Abstract
No algorithm can of course solve the Halting Problem, that is, decide within finite time always correctly whether a given program halts on a certain given input. It might however be able to give correct answers for ‘most’ instances and thus solve it at least approximately. Whether and how well such approximations are feasible highly depends on the underlying encodings and in particular the Gödelization (programming system) which in practice usually arises from some programming language.
We consider BrainF*ck (BF), a simple yet Turing-complete real-world programming language over an eight letter alphabet, and prove that the natural enumeration of its syntactically correct sources codes induces a both efficient and dense Gödelization in the sense of [Jakoby&Schindelhauer’99]. It follows that any algorithm M approximating the Halting Problem for BF errs on at least a constant fraction ε M > 0 of all instances of size n for infinitely many n.
Next we improve this result by showing that, in every dense Gödelization, this constant lower bound ε to be independent of M; while, the other hand, the Halting Problem does admit approximation up to arbitrary fraction δ> 0 by an appropriate algorithm M δ handling instances of size n for infinitely many n. The last two results complement work by [Lynch’74].
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bailey, D.F.: Counting Arrangements of 1’s and -1’s. Mathematics Magazine 69, 128–131 (1996)
Book, R.V.: Telly languages and complexity classes. Information and Control 26(2), 186–193 (1974)
Calude, C.S., Dinneen, M.J., Shu, C.-K.: Computing a Glimpse of Randomness. Experimental Mathematics 11(3), 361–370 (2001)
Calude, C.S., Hertling, P., Khoussainov, B., Wang, Y.: Recursively enumerable reals and Chaitin Ω numbers. Theoretical Computer Science 255, 125–149 (2001)
Chaitin, G.J.: Algorithmic Information Theory. Cambridge University Press, Cambridge (1987)
Goldreich, O.: Combinatorial property testing (a survey). In: Proc. DIMACS Workshop in Randomized Methods in Algorithm Design, pp. 45–59 (1997)
Jakoby, A., Schindelhauer, C.: The Non-Recursive Power of Erroneous Computation. In: Pandu Rangan, C., Raman, V., Sarukkai, S. (eds.) FST TCS 1999. LNCS, vol. 1738, pp. 394–406. Springer, Heidelberg (1999)
Köhler, S.: Zur Approximierbarkeit des Halteproblems in einer praktischen Gödelisierung, Bachelor’s Thesis, University of Paderborn (2004)
Li, M., Vitáni, P.: An Introduction to Kolmogorov Complexity and its Application, 2nd edn. Springer, Heidelberg (1997)
Lynch, N.: Approximations to the Halting Problem. J. Computer and System Sciences 9, 143–150 (1974)
Machtey, M., Young, P.: An Introduction to the General Theory of Algorithms. The Computer Science Library (1978)
Papadimitriou, C.H.: Computational Complexity. Addison-Wesley, Reading (1995)
Rogers Jr., H.: Theory of Recursive Functions and Effective Computability. Mc- Graw Hill, New York (1967)
Rose, G.F., Ullian, J.S.: Approximation of Functions on the Integers. Pacific Journal of Mathematics 13(2), 693–701 (1963)
Schnorr, C.P.: Optimal Enumerations and Optimal Gödel Numberings. Mathematical Systems Theory 8(2), 182–191 (1975)
Smith, C.: A Recursive Introduction to the Theory of Computation. Springer, Heidelberg (1994)
Soare, R.I.: Recursively Enumerable Sets and Degrees. Springer, Heidelberg (1987)
Specker, E.: Nicht konstruktiv beweisbare Sätze der Analysis. J. Symbolic Logic 14(3), 145–158 (1949)
Wikipedia, the free encyclopedia (2005), http://wikipedia.org/wiki/BrainFuck
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Köhler, S., Schindelhauer, C., Ziegler, M. (2005). On Approximating Real-World Halting Problems. In: Liśkiewicz, M., Reischuk, R. (eds) Fundamentals of Computation Theory. FCT 2005. Lecture Notes in Computer Science, vol 3623. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11537311_40
Download citation
DOI: https://doi.org/10.1007/11537311_40
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-28193-1
Online ISBN: 978-3-540-31873-6
eBook Packages: Computer ScienceComputer Science (R0)