U. Schoning. The Power of Counting. In A. L. Selman, editor, Complexity Theory Retrospective,chapter 8, pages 204--223. Springer-Verlag, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....problem in this area, rst posed by Gill in 1977 [Gil77] in the initial paper on probabilistic classes. It implies that PP is also closed under intersection Those interested in further exploring counting classes and the power of counting in complexity theory should consult the papers of Sch oning [Sch90] and Fortnow [For97] 6 Probabilistic Complexity In 1977, Solovay and Strassen [SS77] gave a new kind of algorithm for testing whether a number is prime. Their algorithm ipped coins to help search for a counterexample to primality. They argued that if the number was not prime then with very ....

U. Schoning. The power of counting. In In Alan Selman, editor, Complexity Theory Retrospective, pages 204-223. Springer, New York, 1990.

....of #P under in the above sense is closely related to the relationships between P #P[1] and higher classes such as PH PP and PP PP . 1. Introduction Counting is one of the key notions in computation. Recently, various counting problems have received considerable attention (see, e.g. [Sch90]) and, in order to model them, there have been introduced and extensively studied complexity classes called counting classes , typified by function classes #P [Val79] spanP [KST89] and GapP [FFK94] and language classes PP [Gil77] PhiP [PZ83] C=P [Sim75, Wag86a] and the counting hierarchy ....

U. Schoning, The power of counting, In Complexity Theory Retrospective (A. Selman, ed.), Springer-Verlag (1990), 204--223.

.... work considered the number of (not necessarily distinct) solutions encoded by a nondeterministic computation, and has led via threshold acceptance mechanisms to the work on probabilistic computation [Joh90, Section 4] There has been a recent resurgence of interest in counting for counting s sake [SchU90, Tor91, WagK86a, WagK86b] including the counting of distinct solutions [KST89] which will be the focus in this section. All phylogenetic inference given cost and given limit problems examined in this thesis are in SpanP, and all corresponding optimal cost spanning problems are in Span(NPMV g ....

Schoning, U. The Power of Counting. In A. L. Selman (ed.) Complexity Theory Retrospective, Springer-Verlag, Berlin, 1990. 204--223.

....This implies and generalizes the conjecture that P 6= NP. For a good introduction to complexity theory see, for example, Balc azar et al. 2] Problems related to counting, e.g. How many satisfying truth assignments are there to a given Boolean formula , have also been widely studied (see [15, 12] for example) It has been found [20, 21] that there are counting problems at least as difficult as any problem in PH, and thus (likely) much more difficult than any NP problem. The relationship between quantum computing and counting problems has been previously observed [18, 13, 3] Our result ....

U. Schoning. The Power of Counting. In A. L. Selman, editor, Complexity Theory Retrospective, chapter 8, pages 204--223. Springer-Verlag, 1990.

....important role in computational complexity theory and theoretical computer science. The techniques used in counting complexity have significant applications in circuit complexity and the series of recent results on interactive proof systems. In the first Complexity Theory Retrospective, Schoning [Sch90] gave us a taste of the power of counting. Schoning s survey looked at counting as a technique in complexity theory. We instead will look at counting as a goal and study the techniques (mostly algebraic) that help us understand the computational complexity of counting. In this short survey, we ....

U. Schoning. The power of counting. In A. Selman, editor, Complexity Theory Retrospective, pages 204--223. Springer, New York, 1990.

....Hemachandra s result. Theorem 39. If k 2 then FewP MODZ k P. Proof: Let L 2 FewP via a machine N . By Property 9(ii) there is a machine N 0 such that for every x jaccept(N 0 ; x)j = k jaccept(N;x)j Gamma 1 ( 6j 0 (mod k) if jaccept(N; x)j 1 = 0 if jaccept(N; x)j = 0: Schoning [20] applied similar techniques to the case k = 2. Cai and Hemachandra [7] defined the class Few as follows. Definition 40. A language L belongs to the class Few if there exist a nondeterministic polynomial time machine N , a polynomial q such that jaccept(N; x)j q(jxj) and a polynomial time ....

U. Schoning. The power of counting. In Proceedings of the 3rd Annual Conference on Structure in Complexity Theory, pages 2--18. IEEE Computer Society Press, June 1988.

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U. Schoning. The Power of Counting. In A. L. Selman, editor, Complexity Theory Retrospective,chapter 8, pages 204--223. Springer-Verlag, 1990.

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U. Schoning. The Power of Counting. In A. L. Selman, editor, Complexity Theory Retrospective, chapter 8, pages 204--223. Springer-Verlag, 1990. 16

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U. Schoning. The power of counting. In Complexity Theory Retrospective (A. Selman Ed.), Springer-Verlag (1990), 204\Gamma223.

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