Related papers: Algorithmic information theory: Some recollections
Two philosophical applications of the concept of program-size complexity are discussed. First, we consider the light program-size complexity sheds on whether mathematics is invented or discovered, i.e., is empirical or is a priori. Second,…
A didactical survey of the foundations of Algorithmic Information Theory. These notes are short on motivation, history and background but introduce some of the main techniques and concepts of the field. The "manuscript" has been evolving…
This is a 1971 dissertation. Only its extended abstract was published at the time. While some results appeared in other publications, a number of details remained unpublished and may still have relevance.
Most work on computational complexity is concerned with time. However this course will try to show that program-size complexity, which measures algorithmic information, is of much greater philosophical significance. I'll discuss how one can…
In this paper the author presents some non-conventional thoughts on the complexity of the Universe and the algorithmic reproducibility of the human brain, essentially sparked off by the notion of algorithmic complexity. We must warn that…
This article summarises a Web-book on "Complexity" that was developed to introduce undergraduate students to interesting complex systems in the biological, physical and social sciences, and the common tools, principles and concepts used for…
This essay recounts my personal journey towards a deeper understanding of the mathematical foundations of algorithmic music composition. I do not spend much time on specific mathematical algorithms used by composers; rather, I focus on…
This paper provides some reflections on the field of mathematical software on the occasion of John Rice's 65th birthday. I describe some of the common themes of research in this field and recall some significant events in its evolution.…
This paper covers two topics: first an introduction to Algorithmic Complexity Theory: how it defines probability, some of its characteristic properties and past successful applications. Second, we apply it to problems in A.I. - where it…
One approach to confronting computational hardness is to try to understand the contribution of various parameters to the running time of algorithms and the complexity of computational tasks. Almost no computational tasks in real life are…
Some established and also novel techniques in the field of applications of algorithmic (Kolmogorov) complexity currently co-exist for the first time and are here reviewed, ranging from dominant ones such as statistical lossless compression…
Algorithms for continuous optimization problems have a rich history of design and innovation over the past several decades, in which mathematical analysis of their convergence and complexity properties plays a central role. Besides their…
The paper elaborates an endeavor on applying the algorithmic information-theoretic computational complexity to meta-social-sciences. It is motivated by the effort on seeking the impact of the well-known incompleteness theorem to the…
We discuss views about whether the universe can be rationally comprehended, starting with Plato, then Leibniz, and then the views of some distinguished scientists of the previous century. Based on this, we defend the thesis that…
This is a survey on the computational complexity of nonlinear mixed-integer optimization. It highlights a selection of important topics, ranging from incomputability results that arise from number theory and logic, to recently obtained…
Algorithmic statistics has two different (and almost orthogonal) motivations. From the philosophical point of view, it tries to formalize how the statistics works and why some statistical models are better than others. After this notion of…
We study multivariate problems like function approximation, numerical integration, global optimization and dispersion. We obtain new results on the information complexity $n(\varepsilon,d)$ of these problems. The information complexity is…
10 years ago or so Bill Helton introduced me to some mathematical problems arising from semidefinite programming. This paper is a partial account of what was and what is happening with one of these problems, including many open questions…
This habilitation thesis is intended to be a good introduction to enumeration, the problem of listing solutions. It focuses on the different ways of measuring complexity in enumeration, with a particular emphasis on my contributions to the…
Over the last decades, a class of important mathematical results have required an ever increasing amount of human effort to carry out. For some, the help of computers is now indispensable. We analyze the implications of this trend towards…