Related papers: Computing with and without arbitrary large numbers
Algorithms have been fundamental to recent global technological advances and, in particular, they have been the cornerstone of technical advances in one field rapidly being applied to another. We argue that algorithms possess fundamentally…
Most continuous mathematical formulations arising in science and engineering can only be solved numerically and therefore approximately. We shall always assume that we're dealing with a numerical approximation to the solution. There are two…
In this paper, we will develop a significantly more general notion of classical Ramsey numbers (extending most other graph-theoretic generalizations) and make some preliminary characterizations of these new Ramsey numbers using simple…
We show how to transform into programs the proofs in classical Analysis which use the existence of an ultrafilter on the integers. The method mixes the classical realizability introduced by the author, with the "forcing" of P. Cohen. The…
In the paper we present results to develop an irreducible theory of complex systems in terms of self-organization processes of prime integer relations. Based on the integers and controlled by arithmetic only the self-organization processes…
We study the problem of minimizing the average of a very large number of smooth functions, which is of key importance in training supervised learning models. One of the most celebrated methods in this context is the SAGA algorithm. Despite…
We introduce a model of computation based on read only memory (ROM), which allows us to compare the space-efficiency of reversible, error-free classical computation with reversible, error-free quantum computation. We show that a ROM-based…
Quantum Random Access Memory (QRAM) is a critical component for loading classical data into quantum computers. While constructing a practical QRAM presents several challenges, including the impracticality of an infinitely large QRAM size…
Sampling is a fundamental technique, and sampling without replacement is often desirable when duplicate samples are not beneficial. Within machine learning, sampling is useful for generating diverse outputs from a trained model. We present…
Large language models (LLMs) have recently transformed natural language processing, enabling machines to generate human-like text and engage in meaningful conversations. This development necessitates speed, efficiency, and accessibility in…
Given the complexity of modern software systems, it is of great importance that such systems be able to autonomously modify themselves, i.e., self-adapt, with minimal human supervision. It is critical that this adaptation both results in…
Trials enroll a large number of subjects in order to attain power, making them expensive and time-consuming. Sample size calculations are often performed with the assumption of an unadjusted analysis, even if the trial analysis plan…
Artificial intelligence algorithms are capable of fantastic exploits, yet they are still grossly inefficient compared with the brain's ability to learn from few exemplars or solve problems that have not been explicitly defined. What is the…
In a multi-base representation of an integer (in contrast to, for example, the binary or decimal representation) the base (or radix) is replaced by products of powers of single bases. The resulting numeral system has desirable properties…
Important tasks such as reasoning and planning are fundamentally algorithmic, meaning that solving them robustly requires acquiring true reasoning or planning algorithms, rather than shortcuts. Large Language Models lack true algorithmic…
Many inference scenarios rely on extracting relevant information from known data in order to make future predictions. When the underlying stochastic process satisfies certain assumptions, there is a direct mapping between its exact…
Estimating conditional dependence graphs and precision matrices are some of the most common problems in modern statistics and machine learning. When data are fully observed, penalized maximum likelihood-type estimators have become standard…
Many iterative algorithms in optimization, computational geometry, computer algebra, and other areas of computer science require repeated computation of some algebraic expression whose input changes slightly from one iteration to the next.…
Hypergraphs are structures that can be decomposed or described; in other words they are recursively countable. Here, we get exact and asymptotic enumeration results on hypergraphs by means of exponential generating functions. The number of…
Recursive analysis was introduced by A. Turing [1936], A. Grzegorczyk [1955], and D. Lacombe [1955]. It is based on a discrete mechanical framework that can be used to model computation over the real numbers. In this context the…