Abstract
The purpose of this paper is to survey recent work on how classical asymptotic density interacts with the theory of computability. We have tried to make the survey accessible to those who are not specialists in computability theory and we mainly state results without proof, but we include a few easy proofs to illustrate the flavor of the subject.
This paper is dedicated to Rod Downey in honor of his important contributions to computability theory.
The authors would like to thank the Simons Foundation for its support.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Andrews, U., Cai, M., Diamondstone, D., Jockusch, C., Lempp, S.: Asymptotic density, computable traceability, and \(1\)-randomness. Fundam. Math. 234, 41–53 (2016)
Astor, E., Hirschfeldt, D., Jockusch, C.: Dense computability, upper cones and minimal pairs. In preparation
Barzdin, J.: On a frequency solution to the problem of occurrence in a recursively enumerable set. Proc. Steklov Inst. Math. 133, 49–56 (1973)
Bienvenu, L., Day, A., Hölzl, R.: From bi-immunity to absolute undecidability. J. Symb. Log. 78(4), 1218–1228 (2013)
Cholak, P., Igusa, G.: Bounding a density-\(1\) and quasiminimality in the generic degrees. Preprint
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press and McGraw-Hill (2001). Section 29.3: The simplex algorithm, pp. 790–804
Downey, R.G., Hirschfeldt, D.R.: Algorithmic Randomness and Complexity. Theory and Applications of Computability. Springer, Springer, New York (2010)
Downey, R.G., Jockusch Jr., C.G., Schupp, P.E.: Asymptotic density and computably enumerable sets. J. Math. Log. 13 (2013). 1350005, 43 pp.
Downey, R.G., Jockusch Jr., C.G., McNicholl, T.H., Schupp, P.E.: Asymptotic density and the Ershov Hierarchy. Math. Log. Q. 61, 189–195 (2015)
Dzhafarov, D.D., Igusa, G.: Notions of robust information coding, computability. To appear
Figueira, S., Miller, J.S., Nies, A.: Indifferent sets. J. Log. Comput. 19, 425–443 (2009)
Gurevich, Y.: Average case completeness. J. Comput. Syst. Sci. 42, 346–398 (1991)
Harrison-Trainor, M.: The Gamma questions for many-one degrees. Preprint. arXiv: 1606.05701
Hirschfeldt, D., Jockusch, C., Kuyper, R., Schupp, P.: Coarse reducibility and algorithmic randomness. J. Symb. Log. 81, 1028–1046 (2016)
Hirschfeldt, D.R., Jockusch, C.G., McNicholl, T., Schupp, P.E.: Asymptotic density and the coarse computability bound. Computability 5, 13–27 (2016)
Igusa, G.: Nonexistence of minimal pairs for generic computability. J. Symb. Log. 78, 51–522 (2012)
Igusa, G.: The generic degrees of density-\(1\) sets and a characterization of the hyperarithmetic reals. J. Symb. Log. 80, 1290–1314 (2015)
Jockusch, C.: The degrees of bi-immune sets. Zeitschr. f. math. Logik und Grundlagen d. Math. 15, 135–140 (1969)
Jockusch, C.G., Schupp, P.E.: Generic computability, Turing degrees, and asymptotic density. J. Lond. Math. Soc. 85, 472–490 (2012)
Kapovich, I., Myasnikov, A., Schupp, P., Shpilrain, V.: Generic-case complexity, decision problems in group theory and random walks. J. Algebra 264, 665–694 (2003)
Klee, V., Minty, G.: How good is the simplex algorithm? Inequalities, III (Proceedings of Third Symposium, University of California, Los Angeles, 1969; dedicated to the memory of Theodore S. Motzkin), pp. 159–175. Academic Press, New York (1972)
Kurtz, S.A.: Notions of weak genericity. J. Symb. Log. 48, 764–770 (1983)
Levin, L.: Average case complete problems. SIAM J. Comput. 15, 285–286 (1986)
Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory. Classics in Mathematics. Springer, Heidelberg (2000)
Monin, B.: Asymptotic density, error-correcting codes. https://www.lacl.fr/ benoit.monin/ressources/papers/resolution_gamma.pdf
Monin, B., Nies, A.: A unifying approach to the Gamma question. In: 30th Annual ACM/IEEE Symposium on Logic in Computer Science, LICS 2015, Kyoto, pp. 585–596, 6–10 July 2015
Myasnikov, A., Osin, D.: Algorithmically finite groups. J. Pure Appl. Algebra 215, 2789–2796 (2011)
Myasnikov, A., Rybalov, A.: Generic complexity of undecidable problems. J. Symb. Log. 73, 656–673 (2008)
Myasnikov, A., Shpilrain, V., Ushakov, A.: Non-commutative Cryptography and Complexity of Group-Theoretic Problems. Mathematical Surveys and Monographs, vol. 177. American Mathematical Society (2011)
Nies, A.: Computability and Randomness. Oxford University Press, Oxford (2009)
Novikov, P.S.: On the algorithmic unsolvability of the word problem in group theory. Trudy Math. Inst. Steklov 44, 3–143 (1955)
Post, E.: A variant of a recursively unsolvable problem. Bull. Am. Math. Soc. 52, 264–268 (1946)
Rotman, J.: An Introduction to the Theory of Groups. Graduate Texts in Mathematics, 4th edn. Springer, New York (1995)
Terwijn, S.A., Zambella, D.: Algorithmic randomness and lowness. J. Symb. Log. 66, 1199–1205 (2001)
Woess, W.: Cogrowth of groups and simple random walks. Arch. Math. 41, 363–370 (1983)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2017 Springer International Publishing AG
About this chapter
Cite this chapter
Jockusch, C.G., Schupp, P.E. (2017). Asymptotic Density and the Theory of Computability: A Partial Survey. In: Day, A., Fellows, M., Greenberg, N., Khoussainov, B., Melnikov, A., Rosamond, F. (eds) Computability and Complexity. Lecture Notes in Computer Science(), vol 10010. Springer, Cham. https://doi.org/10.1007/978-3-319-50062-1_30
Download citation
DOI: https://doi.org/10.1007/978-3-319-50062-1_30
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-50061-4
Online ISBN: 978-3-319-50062-1
eBook Packages: Computer ScienceComputer Science (R0)