Abstract
We introduce the logarithmic time counting hierarchy (LCH) as a tool to classify certain combinatorial problems connected to the idea of counting, and consider the model of boolean circuits as a tool to encode in a succinct way instances of these problems. We observe that many natural problems, like “majority” are complete for the different classes of LCH; using this and the fact that as a general rule, the complexity of a problem increases exponentially when its succinct representation is considered, we obtain complete problems for the classes in the polynomial time counting hierarchy. With the help of the succinct encodings, we give sufficient conditions for a problem to be hard for the classes NP and PP. Finally we show another use of the succinct representations, proving translational results for the classes in the counting hierarchy.
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© 1989 Springer-Verlag Berlin Heidelberg
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Torán, J. (1989). Succinct representations of counting problems. In: Mora, T. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1988. Lecture Notes in Computer Science, vol 357. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-51083-4_77
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DOI: https://doi.org/10.1007/3-540-51083-4_77
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