Skip to main content
×
Home
    • Aa
    • Aa

Calibrating Randomness

  • Rod Downey (a1), Denis R. Hirschfeldt (a2), André Nies (a3) and Sebastiaan A. Terwijn (a4)
Abstract

We report on some recent work centered on attempts to understand when one set is more random than another. We look at various methods of calibration by initial segment complexity, such as those introduced by Solovay [125], Downey, Hirschfeldt, and Nies [39], Downey, Hirschfeldt, and LaForte [36], and Downey [31]; as well as other methods such as lowness notions of Kučera and Terwijn [71], Terwijn and Zambella [133], Nies [101, 100], and Downey, Griffiths, and Reid [34]; higher level randomness notions going back to the work of Kurtz [73], Kautz [61], and Solovay [125]; and other calibrations of randomness based on definitions along the lines of Schnorr [117].

These notions have complex interrelationships, and connections to classical notions from computability theory such as relative computability and enumerability. Computability figures in obvious ways in definitions of effective randomness, but there are also applications of notions related to randomness in computability theory. For instance, an exciting by-product of the program we describe is a more-or-less natural requirement-free solution to Post's Problem, much along the lines of the Dekker deficiency set.

Copyright
Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

[1]K. Ambos-Spies and A. Kučera , Randomness in computability theory, Computability Theory and its Applications (Boulder, CO, 1999) (P. A. Cholak , S. Lempp , M. Lerman , and R. A. Shore , editors), Contemporary Mathematics, vol. 257, American Mathematical Society, 2000, pp. 114.

[5]G. Barmpalias , Computably enumerable sets in the Solovay and the strong weak truth table degrees, New Computational Paradigms: First Conference on Computability in Europe, CiE 2005, Amsterdam,The Netherlands, June 8–12, 2005 (S. B. Cooper , B. Löwe , and L. Torenvliet , editors), Lecture Notes in Computer Science, vol. 3526, Springer-Verlag, 2005, pp. 817.

[9]V. Becher , S. Daicz , and G. Chaitin , A highly random number, Combinatorics, Computability and Logic (Constanţa, 2001) (C. S. Calude , M. J. Dinneen , and S. Sburlan , editors), Discrete Mathematics and Theoretical Computer Science, Springer-Verlag, 2001, pp. 5568.

[13]J.-Y. Cai and J. Hartmanis , On Hausdorff and topological dimensions of the Kolmogorov complexity of the real line, Journal of Computer and System Sciences, vol. 49 (1994), pp. 605619.

[14]C. S. Calude , Information and randomness, an algorithmic perspective, Springer-Verlag, 1994, second edition, 2002.

[15]C. S. Calude and R. J. Coles , Program-size complexity of initial segments and domination reducibility, Jewels are forever (J. Karhumäki , H. Maurer , G. Păun , and G. Rozenberg , editors), Springer-Verlag, 1999, pp. 225237.

[17]C. S. Calude , P.H. Hertling , B. Khoussainov , and Y. Wang , Recursively enumerable reals and Chaitin Ω numbers, Theoretical Computer Science, vol. 255 (2001), pp. 125149, extended abstract in STACS 98, Lecture Notes in Computer Science, 1373, Springer-Verlag, Berlin, 1998, pp. 596–606.

[19]C. S. Calude , L. Staiger , and S. A. Terwijn , On partial randomness, Annals of Pure and Applied Logic, vol. 138 (2006), no. 1–3, pp. 2030.

[20]G. J. Chaitin , Atheory of program size formally identical to information theory, Journal of the ACM, vol. 22 (1975), pp. 329340.

[21]G. J. Chaitin , Algorithmic information theory, IBM Journal of Research and Development, vol. 21 (1977), pp. 350–359, 496.

[22]G. J. Chaitin , Algorithmic information theory, Cambridge University Press, 1987.

[23]G. J. Chaitin , Incompleteness theorems for random reals, Advances in Applied Mathematics, vol. 8 (1987), pp. 119146.

[24]A. V. Chernov , An. A. Muchnik , A. E. Romashchenko , A. Shen , and N. K. Vereshchagin , Upper semi-lattice of binary strings with the relation “x is simple conditional to y”, Theoretical Computer Science, vol. 271 (2002), pp. 6995.

[25]P. Cholak , R. Coles , R. Downey , and E. Herrmann , Automorphisms of the lattice of classes: perfect thin classes and a.n.c. degrees, Transactions of the American Mathematical Society, vol. 353 (2001), pp. 48994924.

[27]B. F. Csima and A. Montalbán , A minimal pair of K-degrees, Proceedings of the American Mathematical Society, vol. 134 (2006), pp. 14991502.

[28]G. Davie , Characterising the Martin-Löf random sequences using computably enumerable sets of measure one, Information Processing Letters, vol. 92 (2004), pp. 157160.

[33]R. Downey , E. Griffiths , and G. LaForte , On Schnorr and computable randomness, martingales, and machines, Mathematical Logic Quarterly, vol. 50 (2004), pp. 613627.

[34]R. Downey , E. Griffiths , and S. Reid , On Kurtz randomness, Theoretical Computer Science, vol. 321 (2004), pp. 249270.

