Randomness, computability, and density

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

Volumn 2010, Issue , 2001, Pages 195-205

  Downey, Rod G ^a   Hirschfeldt, Denis R ^a   Nies, André ^b  

^a VICTORIA UNIVERSITY OF WELLINGTON   (New Zealand)

^b UNIVERSITY OF CHICAGO   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

RANDOM PROCESSES; RATIONAL FUNCTIONS; STRUCTURAL PROPERTIES;

COMPUTABLE FUNCTIONS; COMPUTABLY ENUMERABLE; DEGREE STRUCTURE; POSITIVE REAL; RATE OF CONVERGENCE; RELATIVE RANDOMNESS; SOLOVAY DEGREES; SOLOVAY REDUCIBILITY;

COMPUTABILITY AND DECIDABILITY;

EID: 34247233642     PISSN: 03029743     EISSN: 16113349     Source Type: Book Series    
DOI: 10.1007/3-540-44693-1_17     Document Type: Conference Paper

References (33)

1. K. Ambos-Spies, K. Weihrauch, and X. Zheng, Weakly computable real numbers, to appear.
2. P. Boldi and S. Vigna, Equality is a jump, Theoret. Comput. Sci. 219 (1999) 49-64.
3. C. S. Calude, A characterization of c.e. random reals, to appear in Theoret. Comput. Sci.
4. C. S. Calude, Information and Randomness, an Algorithmic Perspective, Monographs in Theoretical Computer Science (Springer-Verlag, Berlin, 1994).
5. C. S. Calude and G. Chaitin, Randomness everywhere, Nature 400 (1999) 319-320.
6. C. S. Calude, R. Coles, P. H. Hertling, and B. Khoussainov, Degree-theoretic aspects of computably enumerable reals, in S. B. Cooper and J. K. Truss (eds.), Models and Computability (Leeds, 1997), vol. 259 of London Math. Soc. Lecture Note Ser. (Cambridge Univ. Press, Cambridge, 1999) 23-39.
7. C. S. Calude, P. H. Hertling, B. Khoussainov, and Y. Wang, Recursively enumerable reals and Chaitin numbers, Centre for Discrete Mathematics and Theoretical Computer Science Research Report Series 59, University of Auckland (October 1997), extended abstract in STACS 98, Lecture Notes in Comput. Sci. 1373, Springer, Berlin, 1998, 596-606.
8. C. S. Calude and A. Nies, Chaitin numbers and strong reducibilities, J. Univ. Comp. Sci. 3 (1998) 1162-1166.
9. G. S. Ceĭtin, A pseudofundamental sequence that is not equivalent to a monotone one, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 20 (1971) 263-271, 290, in Russian with English summary.
10. G. J. Chaitin, A theory of program size formally identical to information theory, J. Assoc. Comput. Mach. 22 (1975) 329-340, reprinted in [13].
11. G. J. Chaitin, Algorithmic information theory, IBM J. Res. Develop. 21 (1977) 350-359, 496, reprinted in [13].
12. G. J. Chaitin, Incompleteness theorems for random reals, Adv. in Appl. Math. 8 (1987) 119-146, reprinted in [13].
13. G. J. Chaitin, Information, Randomness & Incompleteness, vol. 8 of Series in Computer Science, 2nd ed. (World Scientific, River Edge, NJ, 1990).
14. R. J. Coles, R. Downey, and G. LaForte, Randomness and reducibility I, to appear.
15. R. Downey, Computability, definability, and algebraic structures, to appear in Proceedings of the 7th Asian Logic Conference, Taiwan.
16. R. Downey and G. LaForte, Presentations of computably enumerable reals, to appear.
17. C. Ho, Relatively recursive reals and real functions, Theoret. Comput. Sci. 210 (1999) 99-120.
18. K.-I. Ko, On the definition of some complexity classes of real numbers, Math. Systems Theory 16 (1983) 95-100.
19. K.-I. Ko, On the continued fraction representation of computable real numbers, Theoret. Comput. Sci. 47 (1986) 299-313.
20. A. N. Kolmogorov, Three approaches to the quantitative definition of information, Internat. J. Comput. Math. 2 (1968) 157-168.
21. A. H. Lachlan, Recursive real numbers, J. Symbolic Logic 28 (1963) 1-16.
22. L. Levin, On the notion of a random sequence, Soviet Math. Dokl. 14 (1973) 1413-1416.
23. L. Levin, The various measures of the complexity of finite objects (an axiomatic description), Soviet Math. Dokl. 17 (1976) 522-526.
24. M. Li and P. Vitanyi, An Introduction to Kolmogorov Complexity and its Applications, 2nd ed. (Springer-Verlag, New York, 1997).
25. P. Martin-Löf, The definition of random sequences, Inform. and Control 9 (1966) 602-619.
26. H. Rice, Recursive real numbers, Proc. Amer. Math. Soc. 5 (1954) 784-791.
27. C.-P. Schnorr, Process complexity and effective random tests, J. Comput. System Sci. 7 (1973) 376-388.
28. T. Slaman, Randomness and recursive enumerability, to appear.
29. R. Soare, Cohesive sets and recursively enumerable Dedekind cuts, Pacific J. Math. 31 (1969) 215-231.
30. R. J. Solomonoff, A formal theory of inductive inference I, Inform. and Control 7 (1964) 1-22.
31. R. J. Solomonoff, A formal theory of inductive inference II, Inform. and Control 7 (1964) 224-254.
32. 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 (May 1975), unpublished manuscript, IBM Thomas J. Watson Research Center, Yorktown Heights, NY, 215 pages.
33. R. M. Solovay, On random r.e. sets, in A. Arruda, N. Da Costa, and R. Chuaqui (eds.), Non-Classical Logics, Model Theory, and Computability. Proceedings of the Third Latin-American Symposium on Mathematical Logic, Campinas, July 11-17, 1976, vol. 89 of Studies in Logic and the Foundations of Mathematics (North- Holland, Amsterdam, 1977) 283-307.