Counting classes: Thresholds, parity, mods, and fewness Richard Beigel John Gill Ulrich Hertramp Conference paper First Online: 06 June 2005 DOI :
10.1007/3-540-52282-4_31

Part of the
Lecture Notes in Computer Science
book series (LNCS, volume 415) Cite this paper as: Beigel R., Gill J., Hertramp U. (1990) Counting classes: Thresholds, parity, mods, and fewness. In: Choffrut C., Lengauer T. (eds) STACS 90. STACS 1990. Lecture Notes in Computer Science, vol 415. Springer, Berlin, Heidelberg Abstract Counting classes are classes of languages defined in terms of the number of accepting computations of non-deterministic polynomial-time Turing machines. Well known examples of counting classes are NP, co-NP, ⊕P, and PP. Every counting class consists of languages in P^{#P[1]} , the class of languages computable in polynomial time using a single call to an oracle capable of determining the number of accepting paths of an NP machine.

We perform an in-depth investigation of counting classes defined in terms of thresholds and moduli. We show that the computational power of a threshold machine is a monotone function of the threshold. Then we show that the class MODZ_{k} P is at least as large as FewP. Finally, we improve a result of Cai and Hemachandra by showing that recognizing languages in the class Few is as easy as distinguishing uniquely satisfiable formulas from unsatisfiable formulas (or detecting unique solutions, as in [21]).

Research performed at the Johns Hopkins University. Supported in part by the National Science Foundation under grants CCR-8808949 and CCR-8958528.

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CrossRef Google Scholar Authors and Affiliations Richard Beigel John Gill Ulrich Hertramp 1. Dept. of Computer Science New Haven 2. Dept. of Electrical Engineering Stanford University Stanford 3. Institut für Informatik Universität Würzburg, Am Hubland Würzburg