[36]R. Downey , D. R. Hirschfeldt , and G. LaForte , Randomness and reducibility, Journal of Computer and System Sciences, vol. 68 (2004), pp. 96114, extended abstract in Mathematical Foundations of Computer Science 2001 (J. Sgall, A. Pultr, and P. Kolman, editors), Lecture Notes in Computer Science, 2136, Springer-Verlag, 2001, pp. 316–327.

[38]R. Downey , D. R. Hirschfeldt , J. S. Miller , and A. Nies , Relativizing Chaitin's halting probability, Journal of Mathematical Logic, vol. 5 (2005), pp. 167192.

[39]R. Downey , D.R. Hirschfeldt , and A. Nies , Randomness, computability, and density, SIAM Journal on Computing, vol. 31 (2002), pp. 11691183, extended abstract in STACS 2001 Proceedings (A. Ferreira and H. Reichel, editors), Lecture Notes in Computer Science, vol. 2010, Springer-Verlag, 2001, pp. 195–201.

[41]R. Downey , C. Jockusch , and M. Stob , Array nonrecursive sets and multiple permitting arguments, Recursion Theory Week (Oberwolfach, 1989) (K. Ambos-Spies , G. H. Müller , and G. E. Sacks , editors), Lecture Notes in Mathematics, vol. 1432, Springer-Verlag, 1990, pp. 141174.

[43]R. Downey and J. S. Miller , A basis theorem for classes of positive measure and jump inversion for random reals, Proceedings of the American Mathematical Society, vol. 134 (2006), pp. 283288.

[47]P. Gács , Every sequence is reducible to a random one, Information and Control, vol. 70 (1986), pp. 186192.

[49]F. Hausdorff , Dimension und äuβeres Maβ, Mathematische Annalen, vol. 79 (1919), pp. 157179.

[54]C. G. Jockusch Jr., The degrees of bi-immune sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 15 (1969), pp. 135140.

[55]C. G. Jockusch Jr., Three easy constructions of recursively enumerable sets, Logic Year 1979–80 (Proceedings of Seminars and Conferences in Mathematical Logic, University of Connecticut, Storrs, Conn., 1979/80) (M. Lerman , J. H. Schmerl , and R. I. Soare , editors), Lecture Notes in Mathematics, vol. 859, Springer-Verlag, 1981, pp. 8391.

[56]C. G. Jockusch , Jr., M. Lerman , R. I. Soare , and R. Solovay , Recursively enumerable sets modulo iterated jumps and extensions of Arslanov's completeness criterion, The Journal of Symbolic Logic, vol. 54 (1989), pp. 12881323.

[57]C. G. Jockusch Jr., and R. A. Shore , Pseudo-jump operators I: the r.e. case, Transactions of the American Mathematical Society, vol. 275 (1983), pp. 599609.

[58]C. G. Jockusch Jr., and R. I. Soare , classes and degrees of theories, Transactions of the American Mathematical Society, vol. 173 (1972), pp. 3356.

[59]T. Kamae , Subsequences of normal sequences, Israel Journal of Mathematics, vol. 16 (1973), pp. 121149.

[60]H. P. Katseff , Complexity dips in random infinite binary sequences, Information and Control, vol. 38 (1978), pp. 258263.

[62]A. S. Kechris and Y. Moschovakis (editors), Cabal Seminar 76–77, Lecture Notes in Mathematics, vol. 689, Springer-Verlag, 1978.

[63]B. Kjos-Hanssen , A. Nies , and F. Stephan , Lowness for the class of Schnorr random reals, SIAM Journal on Computing, vol. 35 (2006), no. 3, pp. 647657, preliminary results in [11].

[64]K.-I Ko , On the notion of infinite pseudorandom sequences, Theoretical Computer Science, vol. 48 (1986), pp. 933.

[67]A. Kučera , Measure, -classes and complete extensions of PA, Recursion Theory Week (H.-D. Ebbinghaus , G. H. Müller , and G. E. Sacks , editors), Lecture Notes in Mathematics, vol. 1141, Springer-Verlag, 1985, pp. 245259.

[69]A. Kučera , On relative randomness, Annals of Pure and Applied Logic, vol. 63 (1993), pp. 6167.

[70]A. Kučera and T.A. Slaman , Randomness and recursive enumerability, SIAM Journal on Computing, vol. 31 (2001), pp. 199211.

[72]M. Kummer , Kolmogorov complexity and instance complexity of recursively enumerable sets, SIAM Journal on Computing, vol. 25 (1996), pp. 11231143.

[78]M. Li and P. Vitányi , An introduction to Kolmogorov complexity and its applications, 2nd ed., Springer-Verlag, 1997.

[79]D. Loveland , A variant of the Kolmogorov concept of complexity, Information and Control, vol. 15 (1969), pp. 510526.

[80]J.H. Lutz , Category and measure in complexity classes, SIAM Journal on Computing, vol. 19 (1990), pp. 11001131.

[81]J.H. Lutz , Almost everywhere high nonuniform complexity, Journal of Computer and System Sciences, vol. 44 (1992), pp. 220258.

