Abstract
We refine the techniques of Beigel, Gill, Hertrampf [BGH90] who investigated polynomial time counting classes, in order to make them applicable to the case of logarithmic space. We define the complexity classes MOD k L and demonstrate their significance by proving that all standard problems of linear algebra over the finite rings Z/kZ are complete for these classes. We then define new complexity classes LogFew and LogFew NL and identify them as adequate logspace versions of Few and LogFew and Few P. We show that LogFew and LogFew NL is contained in MODZ k L and that LogFew is contained in MOD k L for all k. Also an upper bound for L #L in terms of computation of integer determinants is given from which we conclude that all logspace counting classes are contained in NC 2.
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References
C. Àlvarez and B. Jenner. A Very Hard Log Space Counting Class. 5th Structure in Complexity Theory (IEEE), 154–168, 1990.
E. W. Allender. The Complexity of Sparse Sets in P. Lecture Notes in Computer Science 223, 1st Structure in Complexity Theory Conference, 1–11, 1986.
A. Borodin, S. Cook, P. Dymond, W. Ruzzo, M. Tompa. Two Applications of Complementation via Inductive Counting 3rd Structure in Complexity Theory (IEEE), 116–127, 1988.
J. Balcázar, J. Díaz, and J. Gabarró. Structural Complexity I. EATCS Monographs in Theoretical Computer Science. Springer-Verlag, 1988.
S. J. Berkowitz. On Computing the Determinant in Small Parallel Time Using a Small Number of Processors. Information Processing Letters 18, 147–150, 1984
R. Beigel, J. Gill, and U. Hertrampf. Counting Classes: Thresholds, Parity, Mods, and Fewness. Proceedings of the 7th Symposium on Theoretical Aspects of Computer Science (STACS), Lecture Notes in Computer Science 415, 49–57, 1990.
G. Buntrock, L. A. Hemachandra, and D. Siefkes. Using Inductive Counting to Simulate Nondeterministic Computation. Mathematical Foundations of Computer Science, Lecture Notes in Computer Science 452, 187–194, 1990.
J. Cai and L. A. Hemachandra. On the Power of Parity Polynomial Time. Proceedings of the 6th Symposium on Theoretical Aspects of Computer Science (STACS), Lecture Notes in Computer Science 349, 229–239, 1989
S. A. Cook. A Taxonomy of Problems with Fast Parallel Algorithms. Information and Control 64, 2–22, 1985.
T. Gundermann, N. A. Nasser, and G. Wechsung. A Survey on Counting Classes. 5th Structure in Complexity Theory (IEEE), 140–153, 1990.
N. Immerman. Nondeterministic Space is Closed Under Complement. SIAM Journal on Computing 17, 935–938, 1988
R. Ladner, N. Lynch. Relativization of Questions About Log Space Computability. Mathematical System Theory 10, 19–32, 1976.
K. Lange and P. Rossmanith. Characterizing unambiguous augmented pushdown automata by circuits. Mathematical Foundations of Computer Science, Lecture Notes in Computer Science 452, 399–406, 1990.
C. Meinel. Modified Branching Programs and Their Computational Power. Springer-Verlag Lecture Notes in Computer Science #370, 1989
R. Szelepcsényi. The Method of Forced Enumeration for Nondeterministic Automata. Acta Informatica 26, 279–284, 1988.
J. Torán. Counting the Number of Solutions. Mathematical Foundations of Computer Science, Lecture Notes in Computer Science 452, 121–134, 1990.
L. Valiant. The relative complexity of checking and evaluating. Information Processing Letters, 5, 20–23, 1976.
L. Valiant. The complexity of computing the permanent. Theoretical Computer Science 8, 189–201, 1979.
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© 1991 Springer-Verlag Berlin Heidelberg
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Buntrock, G., Hertrampf, U., Damm, C., Meinel, C. (1991). Structure and importance of logspace-MOD-classes. In: Choffrut, C., Jantzen, M. (eds) STACS 91. STACS 1991. Lecture Notes in Computer Science, vol 480. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0020812
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DOI: https://doi.org/10.1007/BFb0020812
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