Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-23T23:45:28.668Z Has data issue: false hasContentIssue false
  • English
  • Français

An incomplete set of shortest descriptions

Published online by Cambridge University Press:  12 March 2014

Frank Stephan  and
Jason Teutsch
Frank Stephan
Affiliation:
National University of Singapore, Department of Mathematics,10 Lower Kent Ridge Road, Singapore119076, Republic of Singapore, E-mail: fstephan@comp.nus.edu.sg
Jason Teutsch
Affiliation:
Ruprecht-Karls-Universität Heidelberg, Institut Für Informatik, Im Neuenheimer Feld 294, 69120 Heidelberg, Germany, E-mail: teutsch@math.uni-heidelberg.de
Get access Rights & Permissions [Opens in a new window]

Abstract

The truth-table degree of the set of shortest programs remains an outstanding problem in recursion theory. We examine two related sets, the set of shortest descriptions and the set of domain-random strings, and show that the truth-table degrees of these sets depend on the underlying acceptable numbering. We achieve some additional properties for the truth-table incomplete versions of these sets, namely retraceability and approximability. We give priority-free constructions of bounded truth-table chains and bounded truth-table antichains inside the truth-table complete degree by identifying an acceptable set of domain-random strings within each degree.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 77 , Issue 1 , March 2012 , pp. 291 - 307
Copyright
Copyright © Association for Symbolic Logic 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Blum, Manuel, On the size of machines, Information and Control, vol. 11 (1967), pp. 257–265.CrossRefGoogle Scholar
[2]Calude, Cristian S. and Nies, André, Chaitin Ω numbers and strong reducibilities, Journal of Universal Computer Science, vol. 3 (1997), no. 11, pp. 1162–1166 (electronic).Google Scholar
[3]Case, John, Homework assignment for students, Computer and Information Sciences Department, University of Delaware, 1990.Google Scholar
[4]Chaitin, Gregory J., A theory of program size formally identical to information theory, Journal of the ACM, vol. 22 (1975), pp. 329–340.CrossRefGoogle Scholar
[5]Chaitin, Gregory J., Incompleteness theorems for random reals, Advances in Applied Mathematics, vol. 8 ( 1987), no. 2, pp. 119–146.CrossRefGoogle Scholar
[6]Davie, George, Foundations of mathematics—recursion theory question, http://cs.nyu.edu/pipermail/fom/2002-May/005535.html.Google Scholar
[7]Downey, Rodney G. and Hirschfeldt, Denis R., Algorithmic randomness and complexity, Theory and Applications of Computability, Springer, 2010.CrossRefGoogle Scholar
[8]Fenner, Stephen and Schaefer, Marcus, Bounded immunity and btt-reductions, Mathematical Logic Quarterly, vol. 45 (1999), no. 1, pp. 3–21.CrossRefGoogle Scholar
[9]Figueira, Santiago, Stephan, Frank, and Wu, Guohua, Randomness and universal machines, Journal of Complexity, vol. 22 (2006), pp. 738–751.CrossRefGoogle Scholar
[10]Friedman, Harvey, Foundations of mathematics—recursion theory question, http://cs.nyu.edu/pipermail/fom/2002-February/005289.html.Google Scholar
[11]Griffor, Edward R., Handbook of computability theory, Studies in Logic and Foundations of Mathematics, North-Holland, Amsterdam, 1999.Google Scholar
[12]Jain, Sanjay, Stephan, Frank, and Teutsch, Jason, Index sets and universal numberings, CiE 2009: Proceedings of the 5th conference on Computability in Europe, Springer, 2009, pp. 270–279.Google Scholar
[13]Kinber, Efim, On btt-degrees of sets of minimal numbers in Gödel numberings, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 23 (1977), no. 3, pp. 201–212.CrossRefGoogle Scholar
[14]Kummer, Martin, personal communication.Google Scholar
[15]Kummer, Martin, On the complexity of random strings (extended abstract), STACS '96: Proceedings of the 13th annual Symposium on Theoretical Aspects of Computer Science, Springer, 1996, pp. 25–36.CrossRefGoogle Scholar
[16]Kummer, Martin and Stephan, Frank, Recursion-theoretic properties of frequency computation and bounded queries, Information and Computation, vol. 120 (1995), no. 1, pp. 59–77.CrossRefGoogle Scholar
[17]Marandžjan, G. B., On the sets of minimal indices of partial recursive functions, Mathematical Foundations of Computer Science, vol. 74 (1979), pp. 372–374.Google Scholar
[18]Marandžjan, G. B., Selected topics in recursive function theory, Technical Report 1990–75, Technical University of Denmark, August 1990.Google Scholar
[19]Martin-Löf, Per, The definition of random sequences, Information and Control, vol. 9 (1966), pp. 602–619.CrossRefGoogle Scholar
[20]Meyer, Albert R., Program size in restricted programming languages, Information and Control, vol. 21 (1972), pp. 382–394.CrossRefGoogle Scholar
[21]Odifreddi, Piergiorgio, Classical recursion theory, Studies in Logic and the Foundations of Mathematics, vol. 125, North-Holland, Amsterdam, 1989.Google Scholar
[22]Robinson, Robert W., Interpolation and embedding in the recursively enumerable degrees, Annals of Mathematics (2), vol. 93 (1971), pp. 285–314.CrossRefGoogle Scholar
[23]Sacks, Gerald E., Recursive enumerability and the jump operator, Transactions of the Amererican Mathematical Society, vol. 108 (1963), pp. 223–239.CrossRefGoogle Scholar
[24]Sacks, Gerald E., The recursively enumerable degrees are dense, Annals of Mathematics (2), vol. 80 (1964), pp. 300–312.CrossRefGoogle Scholar
[25]Schaefer, Marcus, A guided tour of minimal indices and shortest descriptions, Archive for Mathematical Logic, vol. 37 (1998), no. 8, pp. 521–548.CrossRefGoogle Scholar
[26]Schwarz, Steven, Quotient lattices, index sets, and recursive linear orderings, Ph.D. thesis, University of Chicago, 1982.Google Scholar
[27]Soare, Robert I., Recursively enumerable sets and degrees, Perspectives in Mathematical Logic, Springer, 1987.CrossRefGoogle Scholar
[28]Stephan, Frank, Degrees of computing and learning, Habilitationsschrift, University of Heidelberg, 1999.Google Scholar
[29]Stephan, Frank, The complexity of the set of nonrandom numbers, Randomness and complexity, from Leibnitz to Chaitin (Calude, Cristian S., editor), World Scientific, 2007, pp. 217–230.CrossRefGoogle Scholar
[30]Stephan, Frank and Teutsch, Jason, Immunity and hyperimmunity for sets of minimal indices, Notre Dame Journal of Formal Logic, vol. 49 (2008), no. 2, pp. 107–125.CrossRefGoogle Scholar
[31]Teutsch, Jason, Noncomputable spectral sets, Ph.D. thesis, Indiana University, 2007.Google Scholar
[32]Teutsch, Jason, On the Turing degrees of minimal index sets, Annals of Pure Applied Logic, vol. 148 (2007), no. 1–3, pp. 63–80.CrossRefGoogle Scholar
[33]Zvonkin, A. K. and Levin, L. A., 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), no. 6, pp. 83–124.CrossRefGoogle Scholar