The Infinity Computer and Numerical Computations with Infinite and Infinitesimal Numbers
Yaroslav D. Sergeyev, University of Calabria, Italy
Fuzzy Information in Statistics
Reinhard Viertl, Vienna University of Technology, Austria
"Morphological Computation" - Self-organization, Embodiment, and Biological Inspiration
Rolf Pfeifer, Artificial Intelligence Laboratory Dept. of Informatics, University of Zurich, Switzerland
The Infinity Computer and Numerical Computations with Infinite and Infinitesimal Numbers
Yaroslav D. Sergeyev
University of Calabria
Italy
![]()
Brief Bio
Yaroslav D. Sergeyev is Distinguished Professor at the University of Calabria, Italy (professorship awarded by the Italian Government). He is also Professor (part-time contract) at N.I. Lobachevski Nizhni Novgorod State University, Russia and Affiliated Researcher at the Institute of High Performance Computing and Networking of the Italian National Research Council. He has got his Ph.D. from N.I. Lobachevski Nizhni Novgorod State University and his D.Sc. degree from M.V. Lomonosov State University, Moscow (this degree is Habilitation for Full Professorship in Russian universities). His research interests include numerical analysis, global optimization, infinity computing and calculus, set theory, number theory, fractals, parallel computing, and interval analysis.
Prof. Sergeyev has been awarded several national and international prizes (Pythagoras International Prize in Mathematics, Italy, 2010; Outstanding Achievement Award from the 2010 World Congress in Computer Science, Computer Engineering, and Applied Computing, USA; Lagrange Lecture, Turin University, Italy, 2010; MAIK Prize for the best scientific monograph published in Russian, Moscow, 2008, etc.). He has published more than 60 papers in the leading international journals, 4 books, and 4 patents (his complete list of publications contains almost 200 items). He is a member of editorial boards of 4 international journals and co-editor of 3 special issues. He has given more than 30 plenary and keynote lectures at prestigious international congresses and was a member of Scientific Committees of 40 international congresses. He was Coordinator of numerous national and international research and educational projects. In particular, since 1994 he is Coordinator of the international research and educational program ‘Italian–Russian University’ and in 2011 he was Russian Chairman of the Plenary session of the Italian-Russian Rectors Congress. Software developed under his supervision is used in more than 40 countries of the world. Numerous magazines, newspapers, TV and radio channels have dedicated a lot of space to his research.
Detailed information: http://si.deis.unical.it/~yaro/
Abstract
A new methodology allowing one to execute numerical computations with finite, infinite, and infinitesimal numbers on a new type of a computer – the Infinity Computer – is introduced. A calculator using the Infinity Computer technology is presented during the talk. The new approach is based on the principle ‘The part is less than the whole’ introduced by Ancient Greeks that is applied to all numbers (finite, infinite, and infinitesimal) and to all sets and processes (finite and infinite). It is shown that it becomes possible to write down finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework (different from that of the non-standard Analysis). The new methodology evolves ideas of Cantor and Levi-Civita in a more
applied way and (among other things) introduces infinite integers that possess both cardinal and ordinal properties as usual finite numbers.
Note that foundations of the Set Theory dealing with infinity have been developed starting from the end of the XIX-th century until more or less the first
decades of the XX-th century. Foundations of the classical Analysis dealing both with infinity and infinitesimal quantities have been developed even earlier, more than 200 years ago, with the goal to develop mathematical tools allowing one to solve problems arising in the real world in that remote time. As a result, these parts of Mathematics reflect ideas that people had about Physics (including definitions of the notions continuous and discrete) more than 200 years ago. Thus, these mathematical tools do not include numerous achievements of Physics of the XX-th century. Even the brilliant non-standard Analysis of Robinson made in the middle of the XX-th century has been also directed to a reformulation of the classical Analysis (i.e., Analysis created two hundred years before Robinson) in terms of infinitesimals and not to the creation of a new kind of Analysis that would incorporate new achievements of Physics.
The point of view on infinite and infinitesimal quantities presented in this talk uses strongly two methodological ideas borrowed from the modern Physics:
relativity and interrelations holding between the object of an observation and the tool used for this observation. The latter is directly related to connections
between different numeral systems used to describe mathematical objects and the objects themselves.
The new computational methodology gives the possibility both to execute numerical (not symbolic) computations of a new type and simplifies fields of
Mathematics where the usage of the infinity and/or infinitesimals is necessary (e.g., divergent series, limits, derivatives, integrals, measure theory, probability theory, fractals, etc.). In particular, a number of results related to the First Hilbert Problem and Turing machines are established and a new concept of continuity better reflecting the modern view of physicists on the world around us is introduced. Numerous examples and applications are given: approximation, automatic differentiation, cellular automata, differential equations, linear and non-linear optimization, fractals, percolation,
processes of growth in biological systems, etc.
Fuzzy Information in Statistics
Reinhard Viertl
Vienna University of Technology
Austria
http://www.statistik.tuwien.ac.at/public/viertl
![]()
Brief Bio
Born 1946 in Hall in Tyrol, Austria.
