O. Watanabe and S. Toda, Polynomial time 1-Turing reductions from #PH to #P. Theoritical Computer Science, 100(1):205-221, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....some traditional complexity classes can be characterized by operators. For example NP = 9 P; 2 = 9 8 P, using the existential and universal operators 9 and 8 respectively. They have been used fruitfully to prove (or to give simpler proofs of known) relations between complexity classes [ZF87, To91, WT92, VW93, VW95, VW97]. For example, Toda s Theorem was proved in this framework [To91] In this approach, Zachos and F urer [ZF87] showed that AM can also be characterized by the operator BP, i.e. AM = BP NP. By successfully employing this operator machinery, they were able to give a simple and natural proof that ....

O. Watanabe and S. Toda, Polynomial time 1-Turing reductions from #PH to #P. Theoritical Computer Science, 100(1):205-221, 1992.

....the #P oracle once, the only additional computation that T needs to perform is an arithmetic comparison involving the outcome of g. We denote this class by NP #P[1;comp] Additionally, a stochastic version of Knapsack defined in [9] belongs to this class. Using the results of Toda and Watanabe [12], it is easy to show that NP #P[1;comp] contains the whole Polynomial Hierarchy PH. Algorithm Next Fit Decreasing NFD Input: M = ff1 ; fmg, failure probabilities 1 f i 0. Fault tolerance constraint (1 Gamma ffl) 1 ffl Q i2M f i . Output: A (1 Gamma ffl) fault tolerant ....

S. Toda and O. Watanabe. Polynomial-time 1-Turing reductions from #PH to #P. Theoretical Computer Science 100, pp. 205--221, 1992.

....can be shown to belong to this class. It can be shown that NP #P[1;comp] contains the whole Polynomial Hierarchy PH. Lemma 3. PH NP #P[1;comp] Proof. Given any language L 2 PH, we can decide if an input x is in L by asking an appropriate function in #PH once. Since #PH FP #P[1] [TW92], there exists a polynomial time deterministic Turing machine T d that, on input x, computes an input y x for a function g 2 #P , calls g(y x ) and performs a deterministic computation after getting n = g(y x ) in order to decide if x 2 L. Next, we show that L also belongs to NP #P[1;comp] ....

S. Toda and O. Watanabe. Polynomial-time 1-Turing reductions from #PH to #P. Theoretical Computer Science 100, pp. 205--221, 1992.

....complexity classes can be characterized by operators. For example NP = 9 Delta P; Sigma 2 = 9 Delta 8 Delta P, using the existential and universal operators 9 and 8 respectively. They have been used fruitfully to prove (or to give simpler proofs of known) relations between complexity classes [ZF87, To91, WT92, VW93, VW95, VW97]. For example, Toda s Theorem was proved in this framework [To91] In this approach, Zachos and Furer [ZF87] showed that AM can also be characterized by the operator BP, i.e. AM = BP Delta NP. By successfully employing this operator machinery, they were able to give a simple and natural proof ....

O. Watanabe and S. Toda, Polynomial time 1-Turing reductions from #PH to #P. Theoritical Computer Science, 100(1):205--221, 1992.

....#NP are related [7] by #P span P #NP: Thus, FP #P FP span P FP #NP : But FP #NP FP #PH , where #PH is the class of functions that count the number of accepting computations of polynomial time nondeterministic Turing machines with oracles from PH. Furthermore, Toda and Watanabe [10] show #PH FP #P . Thus, FP #P = FP span P : See also Section 1:8 of Welsh s book [12] We now complete the proof by showing that #CONNECTEDSUBGRAPHS is in FP span P . Let N(G;k) denote k times the number of distinct (up to isomorphism) connected size k subgraphs of G. Since ....

S. Toda and O. Watanabe, Polynomial-time 1-Turing reductions from #PH to #P, Theoretical Computer Science 100 (1992) 205--221.

....(SpanPH) be written #P and #NP (SpanP and SpanNP) and define hardness of functions in the classes of these hierarchies relative to metric reducibility. The following class relations are known: Corollary 19 The following hold: 1. #PH, SpanPH, and FP #P FPSPACE(poly) 2. FPH FP #P [1] TW92, Theorem 5.1] 3. If either #P FPH or FPH #P then PH collapses to a finite level [TW92, Corollary 5.7, Part 1] 4. #PH FP #P [1] TW92, Theorem 4.1] 5. For k 1, #(F Sigma p k ) Span(F Sigma p k ) KST89, Generalization of Proposition 4.7] 6. For k 1, Span(F Sigma p k ) ....

