Related papers: The Limits of Mathematics---Tutorial Version
In this paper, we present a comprehensive system for the treatment of the topic of limits--conceptually, computationally, and formally. The system addresses fundamental linguistic flaws in the standard presentation of limits, which attempts…
We study a class of countably-infinite-dimensional linear programs (CILPs) whose feasible sets are bounded subsets of appropriately defined spaces of measures. The optimal value, optimal points, and minimal points of these CILPs can be…
While Transformer language models (LMs) are state-of-the-art for information extraction, long text introduces computational challenges requiring suboptimal preprocessing steps or alternative model architectures. Sparse attention LMs can…
Mixed Integer Linear Programming (MILP) is a pillar of mathematical optimization that offers a powerful modeling language for a wide range of applications. During the past decades, enormous algorithmic progress has been made in solving…
We consider a minimal extension of the language of arithmetic, such that the bounded formulas provably total in a suitably-defined theory \`a la Buss (expressed in this new language) precisely capture polytime random functions. Then, we…
Automating algorithm configuration is growing increasingly necessary as algorithms come with more and more tunable parameters. It is common to tune parameters using machine learning, optimizing performance metrics such as runtime and…
Understanding the mathematical reasoning capabilities of Large Language Models (LLMs) is a central topic in the study of artificial intelligence. This new domain necessitates the creation of datasets of reasoning tasks for both training and…
Large AI Models (LAIMs), of which large language models are the most prominent recent example, showcase some impressive performance. However they have been empirically found to pose serious security issues. This paper systematizes our…
We introduce a set of eight universal Rules of Inference by which computer programs with known properties (axioms) are transformed into new programs with known properties (theorems). Axioms are presented to formalize a segment of Number…
We present a new algorithm for computing upper bounds on the number of executions of each program instruction during any single program run. The upper bounds are expressed as functions of program input values. The algorithm is primarily…
Matrix completion is a fundamental problem that comes up in a variety of applications like the Netflix problem, collaborative filtering, computer vision, and crowdsourcing. The goal of the problem is to recover a k-by-n unknown matrix from…
In this short communication it is discussed the relation between disentangled states and algorithmic information theory aiming to construct an irreducible sentence whose length increases in a non-polynomial way when the number of qubits…
This article discusses ability of Linear Programming models to be used as solvers of NP-complete problems. Integer Linear Programming is known as NP-complete problem, but non-integer Linear Programming problems can be solved in polynomial…
Algorithmic information theory roots the concept of information in computation rather than probability. These lecture notes were constructed in conjunction with the graduate course I taught at Universit\`a della Svizzera italiana in the…
Researchers often rely on humans to code (label, annotate, etc.) large sets of texts. This kind of human coding forms an important part of social science research, yet the coding process is both resource intensive and highly variable from…
In recent months, large language models (LLMs) have made significant progress in mathematical proof generation, but further advancement is hindered by the lack of a large-scale, high-quality dataset of human-evaluated proofs. While…
In considering the reliability of numerical programs, it is normal to "limit our study to the semantics dealing with numerical precision" (Martel, 2005). On the other hand, there is a great deal of work on the reliability of programs that…
Motivated by certain applications from physics, biochemistry, economics, and computer science, in which the objects under investigation are not accessible because of various limitations, we propose a trial-and-error model to examine…
We present an efficient reduction from the Bounded integer programming (BIP) to the Subspace avoiding problem (SAP) in lattice theory. The reduction has some special properties with some interesting consequences. The first is the new upper…
The interplay of optimizers and architectures in neural networks is complicated and hard to understand why some optimizers work better on some specific architectures. In this paper, we find that the traditionally used sharpness metric does…