S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 21(2):316-- 328, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....every AC circuit can be simulated by a probabilistic depth 2 circuit with a parity gate at the root and 2 AND gates of fanin polylog n on the bottom. Allender and Hertrampf [2] extended this result by reducing the number of random bits used. We also build on techniques of Toda and Ogiwara [16]. Similar results were obtained independently by Tarui [15] who reports that his work was independent of Toda and Ogiwara. 3. Probabilistic Polynomials and Circuits Definition 1. A probabilistic circuit is a circuit with two types of inputs: actual inputs x 1 ; x n from f0; 1g, and ....

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. In Proceedings of the 6th Annual Conference on Structure in Complexity Theory. IEEE Computer Society Press, 1991. To appear.

....each other by polylog time computable monotone projections. 1 Introduction Well known examples of dot operators are the existential operator [33] the counting operator [33] and the BP operator [20] For a survey see the book of Kobler et al. 15] Such operators have been used (for example in [33, 20, 26, 27, 15]) to define new classes and hierarchies, for instance, the counting hierarchy [33] In general, dot operators have been used as a tool to study the relationship between complexity classes. In Section 2 we will generalize this notion of a dot operator so that every language A will determine an ....

S. Toda, M. Ogiwara. Counting classes are at least as hard as the Polynomial-Time Hierarchy, SIAM Journal on Computing 21, 1992, pp. 316--328.

....of cluster operators are included in the images of the corresponding common operators. Inverstigations are limited again to the lowest level of the polynomial time hierarchy. For our considerations we need two lemmata. These results are easy to conclude from wellknown facts, noted in [OH93, Sch89, TO92] by inductive argumentations. 12 Lemma 5.1. The following statements are equivalent. 1. PP Sigma P k . 2. PP Sigma P k 1 = Sigma P k . 3. CH = Sigma P k . Lemma 5.2. The following statements are equivalent. 1. PP UP Sigma P k . 2. PP Sigma P k = UP Sigma P k . 3. CH ....

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial time hierarchy. SIAM Journal on Computing, 21:315--328, 1992.

....of GapP carry over to GapSpanP, because SpanP, like #P, is closed under addition and multiplication. In particular, GapSpanP is closed under addition, multiplication and subtraction. For our other characterizations of GapSpanP, we need the following result from [5] 6] and [9] see also [8]) Lemma 3.3. 5, 6, 9] SpanP = # Delta NP #P NP = # Delta co NP FP Gamma SpanP. Theorem 3.4. GapSpanP = SpanP Gamma FP = FP Gamma SpanP = GapP NP : Proof. It follows from Theorem 3.2 that SpanP Gamma FP GapSpanP and FP Gamma SpanP GapSpanP. On the other hand, we have ....

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. In Proceedings of the Sixth Annual Conference on Structure in Complexity theory, pages 2--12, 1991.

....Zachos [Zac88] has shown that NP BPP implies PH = BPP. Since this result relativizes (i.e. for all A, NP A BPP A implies PH A = BPP A ) we obtain the following corollary from Theorem 3.11. Corollary 3.13 Sigma p 2 BPP path = PH = BPP NP . Toda [Tod91] and Toda and Ogiwara [TO92] showed that PH BPP C for any class C among fPP; C=P; PhiPg. As a consequence, none of these classes can be contained in the polynomial hierarchy unless the polynomial hierarchy collapses. Thus, none of these classes can be contained in BPP path unless the polynomial hierarchy collapses. ....

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomialtime hierarchy. SIAM Journal on Computing, 21(2):316--328, 1992.

....each other by polylog time computable monotone projections. 1 Introduction Well known examples of dot operators are the existential operator [33] the counting operator [33] and the BP operator [20] For a survey see the book of Kobler et al. 15] Such operators have been used (for example in [33, 20, 26, 27, 15]) to define new classes and hierarchies, for instance, the counting hierarchy [33] In general, dot operators have been used as a tool to study the relationship between complexity classes. In Section 2 we will generalize this notion of a dot operator so that every language A will determine an ....

S. Toda, M. Ogiwara. Counting classes are at least as hard as the Polynomial-Time Hierarchy, SIAM Journal on Computing 21, 1992, pp. 316--328.

