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Algorithmic information theory

Published online by Cambridge University Press:  12 March 2014

Michiel van Lambalgen
Michiel van Lambalgen*
Affiliation:
Department of Mathematics and Computer Science, University of Amsterdam, 1018 WB Amsterdam, The Netherlands
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Abstract

We present a critical discussion of the claim (most forcefully propounded by Chaitin) that algorithmic information theory sheds new light on Gödel's first incompleteness theorem.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 54 , Issue 4 , December 1989 , pp. 1389 - 1400
Copyright
Copyright © Association for Symbolic Logic 1989

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References

REFERENCES

[1] Chaitin, G. J., Information theoretic limitations on formal systems, Journal of the Association for Computing Machinery, vol. 21 (1974), pp. 403–424.CrossRefGoogle Scholar
[2] Chaitin, G. J., A theory of program size formally identical to information theory, Journal of the Association for Computing Machinery, vol. 22 (1975), pp. 329–340.CrossRefGoogle Scholar
[3] Chaitin, G. J., Algorithmic information theory, IBM Journal of Research and Development, vol. 21 (1977), pp. 350–359, 496.CrossRefGoogle Scholar
[4] Chaitin, G. J., Gödel's theorem and information, International Journal of Theoretical Physics, vol. 21 (1982), pp. 941–954.CrossRefGoogle Scholar
[5] Chaitin, G. J., Incompleteness theorems for random reals, Advances in Applied Mathematics, vol. 8 (1987), pp. 119–146.CrossRefGoogle Scholar
[6] Chaitin, G. J., Algorithmic information theory, Cambridge Tracts in Computer Science, vol. 1, Cambridge University Press, Cambridge, 1987.CrossRefGoogle Scholar
[7] Choquet, G., Repartition des nombres α(3/2) n mod 1; mesures et ensembles associés, Comptes Rendus des Seances de l'Académie des Sciences, Séries A–B, vol. 290 (1980). pp. A575–A580.Google Scholar
[8] Davis, M., What is a computation? Mathematics today (Steen, L.A., editor), Springer-Verlag, Berlin, 1978, pp. 241–267.Google Scholar
[9] Davis, M., Matijasevič, Y. and Robinson, J., Hubert's tenth problem. Diophantine equations: positive aspects of a negative solution, Mathematical developments arising from Hilbert problems (Browder, F. E., editor), American Mathematical Society, Providence, Rhode Island, 1976, pp. 323–378.CrossRefGoogle Scholar
[10] Dekking, F. M., Regularity and irregularity of sequences generated by automata, Séminaire de theorie des nombres, 1979–1980, Université de Bordeaux I, Talence, 1980, Expose 9.Google Scholar
[11] Ko, Ken-I, On the notion of infinite pseudorandom sequences, Theoretical Computer Science, vol. 48 (1986), pp. 9–33.Google Scholar
[12] Kolmogorov, A. N., On tables of random numbers, Sankhyā Series A, vol. 25 (1963), pp. 369–376.Google Scholar
[13] Kolmogorov, A. N., Three approaches to the definition of the concept of “amount of information”, Selected translations in mathematical statistics and probability, vol. 7, American Mathematical Society, Providence, Rhode Island, 1968, pp. 293–302.Google Scholar
[14] Kolmogorov, A. N., The logical basis for information theory and probability theory, IEEE Transactions on Information Theory, vol. IT-14 (1968), pp. 662–664.CrossRefGoogle Scholar
[15] Kolmogorov, A. N., Combinatorial basis for information theory and probability theory, Russian Mathematical Surveys, vol. 38 (1983), no. 4, pp. 29–40.CrossRefGoogle Scholar
[16] Kolmogorov, A. N., On logical foundations of probability theory, Probability theory and mathematical statistics (proceedings of the fourth USSR-Japan symposium, Toilisi, 1982; Ito, K. and Prokhorov, J. V., editors), Lecture Notes in Mathematics, vol. 1021, Springer-Verlag, Berlin, 1984, pp. 1–5.Google Scholar
[17] Kreisel, G. and Levy, A., Reflection principles and their use for establishing the complexity of axiomatic systems, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 97–142.CrossRefGoogle Scholar
[18] van Lambalgen, M., Random sequences, Ph.D. thesis, Department of Mathematics, University of Amsterdam, Amsterdam, 1987.Google Scholar
[19] Martin-Löf, P., a) Algorithmen und zufällige Folgen, lecture notes, University of Erlangen, Erlangen, 1966. b) Complexity oscillations in infinite binary sequences, Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete , vol. 19 (1971), pp. 225–230.Google Scholar
[20] Martin-Löf, P., The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602–619.CrossRefGoogle Scholar
[21] Rogers, H., Theory of recursive functions and effective compulability, McGraw-Hill, New York, 1967.Google Scholar
[22] Schnorr, C. P., Zufälligkeit und Wahrscheinlichkeit, Lecture Notes in Mathematics, vol. 218, Springer-Verlag, Berlin, 1971.CrossRefGoogle Scholar
[23] Smorynski, C., The incompleteness theorems, Handbook of mathematical logic (Barwise, J., editor), North-Holland, Amsterdam, 1977, pp. 821–865.CrossRefGoogle Scholar
[24] Soare, R.I., Recursively enumerable sets and degrees, Springer-Verlag, Berlin, 1986.Google Scholar
[25] Tymoczko, T. (editor), New directions in the philosophy of mathematics, Birkhäuser, Boston, Massachusetts, 1986.Google Scholar
[26] Vitanyi, P. M. B. and Li, M., Two decades of applied Kolmogorov complexity, Technical Report TR-08-88, Computer Center, Harvard University, Cambridge, Massachusetts, 1988.Google Scholar
[27] Ford, J., Directions in classical chaos, Directions in chaos, vol. 1 (Hao, B. L., editor), World Scientific Publishing, Singapore, 1987, pp. 1–16.Google Scholar