Monads and Quantitative Equational Theories for Nondeterminism and Probability

The monad of convex sets of probability distributions is a well-known tool for modelling the combination of nondeterministic and probabilistic computational effects. In this work we lift this monad from the category of sets to the category of extended metric spaces, by means of the Hausdorff and Kantorovich metric liftings. Our main result is the presentation of this lifted monad in terms of the quantitative equational theory of convex semilattices, using the framework of quantitative algebras recently introduced by Mardare, Panangaden and Plotkin

Faster algorithms for longest common substring

In the classic longest common substring (LCS) problem, we are given two strings S and T, each of length at most n, over an alphabet of size σ, and we are asked to find a longest string occurring as a fragment of both S and T. Weiner, in his seminal paper that introduced the suffix tree, presented an (n log σ)-time algorithm for this problem [SWAT 1973]. For polynomially-bounded integer alphabets, the linear-time construction of suffix trees by Farach yielded an (n)-time algorithm for the LCS problem [FOCS 1997]. However, for small alphabets, this is not necessarily optimal for the LCS problem in the word RAM model of computation, in which the strings can be stored in (n log σ/log n) space and read in (n log σ/log n) time. We show that, in this model, we can compute an LCS in time (n log σ / √{log n}), which is sublinear in n if σ = 2^{o(√{log n})} (in particular, if σ = (1)), using optimal space (n log σ/log n). We then lift our ideas to the problem of computing a k-mismatch LCS, which has received considerable attention in recent years. In this problem, the aim is to compute a longest substring of S that occurs in T with at most k mismatches. Flouri et al. showed how to compute a 1-mismatch LCS in (n log n) time [IPL 2015]. Thankachan et al. extended this result to computing a k-mismatch LCS in (n log^k n) time for k = (1) [J. Comput. Biol. 2016]. We show an (n log^{k-1/2} n)-time algorithm, for any constant integer k > 0 and irrespective of the alphabet size, using (n) space as the previous approaches. We thus notably break through the well-known n log^k n barrier, which stems from a recursive heavy-path decomposition technique that was first introduced in the seminal paper of Cole et al. [STOC 2004] for string indexing with k errors. </p

Faster Algorithms for Longest Common Substring

International audienceIn the classic longest common substring (LCS) problem, we are given two strings S and T , each of length at most n, over an alphabet of size σ, and we are asked to find a longest string occurring as a fragment of both S and T. Weiner, in his seminal paper that introduced the suffix tree, presented an O(n log σ)-time algorithm for this problem [SWAT 1973]. For polynomially-bounded integer alphabets, the linear-time construction of suffix trees by Farach yielded an O(n)-time algorithm for the LCS problem [FOCS 1997]. However, for small alphabets, this is not necessarily optimal for the LCS problem in the word RAM model of computation, in which the strings can be stored in O(n log σ/ log n) space and read in O(n log σ/ log n) time. We show that, in this model, we can compute an LCS in time O(n log σ/ √ log n), which is sublinear in n if σ = 2 o(√ log n) (in particular, if σ = O(1)), using optimal space O(n log σ/ log n). We then lift our ideas to the problem of computing a k-mismatch LCS, which has received considerable attention in recent years. In this problem, the aim is to compute a longest substring of S that occurs in T with at most k mismatches. Flouri et al. showed how to compute a 1-mismatch LCS in O(n log n) time [IPL 2015]. Thankachan et al. extended this result to computing a k-mismatch LCS in O(n log k n) time for k = O(1) [J. Comput. Biol. 2016]. We show an O(n log k−1/2 n)-time algorithm, for any constant k > 0 and irrespective of the alphabet size, using O(n) space as the previous approaches. We thus notably break through the well-known n log k n barrier, which stems from a recursive heavy-path decomposition technique that was first introduced in the seminal paper of Cole et al. [STOC 2004] for string indexing with k errors

Revisiting logical semantics for processes and their distances

Tesis inédita de la Universidad Complutense de Madrid, Facultad de Informática, Departamento de Sistemas Informáticos y Computación, leída el 2-02-2016Esta tesis se enmarca en el amplio campo de la teoría de la concurrencia. Más específicamente, nos centramos en el estudio de las relaciones de similitud entre procesos concurrentes. Comenzamos estudiando la bisimulación, considerada la más importante de estas relaciones, y vemos después cómo podemos extender nuestros resultados al resto de las semánticas de procesos estudiadas durante las últimas décadas. En particular, nuestra contribución a la comunidad científica, se centra en dos puntos principales: – El desarrollo de una caracterización lógica uniforme de las semánticas de procesos: proponemos un esquema lógico común (enmarcado en la conocida lógica modal de Hennessy-Milner) e incluimos las diferentes semánticas en este esquema, enfatizando las diferencias y similitudes entre ellas, que se presentan del modo más claro posible. – La presentación de una nueva noción de distancia, tanto entre procesos finitos como infinitos: la misma se diferencia de las anteriormente propuestas en su carácter global, que acumula las diferencias que aportan los distintos cómputos, en lugar de quedarnos con la máxima de ellas...This thesis can be included in the broad field of concurrency theory. More specifically, we focus on the study of the similarities between concurrent processes. We start from bisimulation, the main of these relations, and then we see how we can extend the obtained results to the rest of the semantics developed along the last years. In particular, our main contributions can be roughly described by the following two items: – The development of a unified logical characterization of process semantics: we propose a common logical scheme (within the framework of the well known Hennessy-Milner Logic) and we set the different semantics in this scheme by emphasizing, in the clearest possible way, the (dis)similarities between them. – We present a new notion of distance for both finite and infinite processes. This novel notion differs from the previously available ones in its global character: instead of taking the maximum disagreement between the two compared processes, it adds all the differences provided by their whole sets of computations...Depto. de Sistemas Informáticos y ComputaciónFac. de InformáticaTRUEunpu

Sobre la verificación automática de autómatas probabilistas distribuidos con información parcial /

Tesis (Doctor en Ciencias de la Computación)--Universidad Nacional de Córdoba, Facultad de Matemática, Astronomía y Física, 2010.En esta tesis desarrollamos algoritmos y técnicas de análisis basadas en model checking para analizar la corrección de sistemas distribuidos con características aleatorias y no deterministas. Una contribución importante es la demostración de que no existe un algoritmo que resuelva el problema de verificación de forma totalmente automática. A pesar de este resultado, presentamos algoritmos que, si bien no pueden determinar la corrección para todos los sistemas y propiedades, sirven para detectar que ciertos sistemas son correctos o incorrectos. Uno de los impedimentos más frecuentes a la hora de verificar PDMs es el problema de la explosión de estado. Este problema, bien conocido y atacado en model checking, se agrava en el ámbito de model checking cuantitativo. Existen trabajos previos que, con el fin de atacar este problema, presentan adaptaciones de las técnicas de reducción orden parcial para model checking cualitativo al caso cuantitativo. Presentamos una nueva adaptación de la técnica de reducción de orden parcial. Nuestra adaptación aprovecha el hecho de que las componentes de un sistema concurrente tienen acceso limitado a la información sobre el estado global del sistema. Concluímos con casos de estudio que muestran las mejoras de nuestros algoritmos y nuestra técnica de orden parcial.Sergio Giro ; dirigido por Pedro R. D'Argenio