....hard classes; for example, from the definition of C=P, it follows that co NP C=P. Also, as an intermediate step in the proof of Toda s theorem it is shown that PH BP. Phi P (the class obtained by applying the BP operator to PhiP) Tod91] In general, it holds that PH BP.Mod k P [TO92] The class UP introduced by Valiant [Val76] is another important complexity class. UP consists of those languages in NP accepted by nondeterministic polynomial time machines having at most one accepting path. Valiant defined UP for studying the relative complexity of checking and evaluating. ....

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 21(2):316--328, 1992.

....assignments to the ith formula of the list produced by Theorem 5.2 on input . Then 1 Y i (1 h( i) 13) equals the characteristic function of SAT. By the closure properties of GapP [FFK94] 13) belongs to GapP. Similarly, we can conditionally derandomize the result by Toda and Ogiwara [TO92] that the polynomial time hierarchy does not add power to GapP in a randomized setting. Applying our techniques to their main lemma yields: Lemma 5.5 Let B be any oracle. If there is a Boolean function f 2 E such that C SAT B f (n) 2 2 n) then GapP NP B is contained in GapP B . This ....

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 21(2):316-328, 1992.

....of having at least one of the reductions being correct is 1 Gamma 1=exp. Moreover, using exact counting they can detect whether such a correct reduction occurred. Thus, in this case, the power of counting allows one to amplify the probability bounds of randomized reductions. For details, see [TO90] As the reader may suspect, there are problems in defining bpp m reductions in non robust classes like D P and co D P . Since D P and co D P do not have the power of exact counting, the amplification technique described above do not work for D P and co D P . The following ....

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. Technical Report 90-09, Department of Computer Science and Information Mathematics, University of ElectroCommunications, July 1990.

....of NP machines. De nition 4.1 ( 44, 45] A function f : N is in #P if there is a nondeterministic polynomial time Turing machine N such that (8x) f(x) #accN (x) where #accN (x) denotes the number of accepting paths of N(x) #P is a tremendously powerful class. Toda ( 42] see also [8, 43, 35]) showed that Turing access to #P suces to accept every set in the polynomial hierarchy. Unfortunately, it is well known that computing power indices is typically #P complete. Prasad and Kelly [34] proved (surprisingly recently) that the Banzhaf index is #P complete, and it has long been known ....

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 21(2):316-328, 1992.

....ZPP(NP) Corollary 4.6. For every k # 1, if C=P # (# P k # # P k ) poly then CH = ZPP(# P k ) Proof. First, since C=P has complete word decreasing self reducible languages [29] C=P # (# P k # # P k ) poly implies C=P # ZPP(# P k ) # PH. Second, since PH # BPP(C=P) [37, 35], C=P # (# P k # # P k ) poly implies PH # (# P k # # P k ) poly and therefore PH collapses to ZPP(# P k ) by Corollary 4.3. Finally, since C=P(PH) # BPP(C=P) 37] it follows that C=P(PH) # PH, and since CH = C=P # C=P(C=P) # . see [38] we get inductively that CH ....

....[29] C=P # (# P k # # P k ) poly implies C=P # ZPP(# P k ) # PH. Second, since PH # BPP(C=P) 37, 35] C=P # (# P k # # P k ) poly implies PH # (# P k # # P k ) poly and therefore PH collapses to ZPP(# P k ) by Corollary 4.3. Finally, since C=P(PH) # BPP(C=P) [37], it follows that C=P(PH) # PH, and since CH = C=P # C=P(C=P) # . see [38] we get inductively that CH # PH (# ZPP(# P k ) Corollary 4.7. Let K # EXP,PSPACE,ModmP , m # 2. If for some k # 1, K # (# P k # # P k ) poly, then K # PH and PH collapses to ZPP(# P k ....

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S. Toda and M. Ogiwara, Counting classes are at least as hard as the polynomial-time hierarchy, SIAM J. Comput., 21 (1992), pp. 316--328.

....satisfying assignments, it answers 0 . An oracle which returns the parity of the number of satisfying assignments has this property, and it follows that NP RP[ Phi P] Toda [Tod89, Tod91] used this to show that the polynomial hierarchy (PH) is contained in BP[ Phi P] Toda and Ogiwara [TO91, TO92] extended this to obtain PH BP[C= P] and related results, while Tarui [Tar91, Tar93] obtained similar results with zero error probability. Allender [All89] used the Valiant Vazirani construction and techniques from the second part of Toda s paper [Tod91] in a circuit setting to prove that AC ....

