Abstract
Fix an optimal Turing machine U and for each n consider the ratio \(\rho ^U_n\) of the number of halting programs of length at most n by the total number of such programs. Does this quantity have a limit value? In this paper, we show that it is not the case, and further characterise the reals which can be the limsup of such a sequence \(\rho ^U_n\). We also study, for a given optimal machine U, how hard it is to approximate the domain of U from the point of view of coarse and generic computability.
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References
Bienvenu, L., Muchnik, A., Shen, A., Vereshchagin, N.: Limit complexities revisited [once more]. Technical report (2012). arxiv:1204.0201
Bienvenu, L., Shen, A.: Random semicomputable reals revisited. In: Dinneen, M.J., Khoussainov, B., Nies, A. (eds.) Computation, Physics and Beyond. LNCS, vol. 7160, pp. 31–45. Springer, Heidelberg (2012)
Calude, C.S., Hertling, P.H., Khoussainov, B., Wang, Y.: Recursively enumerable reals and chaitin omega numbers. In: Meinel, C., Morvan, M. (eds.) STACS 1998. LNCS, vol. 1373, pp. 596–606. Springer, Heidelberg (1998)
Calude, C., Nies, A., Staiger, L., Stephan, F.: Universal recursively enumerable sets of strings. Theoretical Computer Science 412(22), 2253–2261 (2011)
Downey, R., Hirschfeldt, D.: Algorithmic randomness and complexity. Theory and Applications of Computability. Springer, New York (2010)
Downey, R.G., Jockusch Jr., C.G., Schupp, P.E.: Asymptotic density and computably enumerable sets. Journal of Mathematical Logic 13(02) (2013)
Hamkins, J.D., Miasnikov, A.: The halting problem is decidable on a set of asymptotic probability one. Notre Dame Journal of Formal Logic 47(4) (2006)
Köhler, S., Schindelhauer, C., Ziegler, M.: On approximating real-world halting problems. In: Liśkiewicz, M., Reischuk, R. (eds.) FCT 2005. LNCS, vol. 3623, pp. 454–466. Springer, Heidelberg (2005)
Kučera, A., Slaman, T.: Randomness and recursive enumerability. SIAM Journal on Computing 31, 199–211 (2001)
Li, M., Vitányi, P.: An Introduction to Kolmogorov Complexity and Its Applications, 3rd edn. Springer, New York (2007)
Lynch, N.: Approximations to the halting problem. Journal of Computer and System Sciences, 9–143 (1974)
Miller, J.S.: Every 2-random real is Kolmogorov random. Journal of Symbolic Logic 69(3), 907–913 (2004)
Nies, A.: Computability and randomness. Oxford University Press, Oxford Logic Guides (2009)
Schindelhauer, C., Jakoby, A.: The non-recursive power of erroneous computation. In: Pandu Rangan, C., Raman, V., Sarukkai, S. (eds.) FST TCS 1999. LNCS, vol. 1738, p. 394. Springer, Heidelberg (1999)
Claus Peter Schnorr: Optimal enumerations and optimal Gödel numberings. Mathematical Systems Theory 8(2), 181–191 (1974)
Valmari, A.: The asymptotic proportion of hard instances of the halting problem. Technical report, November 2014. arxiv:1307.7066v2
Vereshchagin, N., Uspensky, V., Shen. A.: Kolmogorov complexity and algorithmic randomness (In Russian. See www.lirmm.fr/~ashen for the draft translation.). MCCME (2013)
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Bienvenu, L., Desfontaines, D., Shen, A. (2015). What Percentage of Programs Halt?. In: Halldórsson, M., Iwama, K., Kobayashi, N., Speckmann, B. (eds) Automata, Languages, and Programming. ICALP 2015. Lecture Notes in Computer Science(), vol 9134. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-47672-7_18
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DOI: https://doi.org/10.1007/978-3-662-47672-7_18
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