de números reales y enteros). En segundo lugar, se consideran las configuraciones multiantena para el cálculo de la orientación de un vehículo, presentando un nuevo modelo para la estimación conjunta de posición y rumbo, y proponiendo estimadores deterministas y recursivos basados en la teoría de Lie. Por último, se explora el marco de la estadística robusta para proporcionar nuevas soluciones de navegación precisa, capaces de hacer frente a los ruidos atípicos.Programa de Doctorado en Ciencia y Tecnología Informática por la Universidad Carlos III de MadridPresidente: José Manuel Molina López.- Secretario: Giorgi Gabriele.- Vocal: Fabio Dovi

Application of network coding in satellite broadcast and multiple access channels

Satellite broadcasting and relaying capabilities enable mobile broadcast systems over wide geographical areas, which opens large market possibilities for handheld, vehicular and fixed user terminals. The geostationary (GEO) satellite orbit is highly suited for such applications, as it spares the need for satellite terminals to track the movement of the spacecraft, with important savings in terms of complexity and cost. The large radius of the GEO orbit (more than 40000 km) has two main drawbacks. One is the large free space loss experienced by a signal traveling to or from the satellite, which limits the signal-to-noise ratio (SNR) margins in the link budget with respect to terrestrial systems. The second drawback of the GEO orbit is the large propagation delay (about 250 msec) that limits the use of feedback in both the forward (satellite to satellite terminal) and the reverse (satellite terminal to satellite) link. The limited margin protection causes loss of service availability in environments where there is no direct line of sight to the satellite, such as urban areas. The large propagation delay on its turn, together with the large terminal population size usually served by a GEO satellite, limit the use of feedback, which is at the basis of error-control. In the reverse link, especially in the case of fixed terminals, packet losses are mainly due to collisions, that severely limit the access to satellite services in case a random access scheme is adopted. The need for improvements and further understanding of these setups lead to the development of our work. In this dissertation we study the application of network coding to counteract the above mentioned channel impairments in satellite systems. The idea of using network coding stems from the fact that it allows to efficiently exploit the diversity, either temporal or spatial, present in the system. In the following we outline the original contributions included in each of the chapters of the dissertation. Chapter 3. This chapter deals with channel impairments in the forward link, and specifically with the problem of missing coverage in Urban environments for land mobile satellite (LMS) networks. By applying the Max-flow Min-cut theorem we derive a lower bound on the maximum coverage that can be achieved through cooperation. Inspired by this result, we propose a practical scheme, keeping in mind the compatibility with the DVB-SH standard. We developed a simulator in Matlab/C++ based on the physical layer abstraction and used it to test the performance gain of our scheme with a benchmark relaying scheme that does allow coding at packet level. Chapter 4. The second chapter of contributions is devoted to the information theoretical study of real-time streaming transmissions over fading channels with channel state information at the transmitter only. We introduce this new channel model and propose several transmission schemes, one of which is proved to be asymptotically optimal in terms of throughput. We also provide an upper bound on the achievable throughput for the proposed channel model and compare it numerically with the proposed schemes over a Rayleigh fading channel. Chapter 5. Chapter 5 is devoted to the study of throughput and delay in non-real-time streaming transmission over block fading channels. We derive bounds on the throughput and the delay for this channel and propose different coding techniques based on time-sharing. For each of them we carry out an analytical study of the performance. Finally, we compare numerically the performance of the proposed schemes over a Rayleigh fading channel. Chapter 6. In the last technical chapter we propose a collision resolution method for the return link based on physical layer network coding over extended Galois field (EGF). The proposed scheme extracts information from the colliding signals and achieves important gains with respect to Slotted ALOHA systems as well as with respect to other collision resolution schemes.Una de les característiques mes importants de les plataformes de comunicacions per satèl.lit és la seva capacitat de retransmetre senyals rebuts a un gran número de terminals. Això es fonamental en contextes com la difusió a terminals mòbils o la comunicació entre màquines. Al mateix temps, la disponibilitat d’un canal de retorn permet la creació de sistemes de comunicacions per satèl.lit interactius que, en principi, poden arribar a qualsevol punt del planeta. Els satèl.lits Geoestacionaris son particularment adequats per a complir amb aquesta tasca. Aquest tipus de satèl.lits manté una posició fixa respecte a la Terra, estalviant als terminals terrestres la necessitat de seguir el seu moviment en el cel. Per altra banda, la gran distància que separa la Terra dels satèl.lits Geoestacionaris introdueix grans retrassos en les comunicacions que, afegit al gran número de terminals en servei, limita l’ús de tècniques de retransmissió basades en acknowledgments en cas de pèrdua de paquets. Per tal de sol.lucionar el problema de la pèrdua de paquets, les tècniques més utilitzades son el desplegament de repetidors terrestres, anomenats gap fillers, l’ús de codis de protecció a nivell de paquet i mecanismes proactius de resolució de col.lisions en el canal de retorn. En aquesta tesi s’analitzen i s’estudien sol.lucions a problemes en la comunicació per satèl.lit tant en el canal de baixada com el de pujada. En concret, es consideren tres escenaris diferents. El primer escenari es la transmissió a grans poblacions de terminals mòbils en enorns urbans, que es veuen particularment afectats per la pèrdua de paquets degut a l’obstrucció, per part dels edificis, de la línia de visió amb el satèl.lit. La sol.lució que considerem consisteix en la utilització de la cooperació entre terminals. Una vegada obtinguda una mesura del guany que es pot assolir mitjançant cooperació en un model bàsic de xarxa, a través del teorema Max-flow Min-cut, proposem un esquema de cooperació compatible amb estàndards de comunicació existents. El segon escenari que considerem es la transmissió de vídeo, un tipus de tràfic particularment sensible a la pèrdua de paquets i retards endògens als sistemes de comunicació per satèl.lit. Considerem els casos de transmissió en temps real i en diferit, des de la perspectiva de teoria de la informació, i estudiem diferents tècniques de codificació analítica i numèrica. Un dels resultats principals obtinguts es l’extensió del límit assolible de la capacitat ergòdica del canal en cas que el transmissor rebi les dades de manera gradual, enlloc de rebre-les totes a l’inici de la transmissió. El tercer escenari que considerem es l’accés aleatori al satèl.lit. Desenvolupem un esquema de recuperació dels paquets perduts basat en la codificació de xarxa a nivell físic i en extensions a camps de Galois, amb resultats molt prometedors en termes de rendiment. També estudiem aspectes relacionats amb la implementació pràctica d’aquest esquema