Abstract
How random is a real? Given two reals, which is more random? If we partition reals into equivalence classes of reals of the “same degrees of randomness”, what does the resulting structure look like? The goal of this paper is to look at questions like these, specifically by studying the properties of reducibilities that act as measures of relative randomness, as embodied in the concept of initial-segment complexity. One such reducibility, called domination or Solovay reducibility, was introduced by Solovay [34], and has been studied by Calude, Hertling, Khoussainov, and Wang [8], Calude [3], Kučera and Slaman [22], and Downey, Hirschfeldt, and Nies [15], among others. Solovay reducibility has proved to be a powerful tool in the study of randomness of effectively presented reals. Motivated by certain shortcomings of Solovay reducibility, which we will discuss below, we introduce two new reducibilities and study, among other things, the relationships between these various measures of relative randomness.
The authors’ research was supported by the Marsden Fund for Basic Science.
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References
K. Ambos-Spies and A. Kučera, Randomness in computability theory, in P. A. Cholak, S. Lempp, M. Lerman, and R. A. Shore (eds.), Computability Theory and Its Applications (Boulder, CO, 1999), vol. 257 of Contemp. Math. (Amer. Math. Soc., Providence, RI, 2000) 1–14.
K. Ambos-Spies, K. Weihrauch, and X. Zheng, Weakly computable real numbers, J. Complexity 16 (2000) 676–690.
C. S. Calude, A characterization of c.e. random reals, to appear in Theoret. Comput. Sci.
C. S. Calude, Information and Randomness, an Algorithmic Perspective, Monographs in Theoretical Computer Science (Springer-Verlag, Berlin, 1994).
C. S. Calude and G. Chaitin, Randomness everywhere, Nature 400 (1999) 319–320.
C. S. Calude and R. J. Coles, Program-size complexity of initial segments and domination reducibility, in J. Karhumäki, H. Maurer, G. Păun, and G. Rozenberg (eds.), Jewels Are Forever (Springer-Verlag, Berlin, 1999) 225–237.
C. S. Calude, R. J. 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.
C. S. Calude, P. H. Hertling, B. Khoussainov, and Y. Wang, Recursively enumerable reals and Chaitin Ω 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.
C. S. Calude and A. Nies, Chaitin Ω numbers and strong reducibilities, J. Univ. Comp. Sci. 3 (1998) 1162–1166.
G. S. Ceitin, 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.
G. J. Chaitin, A theory of program size formally identical to information theory, J. Assoc. Comput. Mach. 22 (1975) 329–340, reprinted in [14].
G. J. Chaitin, Algorithmic information theory, IBM J. Res. Develop. 21 (1977) 350–359, 496, reprinted in [14].
G. J. Chaitin, Incompleteness theorems for random reals, Adv. in Appl. Math. 8 (1987) 119–146, reprinted in [14].
G. J. Chaitin, Information, Randomness & Incompleteness, vol. 8 of Series in Computer Science, 2nd ed. (World Scientific, River Edge, NJ, 1990).
R. Downey, D. R. Hirschfeldt, and A. Nies, Randomness, computability, and density, to appear in SIAM J. Comp. (extended abstract in A. Ferreira and H. Reichel (eds.), STACS 2001 Proceedings, Lecture Notes in Computer Science 2010 (Springer, 2001), 195–205).
R. Downey and G. LaForte, Presentations of computably enumerable reals, to appear in Theoret. Comput. Sci.
L. Fortnow, Kolmogorov complexity, to appear in R. G. Downey and D. R. Hirschfeldt (eds.), Aspects of Complexity, Minicourses in Algorithmics, Complexity, and Computational Algebra, NZMRI Mathematics Summer Meeting, Kaikoura, New Zealand, January 7–15, 2000.
C. Ho, Relatively recursive reals and real functions, Theoret. Comput. Sci. 210 (1999) 99–120.
K.-I. Ko, On the definition of some complexity classes of real numbers, Math. Systems Theory 16 (1983) 95–100.
K.-I. Ko, On the continued fraction representation of computable real numbers, Theoret. Comput. Sci. 47 (1986) 299–313.
A. N. Kolmogorov, Three approaches to the quantitative definition of information, Internat. J. Comput. Math. 2 (1968) 157–168.
A. Kučera and T. Slaman, Randomness and recursive enumerability, to appear in SIAM J. Comput.
A. H. Lachlan, Recursive real numbers, J. Symbolic Logic 28 (1963) 1–16.
L. Levin, On the notion of a random sequence, Soviet Math. Dokl. 14 (1973) 1413–1416.
L. Levin, The various measures of the complexity of finite objects (an axiomatic description), Soviet Math. Dokl. 17 (1976) 522–526.
M. Li and P. Vitanyi, An Introduction to Kolmogorov Complexity and its Applications, 2nd ed. (Springer-Verlag, New York, 1997).
P. Martin-Löf, The definition of random sequences, Inform. and Control 9 (1966) 602–619.
H. Rice, Recursive real numbers, Proc. Amer. Math. Soc. 5 (1954) 784–791.
C.-P. Schnorr, Process complexity and effective random tests, J. Comput. System Sci. 7 (1973) 376–388.
R. Soare, Cohesive sets and recursively enumerable Dedekind cuts, Pacific J. Math. 31 (1969) 215–231.
R. Soare, Recursion theory and Dedekind cuts, Trans. Amer. Math. Soc. 140 (1969) 271–294.
R. J. Solomonoff, A formal theory of inductive inference I, Inform. and Control 7 (1964) 1–22.
R. J. Solomonoff, A formal theory of inductive inference II, Inform. and Control 7 (1964) 224–254.
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.
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.
M. van Lambalgen, Random Sequences, PhD Thesis, University of Amsterdam (1987).
D. Zambella, On sequences with simple initial segments, Tech. rep., Institute for Language, Logic and Information, University of Amsterdam (1990).
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Downey, R.G., Hirschfeldt, D.R., La Forte, G. (2001). Randomness and Reductibility. In: Sgall, J., Pultr, A., Kolman, P. (eds) Mathematical Foundations of Computer Science 2001. MFCS 2001. Lecture Notes in Computer Science, vol 2136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44683-4_28
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