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Interweavings of Alan Turing＇s Mathematics and Sociology of Knowledge

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Journal of Mathematics and System Science 2012年第7期2卷 420-429页

作者： Isabel Cafezeiro Ivan da Costa Marques Instituto de Computagao Universidade Federal Fluminense Rua Passo da Patria 24210-240 Brazil and Programa dePos-Graduagao em Historia das Ciencias e das Ttecnicas e Epistemologia Universidade Federal do Rio de Janeiro Programa de Pos-Graduagao em Historia das Ciencias e das Ttecnicas e Epistemologia Universidade Federal do Rio de JaneiroAv Atlantica 22010-000 Brazil

This paper starts from the analysis of how Alan Turing proceeded to build the notion of computability in his famous 1936 text ＂On computable numbers, with an application to the Entscheidungsproblem＂. Looking in detail at his stepwise construction, which starts from the materialities to achieve a satisfactory level of abstraction, it is considered how his way of doing mathematics was one that constructs mathematical knowledge by evading a definite separation between matter and form; in this way, making the world and language come together. Following the same line of reasoning, it is argued in this paper that the abstract and the concrete, the deduction and the induction, the technical and the social as well as the objective and the subjective are unthinkable as pure entities. By considering the controversies and discussions from the mid-nineteenth century until now, it is shown that local （social） elements necessarily participate in what is usually considered ＂technical content＂ or ＂objectivity＂. While Alan Turing was a precursor of what today might be said to be an ＂anthropological approach to mathematical culture＂, unveiling and reviving approaches that enable the axis of authority for mathematics, logic and computing to be shifted, he also opened different paths for the construction of a variety of mathematical knowledge as well.

关键词： computability Alan Tttring knowledge construction.

Understanding the Theory of Karp, Miller and Winograd

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Journal of Applied Mathematics and Physics 2024年第4期12卷 1203-1236页

作者： Athanasios I. Margaris Department of Digital Systems Gaiopolis Campus University of Thessaly Larissa Greece

The objective of this tutorial is to present the fundamental theory of Karp, Miller and Winograd, whose seminal paper laid the foundations regarding the systematic description of the organization of computations in systems of uniform recurrent equations by means of graph structures, via the definition of computability conditions and techniques for the construction of one-dimensional and multi-dimensional scheduling functions. Besides the description of this theory, the paper presents improvements and revisions made by other authors and furthermore, points out the differences regarding the conditions of causality and dependency between the general case of systems of recurrent equations and the special case of multiple nested loops.

关键词： computability Scheduling Computations Recurrent Equations

Representations of Co-regular Subsets on Euclidean Spaces

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Chinese Quarterly Journal of Mathematics 2007年第2期22卷 252-257页

作者：邱玉文王莉萍赵希顺 Institute of Logic and Cognition Sun Yet-sen University Office of Scientific Research and Foreign Affairs Hebi Professional Technology College

For the computability of co-regular subsets in metric spaces, the properties of the co-regular subsets and several reasonable representations on co-regular sets have been suggested in this paper. As last, the ＇weaker... 详细信息

关键词： type-2 theory of effectivity computability computable set representations

Formularless Logic Function

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Open Journal of Discrete Mathematics 2013年第1期3卷 21-24页

作者： Alexey Nikolayevitch Teterin Izhevsk Russia

The concept of computability is defined more exactly and illustrated as an example of Boolean functions and cryptanalysis. To define a Boolean function is not necessary to record its formula. To do that the reduced (compact) description of values is determined in the truth table or in the statement of the problem. We obtain estimates of computation time, the volume of a compact descriptions and the range of variables under which it takes the value 0 or 1, depending polynomially on the number of arguments.

关键词： Complexity of Computation(03D15) Models of Computation(68Q05) Analysis of Algorithms and Problem Complexity(68Q25) Computational Learning Theory(68Q32) Separability Satisfiability Fuzziness Dynamical Systems computability

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智能可计算性思想的发展

科学技术哲学 2017年第5期 117-126页

作者：张寅生蔡曙山

本文从各种“智能”定义中所陈述的共同特征中归纳出智能的根本特征：智能是脑和神经系统通过选择达到系统目标(功能)的能力，人工智能或机器智能则是对生命智能的模仿或特征功能的复制。基于这一定义，本文讨论和总结了智能可计算性思... 详细信息

本文从各种“智能”定义中所陈述的共同特征中归纳出智能的根本特征：智能是脑和神经系统通过选择达到系统目标(功能)的能力，人工智能或机器智能则是对生命智能的模仿或特征功能的复制。基于这一定义，本文讨论和总结了智能可计算性思想的发展脉络，包括哥德尔定义的“广义递归函数”；“选择”的计算活动；递归函数与图灵机；智能可计算的数学表达；智能可计算的核心性质；计算主义与强人工智能等。本文最后引出一些结论。

关键词：智能强人工智能图灵机可计算性递归函数 intelligence strong artificial intelligence Turing Machine computability recursive function

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人工智能与机器学习：算法基础和哲学观点