相关论文: In Memory of Martin Davis

This is yet another version of the course notes in chao-dyn/9407003. Here we change the universal Turing machine that is used to measure program-size complexity so that the constants in our information-theoretic incompleteness theorems are…

Computer science has grown rapidly since its inception in the 1950s and the pioneers in the field are celebrated annually by the A.M. Turing Award. In this paper, we attempt to shed light on the path to influential computer scientists by…

We investigate the computational power of particle methods, a well-established class of algorit hms with applications in scientific computing and computer simulation. The computational power of a compute model determines the class of…

Intelligence necessitates memory. Without memory, humans fail to perform various nontrivial tasks such as reading novels, playing games or solving maths. As the ultimate goal of machine learning is to derive intelligent systems that learn…

This overview article highlights the critical role of mathematics in artificial intelligence (AI), emphasizing that mathematics provides tools to better understand and enhance AI systems. Conversely, AI raises new problems and drives the…

This paper shows that the programming model of Babbage's Analytical Engine, although unconventional, can be harnessed in order to simulate indirect addressing, a capability that was not included in the original instruction set. That is, in…

This paper discusses "computational" systems capable of "computing" functions not computable by predefined Turing machines if the systems are not isolated from their environment. Roughly speaking, these systems can change their finite…

We currently witness the spectacular success of artificial intelligence in both science and public life. However, the development of a rigorous mathematical foundation is still at an early stage. In this survey article, which is based on an…

We advance a Bayesian concept of 'intrinsic asymptotic universality' taking to its final conclusions previous conceptual and numerical work based upon a concept of a reprogrammability test and an investigation of the complex qualitative…

This document is built around a list of thirty-two problems in enumeration of matchings, the first twenty of which were presented in a lecture at MSRI in the fall of 1996. I begin with a capsule history of the topic of enumeration of…

Recent breakthroughs in AI capability have been attributed to increasingly sophisticated architectures and alignment techniques, but a simpler principle may explain these advances: memory makes computation universal. Memory enables…

Topological classification of the 4-manifolds bridges computation theory and physics. A proof of the undecidability of the homeomorphy problem for 4-manifolds is outlined here in a clarifying way. It is shown that an arbitrary Turing…

The purpose of this article is to review the achievements made in the last few years towards the understanding of the reasons behind the success and subtleties of neural network-based machine learning. In the tradition of good old applied…

This is a draft of an article to appear in the October 2022 issue of the Notices of the AMS. In this survey article we explore a fascinating area called descriptive combinatorics and its recently discovered connections to distributed…

Over the past thirty years or so the authors have been teaching various programming for mathematics courses at our respective Universities, as well as incorporating computer algebra and numerical computation into traditional mathematics…

We have defined the Conscious Turing Machine (CTM) for the purpose of investigating a Theoretical Computer Science (TCS) approach to consciousness. For this, we have hewn to the TCS demand for simplicity and understandability. The CTM is…

People solve different problems and know that some of them are simple, some are complex and some insoluble. The main goal of this work is to develop a mathematical theory of algorithmic complexity for problems. This theory is aimed at…

Many algorithms have been developed for enumerating various combinatorial objects in time exponentially less than the number of objects. Two common classes of algorithms are dynamic programming and the transfer matrix method. This paper…

This paper examines whether unitary evolution alone is sufficient to explain emergence of the classical world from the perspective of computability theory. Specifically, it looks at the problem of how the choice related to the measurement…

Vladimir Andreevich Uspensky [1930-2018] was one of the Soviet pioneers of the theory of computation and mathematical logic in general (and my teacher and thesis advisor). This paper is the survey of his mathematical works and their…