Abstract
There are non-context-free languages which are recognizable by randomized pushdown automata even with arbitrarily small error probability. We give an example of a context-free language which cannot be recognized by a randomized pda with error probability smaller than \( \frac{1} {2} - O\left( {\frac{{\log _2 n}} {n}} \right) \) for input size n. Hence nondeterminism can be stronger than probabilism with weakly-unbounded error.
Moreover, we construct two deterministic context-free languages whose union cannot be accepted with error probability smaller than 1/3 − 2−Ω(n), where n is the input length. Since the union of any two deterministic context-free languages can be accepted with error probability 1/3, this shows that 1/3 is a sharp threshold and hence randomized pushdown automata do not have amplification.
One-way two-counter machines represent a universal model of computation. Here we consider the polynomial-time classes of multicounter machines with a constant number of reversals and separate the computational power of nondeterminism, randomization and determinism.
The work of this paper has been supported by the DFG Projects HR 14/6-1 and SCHN 503/2-1.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J. Kaneps, D. Geidmanis, and R. Freivalds, “Tally languages accepted by Monte Carlo pushdown automata”, RANDOM’ 97, Lexture Notes in Computer Science 1269, pp. 187–195.
P. Ďuriš, J. Hromkovič, and K. Inone, “A separation of determinism, Las Vegas and nondeterminism for picture recognition”, Proc. IEEE Conference on Computational Complexity, IEEE 2000, pp. 214–228.
P. Ďuriš, J. Hromkovič, J.D.P. Rolim, and G. Schnitger, “Las Vegas versus determinism for one-way communication complexity, finite automata and polynomial-time computations”, Proc. STACS’97, Lecture Notes in Computer Science 1200, Springer, 1997, pp. 117–128.
M. Dietzfelbinger, M. Kutylowski, and R. Reischuk, “Exact lower bounds for computing Boolean functions on CREW PRAMs”, J. Computer System Sciences 48, 1994, pp. 231–254.
R. Freivalds, “Projections of languages recognizable by probabilistic and alternating multitape automata”, Information Processing Letters 13 (1981), pp. 195–198.
J. Hromkovič, Communication Complexity and Parallel Computing, Springer 1997.
J. Hromkovič, “Communication Protocols — An Exemplary Study of the Power of Randomness”, Handbook on Randomized Computing, (P. Pardalos, S. Kajasekaran, J. Reif, J. Rolim, Eds.), Kluwer Publisher 2001, to appear.
J. Hromkovič, and G. Schnitger, “On the power of randomized pushdown automata”, 5th Int. Conf. Developments in Language Theory, 2001, pp. 262–271.
J. Hromkovič, and G. Schnitger, “On the power of Las Vegas for one-way communication complexity, OBDD’s and finite automata”, Information and Computation, 169, 2001, pp.284–296.
J. Hromkovič, and G. Schnitger, “On the power of Las Vegas II, Two-way finite automata”, Theoretical Computer Science, 262, 2001, pp. 1–24
Immermann, N, “Nondeterministic space is closed under complementation”, SIAM J. Computing, 17 (1988), pp. 935–938.
B. Kalyanasundaram, and G. Schnitger, “The Probabilistic Communication Complexity of Set Intersection”, SIAM J. on Discrete Math. 5(4), pp. 545–557, 1992.
E. Kushilevitz, and N. Nisan, Communication Complexity, Cambridge University Press 1997.
I. Macarie, and M. Ogihara, “Properties of probabilistic pushdown automata”, Technical Report TR-554, Dept. of Computer Science, University of Rochester 1994.
K. Mehlhorn, and E. Schmidt, “Las Vegas is better than determinism in VLSI and distributed computing”, Proc. 14th ACM STOC’82, ACM 1982, pp. 330–337.
I.I. Macarie, and J.I. Seiferas, “Amplification of slight probabilistic advantage at absolutely no cost in space”, Information Processing Letters 72, 1999, pp. 113–118.
A.A. Razborov, “On the distributional complexity of disjointness”, Theor. Comp. Sci. 106(2), pp. 385–390, 1992.
M. Sauerhoff, “On nondeterminism versus randomness for read-once branching programs”, Electronic Colloquium on Computational Complexity, TR 97-030, 1997.
M. Sauerhoff, “On the size of randomized OBDDs and read-once branching programs for k-stable functions”, Proc. STACS’ 99, Lecture Notes in Computer Science 1563, Springer 1999, pp. 488–499.
R. Szelepcsěnyi, “The method of forcing for nondeterministic automata”, Ball. EATCS 33, (1987), pp. 96–100.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hromkovič, J., Schnitger, G. (2003). Pushdown Automata and Multicounter Machines, a Comparison of Computation Modes. In: Baeten, J.C.M., Lenstra, J.K., Parrow, J., Woeginger, G.J. (eds) Automata, Languages and Programming. ICALP 2003. Lecture Notes in Computer Science, vol 2719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45061-0_7
Download citation
DOI: https://doi.org/10.1007/3-540-45061-0_7
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-40493-4
Online ISBN: 978-3-540-45061-0
eBook Packages: Springer Book Archive