....classes of these hierarchies relative to metric reducibility. The following class relations are known: Corollary 19 The following hold: 1. #PH, SpanPH, and FP #P FPSPACE(poly) 2. FPH FP #P [1] TW92, Theorem 5.1] 3. If either #P FPH or FPH #P then PH collapses to a finite level [TW92, Corollary 5.7, Part 1] 4. #PH FP #P [1] TW92, Theorem 4.1] 5. For k 1, #(F Sigma p k ) Span(F Sigma p k ) KST89, Generalization of Proposition 4.7] 6. For k 1, Span(F Sigma p k ) #(F Sigma p k 1 ) KST89, Generalization of Proposition 4.8] 7. For k 1, #(F Sigma ....

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Toda, S. and Watanabe, O. Polynomial-time 1-Turing reductions from #PH to #P. Theoretical Computer Science, 100(1), 205--221, 1992.

....other counting classes that draw ner distinctions. In particular, Valiant s class #NP coincides with the class # coNP of the Hemaspaandra Vollmer framework. As regards complete problems for these higher counting complexity classes, the state of a airs is rather complicated. Toda and Watanabe [TW92] showed if a problem is #P hard via polynomialtime 1 Turing reductions, then it is also # coNP hard and # P k hard, for each k 2, where # P k is the counting version of the class P k at the k th level of the polynomial hierarchy PH. This surprising result yields an abundance of ....

....complexity (i.e. lower than 4 P k 1 complete) Clearly, unless P k 1 collapses to a lower complexity class, no such problem can be # P k complete via parsimonious reductions, which means that a broader class of reductions has to be considered. To this e ect, Toda and Watanabe [TW92] proved the following surprising and quite signi cant result: if a counting problem is #P hard via polynomial time 1 Turing reductions, then it is also # P k complete via the same reductions, for every k 1. Consequently, #perfect matchings is # P k complete via polynomial time 1 Turing ....

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S. Toda and O. Watanabe. Polynomial-time 1-Turing reductions from #PH to #P. Theoretical Computer Science, 100(1):205-221, 1992.

....The counting hierarchy over K is the smallest family of classes containing K and being closed under 9, 8, C. Let CH Delta K be the union of all classes of the counting hierarchy over K, and let CH = CH Delta P. For later use we state the following results from the literature. 2. 10) Proposition [ToWa91]. P #PH = P PP . 2.11) Proposition [Tor88] P PP 9C P. 2.12) Proposition [Tor88] CH = S k0 C k P. 2.13) Proposition [Wag86] FCH = #CH = FP CH . Every combination of a language to function operator with a function to language operator yields a language to language operator. ....

S. Toda, O. Watanabe, Polynomial time 1-Turing reductions from #PH to # P; Theoretical Computer Science, to appear.

....oracle for w. Restricted notions of polynomialtime Turing reductions between counting problems have also been considered. In particular, a polynomial time 1 Turing reduction is a polynomial time Turing reduction in which the Turing machine M is allowed to make at most one call to the oracle for w [Val79a,TW92]. Parsimonious reductions constitute the most restricted notion of reducibility. These are the special case of polynomial time 1 Turing reductions in which v = w g, for some polynomial time computable total function g. In other words, the oracle for w is queried once and no computation is ....

....that #perfect matchings has a parsimonious reduction to #hilbert. Since, as mentioned earlier, #perfect matchings is #P hard under polynomial time 1 Turing reductions (see [Pap94,Zan91] it follows that #hilbert is also #P hard under polynomial time 1 Turing reductions. In a breakthrough paper [TW92], however, Toda and Watanabe proved that if a counting problem is #P hard under polynomial time 1 Turing reductions, then it is also #C hard under such reductions for every class C in the polynomial hierarchy PH. In particular, #hilbert is #NP hard under polynomial time 1 Turing reductions and, ....

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S. Toda and O. Watanabe. Polynomial-time 1-Turing reductions from #PH to #P. Theoretical Computer Science, 100(1):205-221, 1992.

....4.1 and Lemma 4.2, we see that statement 1 implies S 5 qC[ log O(1) n) m B, which directly yields statement (2) 2) 3) is obvious. 3) 1) From Theorem 3.1 and Proposition 4.1, we see that statement 3 implies that relative to all oracles, PSPACE P PP PH . However a result in [TW92] shows that relativizably the equality P PP PH = P PP holds. This implies statement 1. q 5.7 MidbitP vs. PSPACE The class MidbitP was introduced in [GKR 95] Instead of repeating the original definition we simply give a leaf language definition. For background on the class refer to ....

