From Intervals to –? Towards a General Description of Validated Uncertainty

This book is about methodological aspects of uncertainty propagation in data processing. Uncertainty propagation is an important problem: while computer algorithms efficiently process data related to many aspects of their lives, most of these algorithms implicitly assume that the numbers they process are exact. In reality, these numbers come from measurements, and measurements are never 100% exact. Because of this, it makes no sense to translate 61 kg into pounds and get the result—as computers do—with 13 digit accuracy.

In many cases—e.g., in celestial mechanics—the state of a system can be described by a few numbers: the values of the corresponding physical quantities. In such cases, for each of these quantities, we know (at least) the upper bound on the measurement error. This bound is either provided by the manufacturer of the measuring instrument—or is estimated by the user who calibrates this instrument. However, in many other cases, the description of the system is more complex than a few numbers: we need a function to describe a physical field (e.g., electromagnetic field); we need a vector in Hilbert space to describe a quantum state; we need a pseudo-Riemannian space to describe the physical space-time, etc.

To describe and process uncertainty in all such cases, this book proposes a general methodology—a methodology that includes intervals as a particular case. The book is recommended to students and researchers interested in challenging aspects of uncertainty analysis and to practitioners who need to handle uncertainty in such unusual situations.

Motivation and Outline.- A General Description of Measuring Devices: Plan.- A General Description of Measuring Devices: First Step – Finite Set of Possible Outcomes.- A General Description of Measuring Devices: Second Step – Pairs of Compatible Outcomes.- A General Description of Measuring Devices: Third Step – Subsets of Compatible Outcomes.- A General Description of Measuring Devices: Fourth Step – Conditional Statements about Possible Outcomes.- A General Description of Measuring Devices: Fifth Step – Disjunctive Conditional Statements about the Possible Outcomes.- A General Description of Measuring Devices: Summary.- Physical Quantities: A General Description.- Properties of Physical Quantities.- Future Work.

Vladik Kreinovich is Professor of Computer Science at the University of Texas at El Paso. His main interests are representation and processing of uncertainty, especially interval computations and intelligent control. He has published 13 books, 39 edited books, and more than 1,800 papers.

Vladik is Vice-President of the International Fuzzy Systems Association (IFSA), Vice-President of the European Society for Fuzzy Logic and Technology (EUSFLAT), Fellow of International Fuzzy Systems Association (IFSA), Fellow of Mexican Society for Artificial Intelligence (SMIA), and Fellow of the Russian Association for Fuzzy Systems and Soft Computing.

Graçaliz Dimuro received the M.Sc. (1991) and Ph.D. (1998) degrees from the Institute of Informatics of Universidade Federal do Rio Grande´do Sul, Brazil. In 2015, she was POS-DOC of the Brazilian Research Funding Agency CNPq at Universidad Pública de Navarra (UPNA), Spain, and, in 2017, she had a talent grantwith the Institute of Smart Cities of UPNA. She was Visitant Professor at UPNA during 2020-2022. Currently, she is Full Professor with Universidade Federal do Rio Grande, Brazil, and Researcher of level 1 of CNPq. She Member of the council of the Brazilian Society of Computational and Applied Mathematics and Associate Editor of Computational and Applied Mathematics journal (Springer). She was Invited Editor of several journals, e.g., fuzzy sets and systems, applied soft computing, and natural computing. She has reviewed papers for important journals and is Member of several program committees of international conferences. Her H-index (scopus) is 32 (August 2022), having over 2707 citations among 181 published papers. She has been awarded with best paper nominations in important conferences, e.g., NAFIPS 2018 and 2019, and FUZZ-IEEE 2022.

Antônio Carlos da Rocha Costa, Ph.D. in Computer Science (UFRGS, 1993), is Retired Associate Professor (FURG, 2015). He publisheda book: A Variational Basis for the Regulation and Structuration Mechanisms of Agent Societies (Springer, 2019).