On-Line Monitoring for Temporal Logic Robustness

In this paper, we provide a Dynamic Programming algorithm for on-line monitoring of the state robustness of Metric Temporal Logic specifications with past time operators. We compute the robustness of MTL with unbounded past and bounded future temporal operators MTL over sampled traces of Cyber-Physical Systems. We implemented our tool in Matlab as a Simulink block that can be used in any Simulink model. We experimentally demonstrate that the overhead of the MTL robustness monitoring is acceptable for certain classes of practical specifications

Metric 1-spaces

A generalization of metric space is presented which is shown to admit a theory strongly related to that of ordinary metric spaces. To avoid the topological effects related to dropping any of the axioms of metric space, first a new, and equivalent, axiomatization of metric space is given which is then generalized from a fresh point of view. Naturally arising examples from metric geometry are presented

Fast, Linear Time, m-Adic Hierarchical Clustering for Search and Retrieval using the Baire Metric, with linkages to Generalized Ultrametrics, Hashing, Formal Concept Analysis, and Precision of Data Measurement

We describe many vantage points on the Baire metric and its use in clustering data, or its use in preprocessing and structuring data in order to support search and retrieval operations. In some cases, we proceed directly to clusters and do not directly determine the distances. We show how a hierarchical clustering can be read directly from one pass through the data. We offer insights also on practical implications of precision of data measurement. As a mechanism for treating multidimensional data, including very high dimensional data, we use random projections.Comment: 17 pages, 45 citations, 2 figure

Ultrametric and Generalized Ultrametric in Computational Logic and in Data Analysis

Following a review of metric, ultrametric and generalized ultrametric, we review their application in data analysis. We show how they allow us to explore both geometry and topology of information, starting with measured data. Some themes are then developed based on the use of metric, ultrametric and generalized ultrametric in logic. In particular we study approximation chains in an ultrametric or generalized ultrametric context. Our aim in this work is to extend the scope of data analysis by facilitating reasoning based on the data analysis; and to show how quantitative and qualitative data analysis can be incorporated into logic programming.Comment: 19 pp., 5 figures, 3 table

Fast, Linear Time Hierarchical Clustering using the Baire Metric

The Baire metric induces an ultrametric on a dataset and is of linear computational complexity, contrasted with the standard quadratic time agglomerative hierarchical clustering algorithm. In this work we evaluate empirically this new approach to hierarchical clustering. We compare hierarchical clustering based on the Baire metric with (i) agglomerative hierarchical clustering, in terms of algorithm properties; (ii) generalized ultrametrics, in terms of definition; and (iii) fast clustering through k-means partititioning, in terms of quality of results. For the latter, we carry out an in depth astronomical study. We apply the Baire distance to spectrometric and photometric redshifts from the Sloan Digital Sky Survey using, in this work, about half a million astronomical objects. We want to know how well the (more costly to determine) spectrometric redshifts can predict the (more easily obtained) photometric redshifts, i.e. we seek to regress the spectrometric on the photometric redshifts, and we use clusterwise regression for this.Comment: 27 pages, 6 tables, 10 figure

