R. Beigel. Perceptrons, PP, and the Polynomial Hierarchy. Computational Complexity, 4:339--349, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....with nonrelativizable proof techniques. Since these techniques are known to be rare and difficult it is most likely that we are still a long way off from the final solution of these separation questions. The separation results below will be derived on one hand from known oracle constructions [Yao85, Ver92, Bei94, For99] and on the other hand from a new construction that is described in the proof of Theorem 5.15. In particular, in this new relativized world, SBP is not contained in 2 . Since SBP BPP path holds relativizable we will see that our oracle shows that BPP path 6 R and BPP path 6 2 in some ....

....A such that AM coAM . Corollary 5.4 There exists an oracle A such that AM and coAM . Proof : Define A to be the oracle from Theorem 5.3. The corollary follows since SBP PP in all relativized worlds. 2 The following oracle goes back to a construction of Beigel. Theorem 5. 5 ([Bei94]) There exists an oracle A such that P . C 2 fAPP;AWPP;WAPP;BP UP;BPP;P;WPP;SPP;Few;BQP;EQPg. 1. NP 2. 9 BPP 3. MA 6. Proof: Define A to be the oracle from Theorem 5.5 and assume that NP APP . In [Li93a, Li93b] Li proved that APP is low for PP. Since the ....

R. Beigel. Perceptrons, PP, and the polynomial hierarchy. Computational Complexity, 4:339--349, 1994.

....#[M(x) L(x) i 1 Gamma 2 Gammaq(jxj) j total M (x) The proof is analogous to the corresponding proof for BPP. BPP is closed under Turing reductions [Ko82,Zac82] However, no relativizable proof can establish the closure of BPP path under Turing reductions. In particular, Beigel [Bei92] constructed an oracle relative to which P NP is not contained in PP. Since BPP path clearly is contained in PP (and the proof relativizes) it follows that, relative to the same oracle, P NP is not contained in BPP path , and hence, since NP BPP path , BPP path is not closed under Turing ....

R. Beigel. Perceptrons, PP, and the polynomial hierarchy. In Proceedings of the 7th Structure in Complexity Theory Conference, pages 14--19. IEEE Computer Society Press, June 1992.

....an oracle that separates the levels in the PP PH hierarchy, and in fact that there exists an A such that S p;A k 6 PP S p;A k Gamma2 . The fact that our basis, i.e. the One in a box theorem by Minsky and Papert, implies that NP NP 6 PP under an oracle was noted by Fu [4] Beigel [1] has strengthened this separation to obtain that P NP 6 PP under an oracle, and in the last section of this paper we use his result as a basis for a lower bound for perceptrons with bounded weights. Using this lower bound, we get an oracle A such that D p;A k 6 PP S p;A k Gamma2 . We ....

....perceptron is, due to Theorem 8, not powerful enough to compute h n i d and is thus unable to determine if the string 1 n i is in L d (B) for all B. It is therefore possible to set the y B z of length n i such that M B i makes an error on 1 n . 4 Improving the oracle separation Beigel [1] obtained the oracle separation P NP 6 PP. To do this he introduced the language ODD MAX BIT and proved a relation between the bottom fan in, the maximum weight, and the size for perceptrons deciding it. Definition 13 (Beigel [1, Definition 2] ODD MAX BIT is the set of all strings over ....

Richard Beigel. Perceptrons, PP, and the polynomial hierarchy. Computational Complexity, 4(4):339--349, 1994.

....3.14 If #P is closed under division by 2, then P is C= P low and PP low. Proof Since C=P P C= P P C= P and PP P C P P C P, P = SPP implies both C=P P C=P and PP P PP. Q.E.D. Toda [47] has shown that PH P PP ; it is not known whether PH PP (see [6,4]) However, if we assume that #P is closed under division by two, this inclusion indeed holds. Corollary 3.15 If #P is closed under division by 2, then PH PP. Proof Assume that #P is closed under division by 2. Notice that PH P PH BP P PP P . Combining this with Corollary 3.14, ....