S. Toda and O. Watanabe. Polynomial time 1-Turing reductions from #ph to #p. Theoretical Computer Science, 100:205--221, 1992.

....C, p a polynomial and for all x 2 Gamma and y 2 Sigma : R(x ; y) jyj p(jxj) ffl The function # p R is defined by # p R(x) kf y : R(x ; y) gk: With C : P, the class of functions so defined is exactly #P, while with C : PH it equals the class #PH studied by Toda and O. Watanabe [ToWa92]. Note that # p R(x) always takes values between 0 and 2 p(jxj) Gamma 1. We stipulate that the values y of #P functions are encoded as binary strings of length exactly p(jxj) with the most significant bit written leftmost and padded out with leading 0s if necessary. The bits are numbered ....

....that Mod k P is low for MP. In their proof they show implicitly that Mod k P AmpMP, so the lowness for MP (and for AmpMP) follows by the above corollary. By combining Theorem 12 with PH BP[ Phi P] AmpMP one may quickly find a result obtained via lengthy calculations by Toda and O. Watanabe [ToWa92]: Corollary 15. #PH PF #P[1] The following proposition tells us that sets in C Phi P can be one sided amplified . We do not expect all sets in MP to have this property. Proposition 16. For any language D 2 C Phi P, there is a polynomial p such that for all polynomials s, we can find a ....

S. Toda and O. Watanabe. Polynomial time 1-turing reductions from #PH to #P. Theoretical Computer Science, 100(1):205--222, 1992.

....[Tor88, Chapter 2] Valiant s above approach to counting is heavily machine based, rather than predicatebased. In contrast, Toda ( Tod91a] see also the independent [Bur92] proposed a predicatebased definition of a counting hierarchy, and this line has been followed in a number of recent papers [WT92,VW93,Vol94b,Vol94a] One motivation for such a predicate based approach is that counting is most natural in terms of predicates, and this definition allows one greater precision in specifying counting classes. Toda denoted application of his counting operator as NUM Delta, thus avoiding any ....

O. Watanabe and S. Toda. Polynomial time 1-Turing reductions from #PH to #P. Theoretical Computer Science, 100:205--221, 1992.

....(a) For technical details see the complete proof in the Appendix. Corollary 17. a) BP[ Phi P] is low for MP. b) If C=P AmpMP then PP PP = MP. By combining Theorem 16 with PH BP[ Phi P] AmpMP one may quickly obtain a result obtained via lengthy calculations by Toda and O. Watanabe [TW90]: Corollary 18. #PH PF #P[1] 5. The power of more than one bit Definition 19. Let k be a number (or a numerical function of jxj) A language L belongs to the class P #P[k bits] if L is accepted by a deterministic polynomial time bit oracle machine M with some function f 2 #P as oracle, ....

S. Toda and O. Watanabe. Polynomial time 1-turing reductions from #PH to #P. Technical report, University of Electro-Communications, Tokyo, 1990.

.... on an odd number of paths, we obtain R #P m ( PhiP) Similar claims hold for R #P m (C = P) and (with a bit of care) for R #P m (PP) Let us adopt Valiant s [Val79a] standard definition of #NP (informally, #NP = #P) NP ) and the analogous definition of #PH (informally, #PH = #P) PH ) WT92] 3 In this paper, we have discussed a number of R #P m closures of classes. One might naturally wonder whether the R #NP m closures of the classes yield even greater computational power. However, note that from Toda and Watanabe s [WT92] result #PH FP #P[1] we can easily prove ....

....of #PH (informally, #PH = #P) PH ) WT92] 3 In this paper, we have discussed a number of R #P m closures of classes. One might naturally wonder whether the R #NP m closures of the classes yield even greater computational power. However, note that from Toda and Watanabe s [WT92] result #PH FP #P[1] we can easily prove the following proposition, which says that R #NP m closures (or even 3 Vollmer [Vol94] and Toda and Watanabe ( WT92] using the different notation NUM Delta C ) have proposed interesting and different # type classes. Often, though not ....

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O. Watanabe and S. Toda. Polynomial time 1-Turing reductions from #PH to #P. Theoretical Computer Science, 100:205--221, 1992.

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S. Toda and O. Watanabe. Polynomial time 1-Turing reductions from #PH to #P. Theoretical Comput. Sci., 100:205--221, 1992.

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