Related papers: The Limits of Mathematics
We derive upper bounds on the generalization error of a learning algorithm in terms of the mutual information between its input and output. The bounds provide an information-theoretic understanding of generalization in learning problems,…
Lectures given at the Workshop "Hadrons 1994", Gramado, RS, Brasil
These are the revised versions of the papers presented at CICLOPS 2011, a workshop colocated with ICLP 2011.
Chaotic behavior in dynamical systems poses a significant challenge in trajectory control, traditionally relying on computationally intensive physical models. We present a machine learning-based algorithm to compute the minimum control…
This article is the writing notes of a talk on Lie Antialgebras given by the second author at the conference "3Quantum: Algebra Geometry Information" that held in Tallinn in July 2012. The aim of this note is to give a brief survey of the…
We show that real-value approximations of Kolmogorov-Chaitin (K_m) using the algorithmic Coding theorem as calculated from the output frequency of a large set of small deterministic Turing machines with up to 5 states (and 2 symbols), is in…
Chaotic iterations have been introduced on the one hand by Chazan, Miranker [5] and Miellou [9] in a numerical analysis context, and on the other hand by Robert [11] and Pellegrin [10] in the discrete dynamical systems framework. In both…
We present FIMO, an innovative dataset comprising formal mathematical problem statements sourced from the International Mathematical Olympiad (IMO) Shortlisted Problems. Designed to facilitate advanced automated theorem proving at the IMO…
Some open questions in quantum information theory (QIT) are described. Most of them were presented in Banff during the BIRS workshop on Operator Structures in QIT 11-16 February 2007. New material has been added in view of the recent…
Quantum computing is an exciting field with high disruptive potential, but very difficult to access. For this reason, many approaches to teaching quantum computing are being developed worldwide. This always raises questions about the…
We introduce algorithmic information theory, also known as the theory of Kolmogorov complexity. We explain the main concepts of this quantitative approach to defining `information'. We discuss the extent to which Kolmogorov's and Shannon's…
This paper introduces DOoM, a new open-source benchmark designed to assess the capabilities of language models in solving mathematics and physics problems in Russian. The benchmark includes problems of varying difficulty, ranging from…
These notes are based on the lecture courses given at the Ruhr-Universit{\"a}t-Bochum (03--08.02.1997) and at the Universit{\'e} Paul Sabatier (Toulouse, 08-12.01.1996).
This paper studies the low-rank matrix completion problem from an information theoretic perspective. The completion problem is rephrased as a communication problem of an (uncoded) low-rank matrix source over an erasure channel. The paper…
This volume constitutes the pre-proceedings of the 26th International Symposium on Logic-Based Program Synthesis and Transformation (LOPSTR 2016), held on 6-8th September 2016 in Edinburgh, Scotland UK, and co-located with the 18th…
This text provides an introduction to distributed local algorithms -- an area at the intersection of theoretical computer science and discrete mathematics. We collect recent results in the area and demonstrate how they lead to a clean…
This volume contains the proceedings of the Workshop on Partiality and Recursion in Interactive Theorem Provers (PAR 2010) which took place on July 15 in Edinburgh, UK. This workshop was held as a satellite workshop of the International…
There is a cognitive limit in Human Mind. This cognitive limit has played a decisive role in almost all fields including computer sciences. The cognitive limit replicated in computer sciences is responsible for inherent Computational…
This is a collection of notes based on lectures given at IIT Madras in September 2019 and at IFT Madrid in November 2019. It is supposed to be a concise (and therefore not comprehensive) and pragmatic course on applied holography and…
Existence theory in economics is usually in real domains such as the findings of chaotic trajectories in models of economic growth, tatonnement, or overlapping generations models. Computational examples, however, sometimes converge rapidly…