R. Beigel. Polynomial interpolation, threshold circuits, and the polynomial hierarchy. Manuscript, December 1989.

....accepting computation paths. It is easily derived from the de nitions that C=Q PP and coNP C=Q, but it is not known until now whether PP C=Q or whether NP C= Q. Many researchers have intensely studied the computational power of C=Q and PP in order to answer the above unsolved questions [1], 3] 4] 5] 9] 12] 13] 21] 22] 29] 30] 31] 33] 34] 35] These studies fall into two types. On one hand, researchers have tried to gure out the relationships between PH (the polynomial time hierarchy) and C=Q and or PP. In such studies, the lower bound and the upper bound ....

....known results for this type are the following: 1. PH P PP [31] 2. PH BP C=Q and PH BP PP [33] 29] These statements imply that every set in PH is polynomial time randomized many one reducible to a set in C=Q (thus in PP) 3. There is a relativized world in which p 2 6 PP [1]; that is, relativizable proof techniques are meaningless to prove that p 2 PP. 4. There is a relativized world in which NP 6 C= Q[35] that is, relativizable proof techniques are meaningless to prove that NP C=Q. 5. There is a relativized world in which BPP 6 P C=Q , and thus, p ....

R. Beigel: \Polynomial interpolation, threshold circuits, and the polynomial hierarchy", manuscript (December 1989).

....time Turing machines which accept their input if and only if the rightmost accepting path has an odd number. Defining ODDMAXBIT = def Phi (x 1 Delta Delta Delta x n ) fi fi max Phi i fi fi x i = 1 Psi j 1 (mod 2) Psi ; we thus get: BalancedLeaf P (ODDMAXBIT) P NP . Beigel [Bei94] showed that ODDMAXBIT is not represented by the sign of a low degree polynomial with small coefficients. Stated in terms of circuits this means: ODDMAXBIT 62 qC[maj; log O(1) n) Suppose now that for all oracles Y, P NP Y PP Y ; then by Proposition 4.1 and Lemma 4.2, ODDMAXBIT ....

R. Beigel. Perceptrons, PP, and the polynomial hierarchy. Computational Complexity, 4:339--349, 1994.

....if and only if PF NP = PF NP tt . Beigel, Hemachandra,and Wechsung [BHW91] showed that P NP tt PP. Thus, PF NP = PF NP tt implies P NP PP, which suggests that the classes PF NP and PF NP tt are not identical. There is an oracle relative to which P NP is not a subset of PP [Bei94] Recall that P NP tt = P NP (O(log n) Hem89, Wag90, BH91] Indeed, P NP (O(log n) is a natural and robust complexity classes that has natural complete sets [Kre88, KSW87, Kad89, Wag90] Since maxsat is complete for PF NP , the question of whether PF NP PF NP tt is equivalent to ....

R. Beigel. Perceptrons, PP, and the polynomial hierarchy. Computational Complexity, 4:339-- 349, 1994.

....majority of its inputs have value 1. TC 0 is the class of languages accepted by threshold circuits of polynomial size and depth O(1) TC 0 k denotes the class of languages accepted by threshold circuits of depth k. 3 In the meantime, this question has been addressed by Richard Beigel. In [Be 92], Beigel presents an oracle relative to which P NP is not contained in PP. In fact, he shows the stronger result that the inclusion presented in [BHW 91] can not be improved using relativizable proof techniques. The following points explain in part why TC 0 has been the focus of attention. ....

R. Beigel, Perceptrons, PP, and the polynomial hierarchy, Proc. 7th IEEE Structure in Complexity Theory Conference, pp. 14--19.

....the (relativized) simple separation of the levels of the PP PH hierarchy [BU] also can be turned into a strong separation. As a special case, this includes the existence of a PP immune set in P NP (and thus in PH) relative to some oracle, which improves upon a simple separation of Beigel [Bei94]. 2 Preliminaries Fix the two letter alphabet Sigma df = f0; 1g. The set of all strings over Sigma is denoted Sigma , and the set of strings of length n is denoted Sigma n . For any string x 2 Sigma , let jxj denote its length. For any set L Sigma , the complement of L is L ....

....[E L rejects 0 n . By essentially the same arguments, also the very recent result of Berg and Ulfberg [BU] that there is an oracle relative to which the levels of the PP PH = S d0 PP Sigma p d hierarchy separate (which generalizes Beigel s result that (9A) P NP A 6 PP A ] [Bei94]) can be strengthened to level wise strong separations of this hierarchy. The proof of Theorem 4.3 is deferred to the appendix. Theorem 4.3 For any d 1, there exists some oracle F such that P Sigma p;F d contains a PP Sigma p;F d Gamma1 immune set. In particular, P NP F (and thus PH ....