Studies in civil engineering, philosophy, and engineering mathematics. Dipl.-Ing. degree 1972. Doctor of engineering sciences 1974. Assistant at the Technische Hochschule Wien. University docent 1979. Research fellow and visiting lecturer, University of California, Berkeley, 1980-1981. Visiting docent, University of Klagenfurt, 1981-1982. Since 1982 full professor of applied statistics with special emphasis on regional and information sciences, Vienna University of Technology. Visiting professor, University of Innsbruck, 1991-1993. Seasonal instructor, University of Calgary, 2003. Visiting professor, University of Salatiga, Indonesia, 2010, Visiting professor, University of Brno, Czech Republic, 2011.
Member of the International Statistical Institute. Fellow of the Royal Statistical Society, London. President of the Austrian Statistical Society 1987-1995. Invitation to membership in the New York Academy of Sciences, 1999. Election as honorary member of the Austrian Statistical Society, 2008.
Scientific Publications:
Books: Statistical Methods in Accelerated Life Testing (1988), Introduction to Stochastics (in German, 2003), Statistical Methods for Non-Precise Data (1996). Description and Analysis of Fuzzy Information (in German, 2006). Statistical Methods for Fuzzy Data (2011). Editor and co-editor of several other books.
Author of over 100 scientific papers in mathematics and statistics.
Abstract
Data in applications are frequently not precise numbers, vectors, and symbols but more or less non-precise also called fuzzy. All measurement results of continuous quantities are subject to fuzziness. This makes it necessary to describe data by fuzzy models before analyzing them. There are different kinds of uncertainty related to data analysis and especially statistics. The most important are variability, imprecision of data, model uncertainty, and uncertainty of a-priori information. Whereas variability is modeled since a long time by probability models, the quantitative mathematical description of imprecision by so-called fuzzy models was done more recently. Especially in statistical data analysis in a Bayesian context also a-priori information is best modeled by so-called fuzzy probability distributions. Examples of non-precise data and related fuzzy models as well as adapted statistical data analysis will be given in the contribution.
"Morphological Computation" - Self-organization, Embodiment, and Biological Inspiration
Rolf Pfeifer
Artificial Intelligence Laboratory Dept. of Informatics, University of Zurich
Switzerland
Brief Bio
Rolf Pfeifer received his master’s degree in physics and mathematics and his Ph.D. in computer science from the Swiss Federal Institute of Technology (ETH) in Zurich, Switzerland. He spent three years as a post-doctoral fellow at Carnegie-Mellon University and at Yale University in the US. Since 1987 he has been a professor of computer science at the Department of Informatics, University of Zurich, and director of the Artificial Intelligence Laboratory. Visiting professor and research fellow at the Free University of Brussels, the MIT Artificial Intelligence Laboratory in Cambridge, Mass., the Neurosciences Institute (NSI) in San Diego, the Beijing Open Laboratory for Cognitive Science, and the Sony Computer Science Laboratory in Paris. Elected "21st Century COE Professor, Information Science and Technology" at the University of Tokyo in 2004. In 2009 he was also a visiting professor at the Scuola Superiore Sant'Anna in Pisa, at Shanghai Jiao Tong University in China, and he was appointed "Fellow of the School of Engineering" at the University of Tokyo. Currently, he is the Deputy Director of the NCCR Robotics, the "National Competence Center for Research in Robotics" in Switzerland. His research interests are in the areas of embodiment, biorobotics, artificial evolution and morphogenesis, modular robotics, self-assembly and educational technology. He is the author of the books "Understanding Intelligence", MIT Press, 1999 (with C. Scheier), "How the body shapes the way we think: a new view of intelligence," 2007 (with Josh Bongard) MIT Press (popular science style), "Designing intelligence - why brains aren't enough" (short version - with Josh Bongard and Don Berry, e-book), and "La révolution de l'intelligence encorporée" ("The revolution of embodied intelligence"; with Alexandre Pitti) (in French, to appear May 2012). Lecture series: “The ShanghAI Lectures”, a global mixed-reality lecture series on embodied intelligence, broadcast this time from the University of Zurich, and Shanghai Jiao Tong University, China in cooperation with other universities from around the globe (fall term 2012, starting Thursday, 20 September 2012 until December 2012).
World exhibition: ROBOTS ON TOUR - World Congress and Exhibition of Robots, Humanoids, Cyborgs, and more. 9 March 2013, Zurich (Puls 5).
Abstract
Robotics researchers increasingly agree that ideas from biology and self organization can strongly benefit the design of autonomous robots. While in classical robotics there is a clear distinction between the control and the to-be-controlled, in biological systems part of the functionality is incorporated into the morphological and material characteristics: the task of coping with impact in human walking, for example, is taken over by the elasticity of the muscle-tendon system. In other words, part of the "computation" is taken over by the morphology of the agent, thus the term "morphological computation". On the sensory side, the spatial arrangement of the sensors on the body provides information about the stimulus. Biological systems which are - with the exception of the skeleton providing structural support - largely soft and elastic (to varying degrees), capitalize on "morphological computation" by exploiting their morphological and material characteristics as well as processes of self-organization. These ideas play an essential role in the design of next-generation intelligent agents as in the recent field of "soft robotics". Although many challenges remain, concepts from biology and morphological computation will eventually enable researchers to engineer machines for the real world that possess at least some of the desirable properties of biological organisms, such as adaptivity, robustness, and versatility.