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Geoffrion's theorem is a fundamental result from mathematical programming assessing the quality of Lagrangian relaxation, a standard technique to get bounds for integer programs. An often implicit condition is that the set of feasible…
It has been commonly argued, on the basis of Goedel's theorem and related mathematical results, that true artificial intelligence cannot exist. Penrose has further deduced from the existence of human intelligence that fundamental changes in…
The discovery of the infinite integer leads to a partition between finite and infinite numbers. Construction of an infinitesimal and infinitary number system, the Gossamer numbers. Du Bois-Reymond's much-greater-than relations and…
This paper refutes the validity of the polynomial-time algorithm for solving satisfiability proposed by Sergey Gubin. Gubin introduces the algorithm using 3-SAT and eventually expands it to accept a broad range of forms of the Boolean…
For over a decade, the hypercomputation movement has produced computational models that in theory solve the algorithmically unsolvable, but they are not physically realizable according to currently accepted physical theories. While…
We introduce Aristotle, an AI system that combines formal verification with informal reasoning, achieving gold-medal-equivalent performance on the 2025 International Mathematical Olympiad problems. Aristotle integrates three main…
Inspired by the recent pioneering work, dubbed "The Ramanujan Machine" by Raayoni et al. (arXiv:1907.00205), we (automatically) [rigorously] prove some of their conjectures regarding the exact values of some specific infinite continued…
A new concept of an equilibrium in games is introduced that solves an open question posed by A. Neyman.
In this paper, I formalize intelligence measurement in games by introducing mechanisms that assign a real number -- interpreted as an intelligence score -- to each player in a game. This score quantifies the ex-post strategic ability of the…
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…
In the paper "On the interpolation of integer-valued polynomials" (Journal of Number Theory 133 (2013), pp. 4224--4232.) V. Volkov and F. Petrov consider the problem of existence of the so-called $n$-universal sets (related to simultaneous…
In the process of evolution, the brain has achieved such perfection that artificial intelligence systems do not have and which needs its own mathematics. The concept of cognitome, introduced by the academician K.V. Anokhin, as the cognitive…
Over the past two decades, Yuri Gurevich and his colleagues have formulated axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in the new generic framework of abstract state…
We investigate how undecidability enters into computations of classical physical systems and contributes to the increase of entropy and loss of information. In actual computation with finite bit of information capacity we accept…
More than a speculative technology, quantum computing seems to challenge our most basic intuitions about how the physical world should behave. In this thesis I show that, while some intuitions from classical computer science must be…
We endorse and expand upon Susan Schneider's critique of the linear model of AI progress and introduce two novel concepts: "familiar intelligence" and "strange intelligence". AI intelligence is likely to be strange intelligence, defying…
Greedy algorithms are a fundamental category of algorithms in mathematics and computer science, characterized by their iterative, locally optimal decision-making approach, which aims to find global optima. In this review, we will discuss…
In recent years we observed rapid and significant advancements in artificial intelligence (A.I.). So much so that many wonder how close humanity is to developing an A.I. model that can achieve human level of intelligence, also known as…
In this brief note, there is a short recollection of my scientific interactions with the great Russian mathematician Sergey Konstantinovich Godunov.
This paper explores the relationship of artificial intelligence to the task of resolving open questions in mathematics. We first present an updated version of a traditional argument that limitative results from computability and complexity…