可計算數

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、遞迴實數或可計算實數。

等效的定義可以用递归函数、图灵机及λ演算等演算法的正式表示方式而得。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 $a$ 能被某個可計算函數 $f:\mathbb {N} \to \mathbb {Z}$ 以下述方式來近似，那麼 $a$ 就是一個可計算數：給定任何正整數 $n$ ，函數值 $f(n)$ 都滿足

${f(n)-1 \over n}\leq a\leq {f(n)+1 \over n}.$

相關條目

• 可定义数
• 可计算函数、半可計算函數
• 超计算问题

參考資料

• 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, doi:10.1112/plms/s2-42.1.230  (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