fixed point

The study of the dual complexity space, introduced by S. Romaguera and M. P. Schellekens [Quasi-metric properties of complexity spaces, Topol. Appl. 98 (1999), pp. 311-322], constitutes a part of the interdisciplinary research on Computer Science and Topology. The relevance of this theory is given by the fact that it allows one to apply fixed point techniques of denotational semantics to complexity analysis. Motivated by this fact and with the intention of obtaining a mixed framework valid for both disciplines, a new complexity space formed by partial functions was recently introduced and studied by S. Romaguera and O. Valero [On the structure of the space of complexity partial functions, Int. J. Comput. Math. 85 (2008), pp. 631-640]. An application of the complexity space of partial functions to model certain processes that arise, in a natural way, in symbolic computation was given in the aforementioned reference. In this paper, we enter more deeply into the relationship between semantics and complexity analysis of programs. We construct an extension of the complexity space of partial functions and show that it is, at the same time, an appropriate mathematical tool for the complexity analysis of algorithms and for the validation of recursive definitions of programs. As applications of our complexity framework, we show the correctness of the denotational specification of the factorial function and give an alternative formal proof of the asymptotic upper bound for the average case analysis of Quicksort.The first and the third authors acknowledge the support of the Spanish Ministry of Science and Innovation, and FEDER, grant MTM2009-12872-C02-01 (subprogram MTM), and the support of Generalitat Valenciana, grant ACOMP2009/005. The second author acknowledges the support of the Science Foundation Ireland, SFI Principal Investigator Grant 07/IN.1/I977.Romaguera Bonilla, S.; Schellekens, M.; Valero Sierra, Ó. (2011). The complexity space of partial functions: A connection between Complexity Analysis and Denotational Semantics. International Journal of Computer Mathematics. 88(9):1819-1829. https://doi.org/10.1080/00207161003631885S18191829889De Bakker, J. W., & de Vink, E. P. (1998). Denotational models for programming languages: applications of Banach’s Fixed Point Theorem. Topology and its Applications, 85(1-3), 35-52. doi:10.1016/s0166-8641(97)00140-5Emerson, E. A., & Jutla, C. S. (1999). The Complexity of Tree Automata and Logics of Programs. SIAM Journal on Computing, 29(1), 132-158. doi:10.1137/s0097539793304741Flajolet, P., & Golin, M. (1994). Mellin transforms and asymptotics. Acta Informatica, 31(7), 673-696. doi:10.1007/bf01177551García-Raffi, L. M., Romaguera, S., & Sánchez-Pérez, E. A. (2002). Sequence spaces and asymmetric norms in the theory of computational complexity. Mathematical and Computer Modelling, 36(1-2), 1-11. doi:10.1016/s0895-7177(02)00100-0García-Raffi, L. M., Romaguera, S., & Sánchez-Pérez, E. A. (2003). The supremum asymmetric norm on sequence algebras. Electronic Notes in Theoretical Computer Science, 74, 39-50. doi:10.1016/s1571-0661(04)80764-3García-Raffi, L. M., Romaguera, S., Sánchez-Pérez, E. A. and Valero, O. Normed Semialgebras: A Mathematical Model for the Complexity Analysis of Programs and Algorithms. Proceedings of The 7th World Multiconference on Systemics, Cybernetics and Informatics (SCI 2003), Orlando, Florida, USA. Edited by: Callaos, N., Di Sciullo, A. M., Ohta, T. and Liu, T.K. Vol. II, pp.55–58. Orlando, FL: International Institute of Informatics and Systemics.Den Hartog, J. I., de Vink, E. P., & de Bakker, J. W. (2001). Metric Semantics and Full Abstractness for Action Refinement and Probabilistic Choice. Electronic Notes in Theoretical Computer Science, 40, 72-99. doi:10.1016/s1571-0661(05)80038-6Künzi, H.-P. A. (2001). Nonsymmetric Distances and Their Associated Topologies: About the Origins of Basic Ideas in the Area of Asymmetric Topology. History of Topology, 853-968. doi:10.1007/978-94-017-0470-0_3Medina, J., Ojeda-Aciego, M. and Ruiz-Calviño, J. A fixed point theorem for multi-valued functions with an application to multilattice-based logic programming. Applications of Fuzzy Sets Theory: 7th International Workshop on Fuzzy Logic and Applications, WILF 2007, Camogli, Italy, July 7–10, 2007, Proceedings. Edited by: Masulli, F., Mitra, S. and Pasi, G. Vol. 4578, pp.37–44. Berlin: Springer-Verlag. Notes in Artificial IntelligenceO’Keeffe, M., Romaguera, S., & Schellekens, M. (2003). Norm-weightable Riesz Spaces and the Dual Complexity Space. Electronic Notes in Theoretical Computer Science, 74, 105-121. doi:10.1016/s1571-0661(04)80769-2Rodríguez-López, J., Romaguera, S., & Valero, O. (2004). Asymptotic Complexity of Algorithms via the Nonsymmetric Hausdorff Distance. Computing Letters, 2(3), 155-161. doi:10.1163/157404006778330816Rodríguez-López, J., Romaguera, S., & Valero, O. (2008). Denotational semantics for programming languages, balanced quasi-metrics and fixed points. International Journal of Computer Mathematics, 85(3-4), 623-630. doi:10.1080/00207160701210653Romaguera, S., & Schellekens, M. (1999). Quasi-metric properties of complexity spaces. Topology and its Applications, 98(1-3), 311-322. doi:10.1016/s0166-8641(98)00102-3Romaguera, S., & Schellekens, M. (2000). The quasi-metric of complexity convergence. Quaestiones Mathematicae, 23(3), 359-374. doi:10.2989/16073600009485983Romaguera, S., & Schellekens, M. P. (2002). Duality and quasi-normability for complexity spaces. Applied General Topology, 3(1), 91. doi:10.4995/agt.2002.2116Romaguera, S., & Valero, O. (2008). On the structure of the space of complexity partial functions. International Journal of Computer Mathematics, 85(3-4), 631-640. doi:10.1080/00207160701210117Romaguera, S., Sánchez-Pérez, E. A., & Valero, O. (2003). The complexity space of a valued linearly ordered set. Electronic Notes in Theoretical Computer Science, 74, 158-171. doi:10.1016/s1571-0661(04)80772-2Schellekens, M. (1995). The Smyth Completion. Electronic Notes in Theoretical Computer Science, 1, 535-556. doi:10.1016/s1571-0661(04)00029-5Schellekens, M. 1995. “The smyth completion: A common topological foundation for denotational semantics and complexity analysis”. Pittsburgh: Carnegie Mellon University. Ph.D. thesisSeda, A. K., & Hitzler, P. (2008). Generalized Distance Functions in the Theory of Computation. The Computer Journal, 53(4), 443-464. doi:10.1093/comjnl/bxm108Straccia, U., Ojeda-Aciego, M., & Damásio, C. V. (2009). On Fixed-Points of Multivalued Functions on Complete Lattices and Their Application to Generalized Logic Programs. SIAM Journal on Computing, 38(5), 1881-1911. doi:10.1137/070695976Tennent, R. D. (1976). The denotational semantics of programming languages. Communications of the ACM, 19(8), 437-453. doi:10.1145/360303.360308Tix, R., Keimel, K., & Plotkin, G. (2005). RETRACTED: Semantic Domains for Combining Probability and Non-Determinism. Electronic Notes in Theoretical Computer Science, 129, 1-104. doi:10.1016/j.entcs.2004.06.06

