Generalized Distance Functions in the Theory of Computation

Seda, Anthony Karel; Hitzler, Pascal

doi:10.1093/comjnl/bxm108

Cited by 42 publications

(18 citation statements)

References 72 publications

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“…Interesting metrics can also be defined in computation theory with a number of diverse applications [Seda and Hitzler 2008]. In either case, the interest in defining metrics is usually to show that a function is contractive (and, thus, to prove some notion of stability [Sontag 1998] or utilize a fixed-point computation [Seda and Hitzler 2008]) or that we can define an interesting topology [Kopperman 1988].…”

Section: Metrics and Distancesmentioning

confidence: 99%

“…Such topological information is vital in our case as we will demonstrate in Section 2.3. Next, we briefly review the notion of generalized metrics and we refer the reader to [Seda and Hitzler 2008] and the references therein for a more detailed exposition. -A semigroup (V, +) is a set V together with a binary operation + such that (i) the set is closed under + and (ii) + is associative.…”

Section: Metrics and Distancesmentioning

confidence: 99%

“…In our implementation, we are using the Floyd-Warshall algorithm which has running time Θ(|L| 3 ). In order to reason over output trajectories y in the hybrid state space Y , we need to introduce a generalized distance function [Seda and Hitzler 2008]. In the following, we will denote the hybrid space L×Z by H to indicate that a metric is defined over a particular space.…”

Section: Generalized Quasi-metrics For Hybrid Signalsmentioning

confidence: 99%

See 2 more Smart Citations

Probabilistic Temporal Logic Falsification of Cyber-Physical Systems

Abbas

Fainekos

Sankaranarayanan

et al. 2013

ACM Trans. Embed. Comput. Syst.

140

190

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We present a Monte-Carlo optimization technique for finding system behaviors that falsify a Metric Temporal Logic (MTL) property. Our approach performs a random walk over the space of system inputs guided by a robustness metric defined by the MTL property. Robustness is guiding the search for a falsifying behavior by exploring trajectories with smaller robustness values. The resulting testing framework can be applied to a wide class of Cyber-Physical Systems (CPS). We show through experiments on complex system models that using our framework can help automatically falsify properties with more consistency as compared to other means such as uniform sampling.

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Section: Metrics and Distancesmentioning

confidence: 99%

Section: Metrics and Distancesmentioning

confidence: 99%

Section: Generalized Quasi-metrics For Hybrid Signalsmentioning

confidence: 99%

See 1 more Smart Citation

Probabilistic Temporal Logic Falsification of Cyber-Physical Systems

Abbas

Fainekos

Sankaranarayanan

et al. 2013

ACM Trans. Embed. Comput. Syst.

140

190

View full text Add to dashboard Cite

show abstract

“…They show that Random Forests outperform a single CART classifier and perform on par with other ensemble methods like bagging and boosting. On a related remote sensing application, Pal [67] investigated the use of Random Forests for classification tasks and compared their performance with SVM. Pal showed that Random Forests perform equally well to SVM in terms of classification accuracy and training time.…”

Section: Random Forestsmentioning

confidence: 99%

“…Comprehensive background on ordered sets and lattices can be found in [15]. A review of generalized distances and ultrametrics is in [67].…”

Section: The Generalized Ultrametric and Formal Concept Analysismentioning

confidence: 99%

Clusters, Orders, and Trees: Methods and Applications

Aleskerov¹,

Goldengorin²,

Pardalos³

2014

Springer Optimization and Its Applications

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Aims and ScopeOptimization has been expanding in all directions at an astonishing rate during the last few decades. New algorithmic and theoretical techniques have been developed, the diffusion into other disciplines has proceeded at a rapid pace, and our knowledge of all aspects of the field has grown even more profound. At the same time, one of the most striking trends in optimization is the constantly increasing emphasis on the interdisciplinary nature of the field. Optimization has been a basic tool in all areas of applied mathematics, engineering, medicine, economics, and other sciences.The series Springer Optimization and Its Applications publishes undergraduate and graduate textbooks, monographs and state-of-the-art expository work that focus on algorithms for solving optimization problems and also study applications involving such problems. Some of the topics covered include nonlinear optimization (convex and nonconvex), network flow problems, stochastic optimization, optimal control, discrete optimization, multi-objective programming, description of software packages, approximation techniques and heuristic approaches.

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A General Approach to Midpoint Theory and Aggregation of Quasimetrics

Massanet

Valero

2013

Int. J. Intell. Syst.

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Many fields in applied sciences, like Artificial Intelligence and Computer Science, use aggregation methods to provide new generalized metrics from a collection of old ones. Thus, the problem of merging by means of a function a collection of generalized metrics into a single one has been recently studied in depth. Moreover, the midpoint sets for a generalized metric involving fuzzy sets have shown a great potential in medical diagnosis and decision making since it models the concept of "compromise" or "middle way" between two positions. Joining these facts, the aim of this paper is to provide a general framework for the study of midpoint sets for quasimetrics via aggregation theory. In particular, we determine the properties that an aggregation function must satisfy to characterize the midpoint set for a quasimetric generated by means of the fusion of a collection of quasimetrics in terms of the midpoint sets for each of the quasimetrics that are merged. In fact, this study generalizes the results for metrics in this context that are retrieved as a particular case of the exposed theory. Finally, some particular results for generalized metrics defined for fuzzy sets are proved. C 2013 Wiley Periodicals, Inc. MASSANET AND VALEROFrom the interest of information aggregation in the applied sciences, for instance Bioinformatics, Medical Diagnosis, and Decision Making, the notion of midpoint set for a metric to model the concept of "middle way" or "compromise" between two given positions emerges in Mathematics. In 1991, Herburt and Moszyńska provided a general description of the shape of midpoint sets for those metrics that are obtained via the aggregation, in the spirit of Borsík and Doboš, of a finite collection of another ones (see Ref. 6). Concretely, they showed that the midpoint set for a metric induced by aggregation can be expressed as the Cartesian product of the midpoint sets for each of the metrics that are merged. Later, in 2003, the notion of midpoint set was applied, through fuzzy sets, to Medicine by Nieto and Torres in Ref. 11. Concretely, they showed that in practical medicine given two descriptions of a patient, which are represented as fuzzy sets and provided by several expert raters, the new suitable average representation of the patient, as working description, does not reduce, in general, to the Euclidean midpoint. In fact, this working representation can be associated with a wide range of "middle ways" between the original provided patient descriptions (for a fuller treatment of data clustering in Medicine we refer the reader to Ref. 12). Inspired by the fact that the working representation does not concur with the Euclidean midpoint, and thus by the uselessness of Euclidean metric, Nieto and Torres studied, in the aforesaid reference, the shape of midpoint sets for the well-known Hamming metric defined between fuzzy sets. More recently, following the original work of Nieto and Torres, Casasnovas and Rosselló gave an explicit description of the midpoint sets, among others, for the weighted Hamming...

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Generalized Distance Functions in the Theory of Computation

Cited by 42 publications

References 72 publications

Probabilistic Temporal Logic Falsification of Cyber-Physical Systems

Probabilistic Temporal Logic Falsification of Cyber-Physical Systems

Clusters, Orders, and Trees: Methods and Applications

A General Approach to Midpoint Theory and Aggregation of Quasimetrics

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