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Martin-Löf random and PA-complete sets

Published online by Cambridge University Press:  31 March 2017

By
Frank Stephan
Edited by
Zoé Chatzidakis ,
Peter Koepke  and
Wolfram Pohlers
Zoé Chatzidakis
Affiliation:
Université de Paris VII (Denis Diderot)
Peter Koepke
Affiliation:
Rheinische Friedrich-Wilhelms-Universität Bonn
Wolfram Pohlers
Affiliation:
Westfälische Wilhelms-Universität Münster, Germany
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Summary

Abstract. A set A is Martin-Lof random iff the class ﹛A﹜ does not have measure 0. A set A is PA-complete if one can compute relative to A a consistent and complete extension of Peano Arithmetic. It is shown that every Martin-Lof random set either permits to solve the halting problem K or is not PA-complete. This result implies a negative answer to the question of Ambos-Spies and Kucera whether there is a Martin-Lof random set not above K which is also PA-complete.

Introduction. Gacs [3] and Kucera [7, 8] showed that every set can be computed relative to a Martin-Lof random set. In particular, for every set B there is a Martin-Lof random set A such that where K is the halting problem. A can even be chosen such that the reduction from B to A is a weak truth-table reduction, Merkle and Mihailovic [12] give a simplified proof for this fact.

A natural question is whether it is necessary to go up to the degree of in order to find the random set A. Martin-L of random sets can be found below every set which is PA-complete, so there are Martin-Lof random sets in low and in hyperimmune-free Turing degrees. A set A is called PA-complete if one can compute relative to A a complete and consistent extension of the set of first-order formulas provable in Peano Arithmetic. An easier and equivalent definition of being PA-complete is to say that given any partial-recursive and ﹛0, 1﹜-valued function, one can compute relative to A a total extension Ψ of. One can of course choose Ψ such that also Ψ is ﹛0, 1﹜-valued.

Extending all possible ﹛0, 1﹜-valued partial-recursive functions is as difficult as to compute a ﹛0, 1﹜-valued DNR function. A diagonally nonrecursive (DNR) function f satisfies whenever is defined.

Type
Chapter
Information
Logic Colloquium '02 , pp. 342 - 348
Publisher: Cambridge University Press
Print publication year: 2006

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References

[1] Klaus, Ambos-Spies and Antonín, Kučera, Randomness in computability theory Computability Theory and its Applications. Current Trends and Open Problems. Proceedings of a 1999 AMS-IMS-SIAM Joint Summer Research Conference, Boulder, Colorado, USA, June 13–17, 1999 (Peter A. Cholak, Steffen Lempp, Manuel Lerman, and Richard A. Shore, editors), Contemporary Mathematics, vol. 257, American Mathematical Society, Providence, 2000, pp. 1–14.Google Scholar
[2] Marat M., Arslanov, Some generalizations of a fixed-point theorem Soviet Mathematics, (1981), no. 25, pp. 1–10, translated from Izvestiya Vysshikh Uchebnykh Zavedenij, Matematika, (1981), no. 228, pp. 9–16.Google Scholar
[3] Péter, Gács, Every sequence is reducible to a random one Information and Control, vol. 70 (1986), no. 2-3, pp. 186–192.
[4] Carl G., Jockusch, Jr., Manuel, Lerman, Robert I., Soare, and Robert M., Solovay, Recursively enumerable sets modulo iterated jumps and extensions of Arslanov's completeness criterion The Journal of Symbolic Logic, vol. 54 (1989), no. 4, pp. 1288–1323.
[5] Carl G., Jockusch, Jr. and Robert I., Soare, Π01 classes and degrees of theories Transactions of the American Mathematical Society, vol. 173 (1972), pp. 33–56.
[6] Steven M., Kautz, Degrees of Random Sets, Ph.D. thesis, Cornell University, 1991.
[7] Antonín, Kučera, Measure, Π01-classes and complete extensions of PA, Recursion Theory Week, Proceedings of a Conference Held in Oberwolfach, West Germany, April 15–21, 1984 (Heinz-Dieter, Ebbinghaus, Gert H., Müller, and Gerald E., Sacks, editors), Lecture Notes in Mathematics, vol. 1141, Springer, Berlin, 1985, pp. 245–259.
[8] Antonín, Kučera, Onthe use of diagonally nonrecursive functions Logic Colloquium 1987. Proceedings of the Colloquium Held in Granada, Spain, July 20–25, 1987 (Heinz-Dieter, Ebbinghaus, José Fernandez-Prida, Manuel Garrido, Daniel Lascar, and Mario Rodrıguez Artalejo, editors), Studies in Logic and the Foundations of Mathematics, vol. 129, North-Holland, Amsterdam, 1989, pp. 219–239.
[9] Antonín, Kučera and Theodore A., Slaman, Randomness and recursive enumerability SIAM Journal on Computing, vol. 31 (2001), no. 1, pp. 199–211.
[10] Ming, Li and Paul, Vitányi, An Introduction to Kolmogorov Complexity and its Applications, 2 ed., Springer, New York, 1997.
[11] Per, Martin-Löf, The definition of random sequences Information and Computation, vol. 9 (1966), pp. 602–619.
[12] Wolfgang, Merkle and Nenad, Mihailović, On the construction of effective random sets Mathematical Foundations of Computer Science 2002, 27th International Symposium, MFCS 2002, Warsaw, Poland, August 26–30, 2002, Proceedings (Krzysztof Diks and Wojciech Rytter, editors), Lecture Notes in Computer Science, vol. 2420, Springer, Berlin, 2002, pp. 568–580.
[13] Piergiorgio, Odifreddi, Classical Recursion Theory, North-Holland, Amsterdam, 1989.
[14] DanaScott and Stanley, Tennenbaum, On the degrees of complete extensions of arithmetic Notices of the American Mathematical Society, vol. 7 (1960), pp. 242–243.
[15] Robert I., Soare, Recursively Enumerable Sets and Degrees, Springer, Berlin, 1987.

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