可計算數

可以用會結束的演算法計算有限次數到任意精確度的實數
各种各样的
基本

延伸
其他

圓周率
自然對數的底
虛數單位
無限大

可計算數(英語:computable numbers),是数学名詞,是指可用有限次、會結束的算法計算到任意精確度的实数。可計算數也被稱為遞迴數遞迴實數可計算實數

等效的定義可以用递归函数图灵机λ演算等演算法的形式表示法而得。可計算數形成實閉域,可以在許多數學應用上取代实数

定義

如果一個實數 能被某個可計算函數   以下述方式來近似,那麼   就是一個可計算數:給定任何正整數 ,函數值 都滿足:

 

不可計算數

非可計算的實數即為不可計算數。1975年,計算機學家格里高里·柴廷英语Gregory Chaitin做了一個有趣的實驗:選擇任意一種程式語言,隨意輸入一段程式碼,該程式碼能夠成功運行並且能夠在有限時間內終止的機率即為柴廷常數,這個數為一個經典的不可計算數。[1]

相關條目

相關書籍

參考資料

引用

  1. ^ 比根号2更“无理”的数 | 科学人 | 果壳网 科技有意思. 2011-03-09 [2018-06-30]. (原始内容存档于2019-06-05). 

來源

  • Oliver Aberth 1968, Analysis in the Computable Number Field, Journal of the Association for Computing Machinery (JACM), vol 15, iss 2, pp 276–299. This paper describes the development of the calculus over the computable number field.
  • Errett Bishop and Douglas Bridges, Constructive Analysis, Springer, 1985, ISBN 0-387-15066-8
  • Douglas Bridges and Fred Richman. Varieties of Constructive Mathematics, Oxford, 1987.
  • Jeffry L. Hirst, Representations of reals in reverse mathematics, Bulletin of the Polish Academy of Sciences, Mathematics, 55, (2007) 303–316.
  • 马文·闵斯基 1967, Computation: Finite and Infinite Machines, Prentice-Hall, Inc. Englewood Cliffs, NJ. No ISBN. Library of Congress Card Catalog No. 67-12342. His chapter §9 "The Computable Real Numbers" expands on the topics of this article.
  • E. Specker, "Nicht konstruktiv beweisbare Sätze der Analysis" J. Symbol. Logic, 14 (1949) pp. 145–158
  • Turing, A.M., On Computable Numbers, with an Application to the Entscheidungsproblem, Proceedings of the London Mathematical Society, 2 42 (1), 1936, 42 (1): 230–651937 [2018-08-22], doi:10.1112/plms/s2-42.1.230, (原始内容存档于2004-04-03)  (and Turing, A.M., On Computable Numbers, with an Application to the Entscheidungsproblem: A correction, Proceedings of the London Mathematical Society, 2 43 (6), 1938, 43 (6): 544–61937, doi:10.1112/plms/s2-43.6.544 ). Computable numbers (and Turing's a-machines) were introduced in this paper; the definition of computable numbers uses infinite decimal sequences.
  • Klaus Weihrauch 2000, Computable analysis, Texts in theoretical computer science, Springer, ISBN 3-540-66817-9. §1.3.2 introduces the definition by nested sequences of intervals converging to the singleton real. Other representations are discussed in §4.1.
  • Klaus Weihrauch, A simple introduction to computable analysis
  • H. Gordon Rice. "Recursive real numbers." Proceedings of the American Mathematical Society 5.5 (1954): 784-791.
  • V. Stoltenberg-Hansen, J. V. Tucker "Computable Rings and Fields" in Handbook of computability theory edited by E.R. Griffor. Elsevier 1999