C. Calude, P. Hertling, B. Khoussainov, Y. Wang, Recursively enumerable reals and Chaitin's number, in STACS '98, Lecture Notes in Comput. Sci. 1373 (1988), 596-606. Home/Search Document Not in Database Summary Related Articles Check |
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Computability, Definability and Algebraic Structures - Downey (1999) (Correct)
....of a computable sequence of rationals. That is, we have a = lim s q s such that we can compute q i for all i. But note that although we know that the sequence converges, we don t computably know how fast, at least in general. The following theorem is from Calude, Hertling, Khoussainov and Wang [10]. A set A is called pre x free i for all 2 A, and all with an initial segment of , 62 A. Theorem 6.1 (Calude et al. 10] The following are equivalent. i) a is computably enumerable. These de nitions de ne the relevant real up to equivalence. Pre x free sets are considered ....
....we know that the sequence converges, we don t computably know how fast, at least in general. The following theorem is from Calude, Hertling, Khoussainov and Wang [10] A set A is called pre x free i for all 2 A, and all with an initial segment of , 62 A. Theorem 6. 1 (Calude et al. [10]) The following are equivalent. i) a is computably enumerable. These de nitions de ne the relevant real up to equivalence. Pre x free sets are considered for technical reasons since if a set A is pre x free then, by Kraft s inequality, we know that n2A 2 converges. Pre x free sets ....
[Article contains additional citation context not shown here]
Calude, C., P. Hertling, B. Khoussainov, and Y. Wang, Recursively enumerable reals and Chaitin's number, in STACS'98, Springer Lecture Notes in Computer Science, Vol. 1373, 1998, 596-606.
Presentations of Computably Enumerable Reals - Downey, LaForte (2000) (2 citations) (Correct)
....= n2A 2 for a computably enumerable A N strongly computably enumerable. The other way that one can think of reals is as limits of Cauchy sequences: that is, lim s fq s : q s 2 Bg: The situation was recently clari ed by Calude, Khoussainov, Hertling and Wang. Theorem 1 (Calude et.al. [4]) The following are equivalent for a real . i) is computably enumerable. ii) The lower Dedekind cut of is computably enumerable. iii) There is an in nite computably enumerable pre x free set W such that = iv) There is a computable pre x free set W such that ....
....f(k; s) 1 and f(k; s 1) 0, then there exists k such that f(k ; s) 0 and f(k ; s 1) 1. b) a 1 a 2 : where a i = lim s f(i; s) vi) There is a computable increasing sequence of rationals with limit . Although the apparently stronger (iv) is not explicitly stated in [4], it follows from (iii) since there are always an in nite number of strings we can add at any particular stage in the enumeration. Hence we can rule out larger and larger subsets of f0; 1g thereby making the complement of W computably enumerable as well. We refer to the approximation in (v. ....
[Article contains additional citation context not shown here]
Calude, C., Hertling, P., Khoussainov, B., Wang, Y., Recursively enumerable reals and Chaitin's number, in STACS '98, Springer Lecture Notes in Computer Science, Vol 1373, 1998, 596-606.
Some Computability-Theoretical Aspects of Reals and Randomness - Downey (2001) (2 citations) (Correct)
....i for all 2 A, and all with an initial segment of , 62 A. Pre x free sets are considered for technical reasons since if a set A is pre x free then, as we soon see, by Kraft s inequality, we know that n2A 2 converges and conversely. Theorem 2 (Calude, Khoussainov, Hertling, Wang [8], Soare [50] The following are equivalent. i) is the limit of a computable enumerable monotone increasing (in the real ordering) sequence of rationals. ii) is computably enumerable. iii) There is an in nite computably enumerable pre x free set A with = iv) There is a computable ....
Calude, C., Hertling, P., Khoussainov, B., Wang, Y., Recursively enumerable reals and Chaitin's number, in STACS '98, Springer Lecture Notes in Computer Science, Vol 1373, 1998, 596-606.
Structural Properties of D.C.E. Degrees and Presentations of C.E.. - Wu (2002) (Correct)
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C. Calude, P. Hertling, B. Khoussainov, Y. Wang, Recursively enumerable reals and Chaitin's number, in STACS '98, Lecture Notes in Comput. Sci. 1373 (1988), 596-606.
Trivial Reals - Downey, Hirschfeldt, Nies, Stephan (2002) (3 citations) (Correct)
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Calude, C., P. Hertling, B. Khoussainov, B., and Y. Wang, Recursively enumerable reals and Chaitin's # numbers, Theoretical Computer Science 255 (2001), 125--149.
Computably Enumerable Reals and Uniformly Presentable Ideals - Downey, Terwijn (2002) (Correct)
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Calude, C., Hertling, P., Khoussainov, B., Wang, Y., Recursively enumerable reals and Chaitin's number, in STACS '98, Springer Lecture Notes in Computer Science 1373 (1998) 596-606.
The Kolmogorov Complexity of Random Reals - Yu, Ding, Downey (2003) (Correct)
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C. Calude, P. Hertling, B. Khoussainov, Y. Wang, Recursively enumerable reals and Chaitin's number, in STACS '98, Lecture Notes in Computer Science, Vol 1373, Springer-Verlag, 1998, 596-606.
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