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 21:316--328, 1992.

....has no satisfying assignments, it answers 0 . An oracle which returns the parity of the number of satisfying assignments has this property, and it follows that NP RP[ Phi P] Toda [Tod89, Tod91] used this to show that the polynomial hierarchy (PH) is contained in BP[ Phi P] Toda and Ogiwara [TO91, TO92] extended this to obtain PH BP[C= P] and related results, while Tarui [Tar91, Tar93] obtained similar results with zero error probability. Allender [All89] used the Valiant Vazirani construction and techniques from the second part of Toda s paper [Tod91] in a circuit setting to prove that ....

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. In The Proceedings of the 6th Annual IEEE Conference on Structure in Complexity Theory, page to appear, 1991.

....survey by Fortnow [For97] for background on these counting classes. Theorem 5.2 implies: 8 Corollary 5.4 If there is a Boolean function f 2 E such that C SAT f (n) 2 2 Omega Gamma n) then NP is contained in SPP. Similarly, we can conditionally derandomize the result by Toda and Ogiwara [TO92] that the polynomial time hierarchy does not add power to GapP in a randomized setting. Applying our techniques to their main lemma yields: Lemma 5.5 Let B be any oracle. If there is a Boolean function f 2 E such that C SAT B f (n) 2 2 Omega Gamma n) then GapP NP B is contained in ....

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 21(2):316--328, 1992.

....( MODm ; m) N Omega Gamma18 ( Omega Gamma N) is m is square free) MODmP is defined to generalize the definition of PhiP. A language L belongs to MODmP if there exists a nondeterministic polynomial time machine M such that x 2 L ( the number of accepting paths of M(x) is non zero modulo m [1, 23, 21] Using our lower bounds we construct an oracle such that: MOD n P is closed under complement and union iff n is a prime power, and MOD n P 6ae MODmP iff n has a prime divisor that is not a divisor of m This oracle is consistent with the known structure of these classes. A MODm polynomial of ....

....O(1) by a folklore theorem [14, 10, 11, 8, 19] 4 An oracle for the conjectured relations among MODmP classes The class MODmP is a generalization of Papadimitrou and Zachos s counting class PhiP. First developed by Cai and Hemachandra [12] these classes have since been studied by many others [9, 10, 14, 1, 23, 21]. It is known that MODmP = MODm 0 P where m 0 is the product of all distinct prime divisors of m [14] that MOD n P MODmP if every prime divisor of n is a divisor of m [14] that MODmP is closed under polynomial time Turing reductions if m is a power of a prime [10] that MODmP is closed ....

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. In Proceedings of the 6th Annual Conference on Structure in Complexity Theory. IEEE Computer Society Press, 1991. To appear.

....the proof by Karp and Lipton [KL80] that if NP P=poly then p 2 = p 2 . One can verify that in each case we get a p 2 expression for L. 2 Whether we can eliminate the advice bit remains an interesting open question. Proof of Theorem 5. 3(4) p 2 = UP NP[1] Toda and Ogihara [TO92] show that UP NP = UP NP[1] Hence we only need to prove that p 2 = UP NP . Consider L, P and the y as in the proof of Theorem 5.3(3) Consider a formula y that encodes y is satisfiable or there is some w y such that h( y ; w ) for some or h( y ; w ) for ....

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 21(2):316--328, 1992.

....acceptance possibility ) and which is complete for NQP (a quantum analog of NP) is hard for the polynomialtime hierarchy. This is done by showing that NQP is precisely the exact counting class [23] coC= P: Theorem 1.1 NQP = coC=P. coC=P, in turn, is hard for PH under randomized reductions [20, 21], and may still be hard even if P = NP. Thus Corollary 1.2 The problem of determining if the acceptance probability of a quantum computation is non zero (QAP ) is hard for the polynomial time hierarchy under polynomial time randomized reductions. We will see in Section 4 that Theorem 1.1 is ....

....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 further strengthens the connections ....

[Article contains additional citation context not shown here]

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 21(2):316-- 328, 1992.