From Formal Requirement Analysis to Testing and Monitoring of Cyber-Physical Systems

abstract: Cyber-Physical Systems (CPS) are being used in many safety-critical applications. Due to the important role in virtually every aspect of human life, it is crucial to make sure that a CPS works properly before its deployment. However, formal verification of CPS is a computationally hard problem. Therefore, lightweight verification methods such as testing and monitoring of the CPS are considered in the industry. The formal representation of the CPS requirements is a challenging task. In addition, checking the system outputs with respect to requirements is a computationally complex problem. In this dissertation, these problems for the verification of CPS are addressed. The first method provides a formal requirement analysis framework which can find logical issues in the requirements and help engineers to correct the requirements. Also, a method is provided to detect tests which vacuously satisfy the requirement because of the requirement structure. This method is used to improve the test generation framework for CPS. Finally, two runtime verification algorithms are developed for off-line/on-line monitoring with respect to real-time requirements. These monitoring algorithms are computationally efficient, and they can be used in practical applications for monitoring CPS with low runtime overhead.Dissertation/ThesisDoctoral Dissertation Computer Science 201

Semi-lipschitz functions, best approximation, and fuzzy quasi-metric hyperspaces

En los últimos años se ha desarrollado una teoría matemática que permite generalizar algunas teorías matemáticas clásicas: hiperespacios, espacios de funciones, topología algebraica, etc. Este hecho viene motivado, en parte, por ciertos problemas de análisis funcional, concentración de medidas, sistemas dinámicos, teoría de las ciencias de la computación, matemática económica, etc. Esta tesis doctoral está dedicada al estudio de algunas de estas generalizaciones desde un punto de vista no simétrico. En la primera parte, estudiamos el conjunto de funciones semi-Lipschitz; mostramos que este conjunto admite una estructura de cono normado. Estudiaremos diversos tipos de completitud (bicompletitud, right k-completitud, D-completitud, etc), y también analizaremos cuando la casi-distancia correspondiente es balanceada. Además presentamos un modelo adecuado para el computo de la complejidad de ciertos algoritmos mediante el uso de normas relativas. Esto se consigue seleccionando un espacio de funciones semi-Lipschitz apropiado. Por otra parte, mostraremos que estos espacios proporcionan un contexto adecuado en el que caracterizar los puntos de mejor aproximación en espacios casi-métricos. El hecho de que varias hipertopologías hayan sido aplicadas con éxito en diversas áreas de Ciencias de la Computación ha contribuido a un considerable aumento del interés en el estudio de los hiperespacios desde un punto de vista no simétrico. Así, en la segunda parte de la tesis, estudiamos algunas condiciones de mejor aproximación en el contexto de hiperespacios casi-métricos. Por otro lado, caracterizamos la completitud de un espacio uniforme usando la completitud de Sieber-Pervin, la de Smyth y la D-completitud de su casi-uniformidad superior de Hausdorff-Bourbaki, definida en los subconjuntos compactos no vacíos. Finalmente, introducimos dos nociones de hiperespacio casi-métrico fuzzy.Sánchez Álvarez, JM. (2009). Semi-lipschitz functions, best approximation, and fuzzy quasi-metric hyperspaces [Tesis doctoral no publicada]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/5769Palanci