Skip to main content Accessibility help
×
Home

A LEARNING-THEORETIC CHARACTERISATION OF MARTIN-LÖF RANDOMNESS AND SCHNORR RANDOMNESS

  • FRANCESCA ZAFFORA BLANDO (a1)

Abstract

Numerous learning tasks can be described as the process of extrapolating patterns from observed data. One of the driving intuitions behind the theory of algorithmic randomness is that randomness amounts to the absence of any effectively detectable patterns: it is thus natural to regard randomness as antithetical to inductive learning. Osherson and Weinstein [11] draw upon the identification of randomness with unlearnability to introduce a learning-theoretic framework (in the spirit of formal learning theory) for modelling algorithmic randomness. They define two success criteria—specifying under what conditions a pattern may be said to have been detected by a computable learning function—and prove that the collections of data sequences on which these criteria cannot be satisfied correspond to the set of weak 1-randoms and the set of weak 2-randoms, respectively. This learning-theoretic approach affords an intuitive perspective on algorithmic randomness, and it invites the question of whether restricting attention to learning-theoretic success criteria comes at an expressivity cost. In other words, is the framework expressive enough to capture most core algorithmic randomness notions and, in particular, Martin-Löf randomness—arguably, the most prominent algorithmic randomness notion in the literature? In this article, we answer the latter question in the affirmative by providing a learning-theoretic characterisation of Martin-Löf randomness. We then show that Schnorr randomness, another central algorithmic randomness notion, also admits a learning-theoretic characterisation in this setting.

Copyright

Corresponding author

*DEPARTMENT OF PHILOSOPHY AND LOGICAL DYNAMICS LAB (CSLI) STANFORD UNIVERSITY STANFORD, CA 94305-2155, USA E-mail: fzaffora@stanford.edu

References

Hide All
[1]Chaitin, G. J. (1966). On the length of programs for computing finite binary sequences. Journal of the Association for Computing Machinery, 13, 547569.
[2]Gaifman, H. & Snir, M. (1982). Probabilities over rich languages, testing and randomness. The Journal of Symbolic Logic, 47, 495548.
[3]Gold, E. (1967). Language identification in the limit. Information and Control, 10, 447474.
[4]Kolmogorov, A. N. (1965). Three approaches to the quantitative definition of information. Problems of Information Transmission, 1, 17.
[5]Kurtz, S. A. (1981). Randomness and Genericity in the Degrees of Unsolvability. Ph.D. Dissertation, University of Illinois at Urbana-Champaign.
[6]Levin, L. A. (1973). On the notion of a random sequence. Soviet Mathematics Doklady, 14, 14131416.
[7]Levin, L. A. (1976). Uniform tests of randomness. Soviet Mathematics Doklady, 17 (2), 337340.
[8]Martin-Löf, P. (1966). The definition of a random sequence. Information and Control, 9, 602619.
[9]Miyabe, K. (2013). L 1-Computability, layerwise computability and Solovay reducibility. Computability, 2, 1529.
[10]Osherson, D., Stob, M., & Weinstein, S. (1986). Systems that Learn (first edition). Cambridge, MA: MIT Press.
[11]Osherson, D. & Weinstein, S. (2008). Recognizing strong random reals. The Review of Symbolic Logic, 1(1), 5663.
[12]Putnam, H. (1963). ‘Degree of confirmation’ and inductive logic. In Schilpp, P. A., editor. The Philosophy of Rudolf Carnap. La Salle, III: Open Court, pp. 761783.
[13]Schnorr, C. P. (1971a). A unified approach to the definition of a random sequence. Mathematical Systems Theory, 5, 246258.
[14]Schnorr, C. P. (1971b). Zufälligkeit und Wahrscheinlichkeit: Eine algorithmische Begründung der Wahrscheinlichkeitstheorie. Lecture Notes in Mathematics, Vol. 218. Berlin, Heidelberg: Springer-Verlag.
[15]Solomonoff, R. J. (1964). A formal theory of inductive inference, I and II. Information and Control, 7, 1–22 and 224254.
[16]Ville, J. (1939). Étude critique de la notion de collectif. Monographies des probabilités. Paris: Gauthiers-Villars.

Keywords

A LEARNING-THEORETIC CHARACTERISATION OF MARTIN-LÖF RANDOMNESS AND SCHNORR RANDOMNESS

  • FRANCESCA ZAFFORA BLANDO (a1)

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.