....N) if m is square free) The complexity class MODmP is defined to generalize the definition of PhiP. A language L belongs to MODmP if there exists a nondeterministic polynomialtime machine M such that x 2 L iff the number of accepting paths of M(x) is non zero modulo m (Babai and Fortnow 1990, Toda and Ogiwara 1992, Tarui 1993) In Section 4, we use our lower bounds to construct an oracle such that MOD n P is closed under complementation and union iff n is a prime power, and MOD n P MODmP iff all prime divisors of n are divisors of m. This oracle is consistent with the known structure of these classes. ....

.... MODmP is a generalization of the counting class PhiP (Papadimitriou and Zachos 1983, Goldschlager and Parberry 1986) First developed by Cai and Hemachandra (1990) these classes have since been studied by many others (Beigel 1991, Beigel and Gill 1992, Hertrampf 1990, Babai and Fortnow 1990, Toda and Ogiwara 1992, Tarui 1993) It is known that MODmP = MODm 0 P where m 0 is the product of all distinct prime divisors of m (Hertrampf, 1990) that MOD n P MODmP if every prime divisor of n is a divisor of m (Hertrampf, 1990) that MODmP is closed under polynomial time Turing reductions if m is a power of ....

S. Toda and M. Ogiwara, Counting classes are at least as hard as the polynomialtime hierarchy. SIAM J. Comput. 21 (1992), 316-328.

.... of the past 20 years of research in theoretical computer science has been devoted to proving that NP complete problems (and thus #P complete functions) cannot be feasibly computed unless a wide range of implausible consequences occur (see the survey [Sip92] For example, Toda ( Tod91] see also [TO92,Gup95,RR95] has shown that Turing access to #P subsumes the entire polynomial hierarchy. However, a dynamic programming approach will allow us to perform an exact computation of both the Banzhaf power index and the Shapley Shubik power index even for the relatively large inputs of the censuses ....

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomialtime hierarchy. SIAM Journal on Computing, 21(2):316--328, 1992.

..... 3 5. 51] C C P = C C= P and C= C P = C= C= P. 3 The reader is cautioned that there is an exceedingly minor, easily corrected, arithmetic error in Simon s proof [45, p. 94] 6 6. CH = S k 0 C C z k P. 4 7. 26] PP UP = PP and C=P UP = C= P. 8. [50] PP PH BP PP and P PH BP P. 9. 43] BP PH PH. 10. 16] PP is closed under truth table reductions. 11. 21] C=P is closed under positive truth table reductions. Next, we de ne the closure properties that we will consider. De nition 2.8 1. A function class F is closed under ....

S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. In Proceedings of the 6th Structure in Complexity Theory Conference, pages 2-12. IEEE Computer Society Press, June/July 1991.

.... the latter type of study asks a question Are there any two di erent reducibility notions r and s such that every set that is r reducible to a set in K is s reducible to a set in K and vice versa Along these ways of investigations, there have been proven many results for many complexity classes [1,3,4,5,8,10,11,12,14,18,21,22,27,30,31]. Nevertheless, surprisingly, these studies were on particular complexity classes, and no generalized results have been proven. We do believe that one could extract some essence of the proof techniques exposed in this urry of results. Here in this paper, we generalize the results concerning NP ....

.... classes called counting classes have been introduced, such as PP[9; 23] C= P[23,32] P[20] and MOD k P[2,6] Recently, Toda and Ogiwara, and independently Tarui, showed that the polynomial time hierarchy is polynomial time randomized many one reducible to a set in C= P, PP and MOD k P [24,27]. These results tell us that counting classes are in some sense at least as hard as the polynomial time hierarchy, and thus, widely increased the importance of these classes. C=P can be considered as an extension of coNP. More precisely, C=P is de ned as follows: De nition 4.4 [23,32] A set L is ....

S. Toda and M. Ogiwara, Counting classes are at least as hard as the polynomial-time hierarchy, SIAM J. Comput. 21 (1992), 316-328.

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S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 21(2):316-- 328, 1992.

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S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 21(2):316--328, 1992.

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S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 21(2):316-- 328, 1992.

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S. Toda and M. Ogiwara. Counting classes are at least as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 21(2):316-328, 1992.

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