[82]J.H. Lutz , Dimension in complexity classes, SIAM Journal on Computing, vol. 32 (2003), pp. 12361259, extended abstract in 15th Annual IEEE Conference on Computational Complexity (Florence, 2000) (F. Titsworth, editor), IEEE Computer Society, Los Alamitos, CA, 2000, pp. 158–169).

[83]J.H. Lutz , The dimensions of individual strings and sequences, Information and Computation, vol. 187 (2003), pp. 4979, preliminary version: Gales and the constructive dimension of individual sequences, in Proceedings of the 27th International Colloquium on Automata, Languages, and Programming, (U. Montanari, J.D.P. Rolim, and E. Welzl, editors), Springer-Verlag, 2000, pp. 902–913.

[84]J.H. Lutz , Effective fractal dimensions, Mathematical Logic Quarterly, vol. 51 (2005), pp. 6272.

[85]P. Martin-Löf , The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602619.

[86]E. Mayordomo , A Kolmogorov complexity characterization of constructive Hausdorff dimension, Information Processing Letters, vol. 84 (2002), pp. 13.

[90]W. Merkle , J.S. Miller , A. Nies , J. Reimann , and F. Stephan , Kolmogorov-Loveland randomness and stochasticity, Annals of Pure and Applied Logic, vol. 138 (2006), no. 1–3, pp. 183210, preliminary version in STACS 2005, Lecture Notes in Computer Science, vol. 3404, Springer-Verlag, 2005, pp. 422–433.

[96]W. Miller and D. A. Martin , The degrees of hyperimmune sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 159166.

[97]A. A. Muchnik , A. L. Semenov , and V. A. Uspensky , Mathematical metaphysics of randomness, Theoretical Computer Science, vol. 207 (1998), pp. 263317.

[98]A. Nabutovsky and S. Weinberger , The fractal nature of Riem/Diff I, Geometriae Dedicata, vol. 101 (2003), pp. 154.

[99]A. Nies , Effectively dense Boolean algebras and their applications, Transactions of the American Mathematical Society, vol. 352 (2000), pp. 49895012.

[100]A. Nies , Lowness properties and randomness, Advances in Mathematics, vol. 197 (2005), pp. 274305.

[108]J. C. Oxtoby , Measure and category, 2nd ed., Springer-Verlag, 1980.

[109]J. Raisonnier , A mathematical proof of S. Shelah's theorem on the measure problem and related results, Israel Journal of Mathematics, vol. 48 (1984), pp. 4856.

[116]C.-P. Schnorr , A unified approach to the definition of a random sequence, Mathematical Systems Theory, vol. 5 (1971), pp. 246258.

, , Lecture Notes in Mathematics, vol. , , .[117]C.-P. Schnorr Zufälligkeit und Wahrscheinlichkeit 218 Springer-Verlag1971

[118]C.-P. Schnorr , Process complexity and effective random tests, Journal of Computer and System Sciences, vol. 7 (1973), pp. 376388.

[119]C. E. Shannon , The mathematical theory of communication, Bell System Technical Journal, vol. 27 (1948), pp. 379–423, 623656.

[120]J. H. Silver , Counting the number of equivalence classes of Borel and coanalytic equivalence relations, Annals of Mathematical Logic, vol. 18 (1980), pp. 128.

[121]R. I. Soare , Recursively enumerable sets and degrees, Springer-Verlag, 1987.

[124]R. Solovay , A model of set theory in which every set of reals is Lebesgue measurable, Annals of Mathematics, vol. 92 (1970), pp. 156.

[126]L. Staiger , Kolmogorov complexity and Hausdorff dimension, Information and Computation, vol. 103 (1993), pp. 159194.

[127]L. Staiger , A tight upper bound on Kolmogorov complexity and uniformly optimal prediction, Theory of Computing Systems, vol. 31 (1998), pp. 215229.

[128]L. Staiger , Constructive dimension equals Kolmogorov complexity, Information Processing Letters, vol. 93 (2005), pp. 149153, preliminary version: Research Report CDMTCS-210, University of Auckland, January 2003.

[130]K. Tadaki , A generalization of Chaitin's halting probability Ω and halting self-similar sets, Hokkaido Mathematical Journal, vol. 31 (2002), pp. 219253.

[137]R. von Mises , Grundlagen der Wahrscheinlichkeitsrechnung, Mathematische Zeitschrift, vol. 5 (1919), pp. 5299.

[139]Y. Wang , A separation of two randomness concepts, Information Processing Letters, vol. 69 (1999), pp. 115118.

[141]L. Yu and D. Ding , There are 20 many H-degrees in the random reals, Proceedings of the American Mathematical Society, vol. 132 (2004), pp. 24612464.

[143]L. Yu , D. Ding , and R. Downey , The Kolmogorov complexity of random reals, Annals of Pure and Applied Logic, vol. 129 (2004), pp. 163180.

[145]A. K. Zvonkin and L. A. Levin , The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms, Russian Mathematical Surveys, vol. 25 (1970), pp. 83124.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of Symbolic Logic
  • ISSN:
  • EISSN:
  • URL: /core/journals/bulletin-of-symbolic-logic
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×