Computing a glimpse of randomness

Experimental Mathematics

Volumn 11, Issue 3, 2002, Pages 361-370

  Calude, Cristian S ^a   Dinneen, Michael J ^a   Shu, Chi Kou ^a  

^a UNIVERSITY OF AUCKLAND   (New Zealand)

Author keywords

Algorithmic randomness; Chaitin Omega number; Halting problem

Indexed keywords

EID: 0036942286     PISSN: 10586458     EISSN: None     Source Type: Journal    
DOI: 10.1080/10586458.2002.10504481     Document Type: Article

References (35)

1. [Bennett and Gardner 79] C. H. Bennett, M. Gardner. "The random number omega bids fair to hold the mysteries of the universe." Scientific American 241 (1979), 20-34.
2. [Bridges 94] D. S. Bridges. Computability - A Mathematical Sketchbook, Springer Verlag, Berlin, 1994.
3. [Calude 94] C. S. Calude. Information and Randomness. An Algorithmic Perspective. Springer-Verlag, Berlin, 1994.
4. [Calude 00] C. S. Calude. "A glimpse into algorithmic information theory." in Logic, Language and Computation, P. Blackburn, N. Braisby, L. Cavedon, A. Shimojima (eds.), Volume 3, CSLI Series, pp. 67-83, Cambridge University Press, Cambridge, 2000.
5. [Calude 02a] C. S. Calude. "A characterization of c.e. random reals." Theoret. Comput. Sci., 217 (2002), 3-14.
6. [Calude 02b] C. S. Calude. "Chaitin Ω numbers, Solovay machines and incompleteness." Theoret. Comput. Sci. 284 (2002), 269-277.
7. [Calude and Chaitin 99] C. S. Calude and G. J. Chaitin. "Randomness everywhere." Nature, 400:22 (July 1999), 319-320.
8. [Calude et al. 00] C. S. Calude, M. J. Dinneen, and C. Shu. "Computing 80 Initial Bits of A Chaitin Omega Number: Preliminary Version." CDMTCS Research Report 146 (2000)
9. [Calude et al. 01] C. S. Calude, P. Hertling, B. Khoussainov, and Y. Wang. "Recursively enumerable reals and Chaitin Ω numbers." in Proceedings of the 15th Symposium on Theoretical Aspects of Computer Science (Paris), M. Morvan, C. Meinel, D. Krob (eds.), pp. 596-606, Springer-Verlag, Berlin, 1998. Full paper in Theoret. Comput. Sci. 255 (2001), 125-149.
10. [Calude et al. 01] C. S. Calude, P. Hertling, B. Khoussainov, and Y. Wang. "Recursively enumerable reals and Chaitin Ω numbers." in Proceedings of the 15th Symposium on Theoretical Aspects of Computer Science (Paris), M. Morvan, C. Meinel, D. Krob (eds.), pp. 596-606, Springer-Verlag, Berlin, 1998. Full paper in Theoret. Comput. Sci. 255 (2001), 125-149.
11. [Calude and Jürgensen 94] C. Calude and H. Jürgensen. "Randomness as an invariant for number representations." in Results and Trends in Theoretical Computer Science, H. Maurer, J. Karhumäki, G. Rozenberg (eds.), pp. 44-66, Springer-Verlag, Berlin, 1994.
12. [Casti 97] J. L. Casti. "Computing the uncomputable." The New Scientist, 154/2082 (17 May 1997), 34.
13. [Chaitin 75] G. J. Chaitin. "A theory of program size formally identical to information theory." J. Assoc. Comput. Mach. 22 (1975), 329-340. (Reprinted in: [Chaitin 90b], 113-128)
14. [Chaitin 90a] G. J. Chaitin. Algorithmic Information Theory, Cambridge University Press, Cambridge, 1987. (Third printing 1990)
15. [Chaitin 90b] G. J. Chaitin. Information, Randomness and Incompleteness, Papers on Algorithmic Information Theory, World Scientific, Singapore. 1987. (2nd ed., 1990)
16. [Chaitin 97] G. J. Chaitin. The Limits of Mathematics. Springer-Verlag, Singapore, 1997.
17. [Chaitin 99] G. J. Chaitin. The Unknowable, Springer-Verlag, Singapore, 1999.
18. [Chaitin 00a] G. J. Chaitin. Exploring Randomness, Springer-Verlag, London, 2000.
19. [Chaitin 00b] G. J. Chaitin. Personal communication to C. S. Calude, November 2000.
20. [Chaitin 01] G. J. Chaitin. Personal communication to C. S. Calude, December 2001.
21. [Downey 02] R. G. "Downey. Some Computability-Theoretical Aspects of Reals and Randomness." CDMTCS Research Report 173 (2002).
22. [Hertling and Weihrauch 98] P. Hertling and K. Weihrauch. "Randomness spaces." in Automata, Languages and Programming, Proceedings of the 25th International Colloquium. ICALP'98 (Aalborg, Denmark), K. G. Larsen. S. Skyum, and G. Winskel (eds.), pp. 796-807, Springer-Verlag, Berlin, 1998.
23. [Kučera and Slaman 01] A. Kucera and T. A. Slaman. "Randomness and recursive enumerability." SIAM J. Comput., 31:1 (2001), 199-211.
24. [Martin-Löf 66] P. Martin-Löf. Algorithms and Random Sequences, Erlangen University, Nürnberg, Erlangen, 1966.
25. [Martin-Löf 66] P. Martin-Löf. "The definition of random sequences." Inform. and Control 9 (1966), 602-619.
26. [Marxen and Buntrock 90] H. Marxen and J. Buntrock. "Attacking the busy beaver 5." Bull EATCS 40 (1990), 247-251.
27. [Odifreddi 99] P. Odifreddi. Classical Recursion Theory, North-Holland, Amsterdam, Vol.1, 1989, Vol. 2, 1999.
28. [Shu 03] C. Shu. Computing Exact Approximations of a Chaitin Omega Number. Ph.D. Thesis. University of Auckland, New Zealand, 2003.
29. [Scare 69] R. I. Scare. "Recursion theory and Dedekind cuts." Trans. Amer. Math. Soc. 140 (1969), 271-294.
30. [Scare 87] R. I. Scare. Recursively Enumerable Sets and Degrees. Springer-Verlag, Berlin, 1987.
31. [Solovay 75] R. M. Solovay. Draft of a paper (or series of papers) on Chaitin's work … done for the most part during the period of Sept.-Dec. 1974, unpublished manuscript, IBM Thomas J. Watson Research Center. Yorktown Heights, New York, May 1975, 215 pp.
32. [Solovay 00] R. M. Solovay. "A version of Ω for which ZFC can not predict a single bit." in Finite Versus Infinite. Contributions to an Eternal Dilemma. C.S. Calude, G. Pǎun (eds.), pp. 323-334, Springer-Verlag, London, 2000.
33. [Staiger 91] L. Staiger. "The Kolmogorov complexity of real numbers." in Proc. Fundamentals of Computation Theory, Lecture Notes in Comput. Sci. No. 1684, G. Ciobanu and Gh. Pǎun (eds.), pp. 536-546, Springer-Verlag, Berlin, 1999.
34. [Stewart 91] I. Stewart. "Deciding the undecidable." Nature 352 (1991), 664-665.
35. [Weihrauch 87] K. Weihrauch. Computability, Springer-Verlag, Berlin, 1987.