## Notre Dame Journal of Formal Logic

### Program Size Complexity for Possibly Infinite Computations

#### Abstract

We define a program size complexity function as a variant of the prefix-free Kolmogorov complexity, based on Turing monotone machines performing possibly unending computations. We consider definitions of randomness and triviality for sequences in relative to the complexity. We prove that the classes of Martin-Löf random sequences and -random sequences coincide and that the -trivial sequences are exactly the recursive ones. We also study some properties of and compare it with other complexity functions. In particular, is different from , the prefix-free complexity of monotone machines with oracle A.

#### Article information

Source
Notre Dame J. Formal Logic Volume 46, Number 1 (2005), 51-64.

Dates
First available: 31 January 2005

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1107220673

Digital Object Identifier
doi:10.1305/ndjfl/1107220673

Mathematical Reviews number (MathSciNet)
MR2131546

Zentralblatt MATH identifier
02186750

#### Citation

Becher, Verónica; Figueira, Santiago; Nies, André; Picchi, Silvana. Program Size Complexity for Possibly Infinite Computations. Notre Dame Journal of Formal Logic 46 (2005), no. 1, 51--64. doi:10.1305/ndjfl/1107220673. http://projecteuclid.org/euclid.ndjfl/1107220673.

#### References

• [1] Becher, V., S. Daicz, and G. Chaitin, "A highly random number", pp. 55--68 in Combinatorics, Computability and Logic: Proceedings of the Third Discrete Mathematics and Theoretical Computer Science Conference (DMTCS'01), edited by C. S. Calude and M. J. Dineen and S. Sburlan, Springer-Verlag, London, 2001.
• [2]špace-1pt Chaitin, G. J., "A theory of program size formally identical to information theory", Journal of the Association for Computing Machinery, vol. 22 (1975), pp. 329--40.
• [3]špace-1pt Chaitin, G. J., "Algorithmic entropy of sets", Computers & Mathematics with Applications, vol. 2 (1976), pp. 233--45.
• [4]špace-1pt Chaitin, G. J., "Information-theoretic characterizations of recursive infinite strings", Theoretical Computer Science, vol. 2 (1976), pp. 45--48.
• [5]špace-1pt Downey, R. G., D. R. Hirschfeldt, A. Nies, and F. Stephan, "Trivial reals", pp. 103--31 in Proceedings of the 7th and 8th Asian Logic Conferences, Singapore University Press, Singapore, 2003.
• [6]špace-1pt Ferbus-Zanda, M., and S. Grigorieff, "Church, cardinal and ordinal representations of integers and kolmogorov complexity", in preparation, 2003.
• [7]špace-1pt Gacs, P., "On the symmetry of algorithmic information", Soviet Mathematics, Doklady (Akademiia Nauk SSSR. Doklady), vol. 15 (1974), pp. 1477--80.
• [8]špace-1pt Levin, L. A., "The concept of a random sequence", Doklady Akademii Nauk SSSR, vol. 212 (1973), pp. 548--50.
• [9]špace-1pt Levin, L. A., "Laws on the conservation (zero increase) of information, and questions on the foundations of probability theory", Problemy Peredači Informacii, vol. 10 (1974), pp. 30--35.
• [10]špace-1pt Li, M., and P. Vitanyi, An Introduction to Kolmogorov Complexity and Its Applications, 2d edition, Graduate Texts in Computer Science. Springer-Verlag, New York, 1997.
• [11]špace-1pt Martin-Löf, P., "The definition of random sequences", Information and Control, vol. 9 (1966), pp. 602--19.
• [12]špace-1pt Schnorr, C.-P., "Process complexity and effective random tests", Journal of Computer and System Sciences, vol. 7 (1973), pp. 376--88. Fourth Annual ACM Symposium on the Theory of Computing (Denver, Colo., 1972).
• [13]špace-1pt Solovay, R. M., "Draft of a paper (or series of papers) on Chaitin's work done for the most part during the period Sept. to Dec. 1974", 1974.
• [14]špace-1pt Uspensky, V. A., and A. Shen, "Relations between varieties of Kolmogorov complexities", Mathematical Systems Theory, vol. 29 (1996), pp. 271--92.
• [15]špace-1pt Zvonkin, A. K., and L. A. Levin, The complexity of finite objects and the basing of the concepts of information and randomness on the theory of algorithms,'' Uspekhi Matematicheskikh Nauk, vol. 25 (1970), pp. 85--127.