Abstract
Complexity classes, which are defined via finite commutative monoids, can be considered as (very) regular counting classes. These include well-known classes like NP, coNP, ⊕P, other MOD-classes, but also the classes of finite acceptance type, and many more.
In these cases, the acceptance mechanism can be defined by a regular leaf language, where acceptance really depends only on the number of occurrences of the various letters in the actual leafstring. In other words, the acceptance mechanism is given by a symmetric regular language. Generally all classes described in this way are the so called eventually periodic counting classes.
In this paper we relax the symmetry condition on the regular leaf language: We allow all regular leaf languages, but we admit only machines, which on all input words will only produce symmetric leafstrings, which means all appearing leaf strings will either under all permutations belong to the acceptance language, or under all permutations not belong to the acceptance language.
We give an exact characterization of all complexity classes, which can be described in this manner. It turns out that besides the classes obtained via finite commutative monoids, we also can describe promise classes like UP or MODZ2P in this way.
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
R. Beigel, J. Gill, U. Hertrampf, Counting classes: thresholds, parity, mods, and fewness; Proceedings of the 7th Symp. on Theoretical Aspects of Computer Science (1990), LNCS 415, pp. 49–57.
R. Beigel, H. Straubing, The power of local self-reductions; Proceedings of the 10th IEEE Structure in Complexity Theory Conference (1995), pp. 277–285.
D. P. Bovet, P. Crescenzi, R. Silvestri, A uniform approach to define complexity classes; Theoretical Computer Science104 (1992), pp. 263–283.
J. Cai, M. Furst, PSPACE survives constant-width bottlenecks; International Journal of Foundations of Computer Science2 (1991), pp. 67–76.
J. Cai, L. Hemachandra, On the power of parity polynomial time; Proceedings of the 6th Symp. on Theoretical Aspects of Computer Science (1989), LNCS 349, pp. 229–239.
V. Diekert, Combinatorics on Traces; LNCS 454, Springer-Verlag Berlin Heidelberg (1990).
S. Eilenberg, M. P. Schützenberger, Rational sets in commutative monoids; Journal of Algebra13 (1969), pp. 173–191.
T. Gundermann, N.A. Nasser, G. Wechsung, A survey on counting classes; Proceedings of the 5th Structure in Complexity Theory Conference (1990), pp. 140–153.
L. A. Hemaspaandra, M. Ogihara, Universally serializable computation; Journal of Computer and System Sciences55(3) (1997), pp. 547–560.
L. A. Hemachandra, A. Hoene, On sets with efficient implicit membership tests; SIAM Journal on Computing, 20 (1991), pp. 1148–1156.
U. Hertrampf, Classes of bounded counting type and their inclusion relations; Proceedings of the 12th Symp. on Theoretical Aspects of Computer Science (1995), LNCS 900, pp. 60–70.
U. Hertrampf, Acceptance by transformation monoids (with an application to local self reductions); Proceedings of the 12th IEEE Conference on Computational Complexity (1997), pp. 213–224.
U. Hertrampf, Polynomial time machines equipped with word problems over algebraic structures as their acceptance criteria; Proceedings of the 11th International Symposium on Fundamentals of Computation Theory (1997), pp. 233–244.
U. Hertrampf, C. Lautemann, T. Schwentick, H. Vollmer, K.W. Wagner, On the power of polynomial time bit-reductions; Proceedings of the 8th Structure in Complexity Theory Conference (1993), pp. 200–207.
B. Jenner, P. McKenzie, D. Thérien, Logspace and logtime leaf languages; Proceedings of the 9th Structure in Complexity Theory Conference (1994), pp. 242–254.
M. Ogihara, On serializable languages; International Journal of Foundations of Computer Science5 (1994), pp. 303–318.
L. G. Valiant, The complexity of computing the permanent; Theoretical Computer Science8 (1979), pp. 189–201.
H. Veith, Succinct representations and leaf languages; Proceedings of the 11th IEEE Conference on Computational Complexity (1996), pp. 118–126.
N. K. Vereshchagin, Relativizable and non-relativizable theorems in the polynomial theory of algorithms; Izvestija Rossijskoj Akademii Nauk57 (1993), pp. 51–90. In Russian.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1999 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hertrampf, U. (1999). Generalized Regular Counting Classes. In: Kutyłowski, M., Pacholski, L., Wierzbicki, T. (eds) Mathematical Foundations of Computer Science 1999. MFCS 1999. Lecture Notes in Computer Science, vol 1672. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-48340-3_38
Download citation
DOI: https://doi.org/10.1007/3-540-48340-3_38
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-66408-6
Online ISBN: 978-3-540-48340-3
eBook Packages: Springer Book Archive