R. Beigel. Perceptrons, PP, and the polynomial hierarchy. Computational Complexity, 4:339--349, 1994.

....seen, the inclusion coNP NP=OptP implies PH = P NP , we can state the following theorem. Theorem 4.1. PH 6= P NP ) NP=OptP 6= P= OptP. Furthermore, by the recent result of Toda [39] that PH P PP , it follows that P NP[O(logn) 6= PP and P NP 6= PP= OptP unless PH = P NP . Beigel [7] constructed an oracle A such that P NP A Gamma PP A 6= Since P NP[O(logn) PP [9] oracle A also separates P NP[O(logn) and P NP (for a direct proof see [14] Cai et al. 15] showed the existence of an oracle A such that relative to A the boolean hierarchy is infinite, i.e. 8 ....

R. Beigel. Perceptrons, PP, and the polynomial hierarchy. In Proceedings 7th Structure in Complexity Theory Conference, p. 14-19. IEEE Computer Society, 1989.

....is known, for example, that NC 1 = BF [11] so we ask whether PP and PL are closed under Booleanformula reductions. However, it appears that BF is too large to be equal to PP, because of the following lower bound which we prove: n= log log n T BF Therefore, Beigel s oracle A from [3], which makes log n log log n T 6 PP , also makes Thus the answer to whether PP is closed under Boolean formula reductions will require nonrelativizing techniques. While we are unable to determine whether PP, we do prove a slightly weaker upper bound on BF Finally, we ....

....g, where jT i j is the size of T i . v t (log n)v 1 : log n)v t Gamma1 = log n) v 1 : v t Gamma1 ) log n) t Gamma 1)v t Gamma1 (log n) t Gamma 1) log n) t Gamma 1) Since t f(n) O(logn= loglog n) we have v t = n . Corollary 39. P Theorem 40 ([3]) If f(n) 6= O(logn) there is an oracle A such that P . 6= PP By Theorem 40 and Corollary 41, it is impossible to prove that PP is closed under BF reductions by relativizable methods. Although it is well known that NC 1 = BF, by combining Theorem 38 and Theorem 40, we see now that ....

R. Beigel. Perceptrons, PP, and the polynomial hierarchy. In Proceedings of the 7th Annual Conference on Structure in Complexity Theory, pages 14--19, 1992.

....wonder whether that result is tight or whether P = PP for some function f(n) 6= O(log n) Since our proof techniques are valid for computation relative to an oracle, we turn to relativized complexity for insight. There is an oracle A for which P f(n) T PP if and only if f(n) O (log n) [5]. Since NP PP for all B, with that same A we have ( f(n) O (log n) This is circumstantial evidence that our collapse is the best possible, which is surprising, because we have come to expect that collapses translate upward. For example, if P 2 T = NP then PH = NP [20] On the ....

....and only if f(n) O(log n) By combining the results of this paper with some previously known lower bounds we can prove relativized separations all the way up the query hierarchies. and g(n) 6= O(f(n) Then there is an oracle A such that Proof: We take the following definitions from [5]. ffl CHUNK(A; n; f) is the set containing the lexicographically first f(n) strings, starting from 0 , that belong to A. ffl max(B) is the lexicographically greatest element of B, if it exists; the empty string, otherwise. ffl ODD MAX ELEMENT f is f0 : max(CHUNK(A; n; f) ends in a 1g. ....

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Richard Beigel. Polynomial interpolation, threshold circuits, and the polynomial hierarchy. YALEU/DCS/TR 843, Yale University, Dept. of Computer Science, January 1991.

....numbers, so 1 o(1) which is false. Therefore (log s) n Omega Gamma49 , so s = 2 n Omega Gamma4 : Instead of using Smolensky s idea, one could use the result from [3] that a small order perceptron cannot even approximate parity. 5. A Suggestion on Terminology Many papers [8, 9, 1, 5, 19, 6] have considered circuits whose bottom level consists of AND gates or OR gates having fanin polylog n, while gates at other levels have unbounded fanin. Perceptrons have a single threshold gate at the root and polylog fanin AND gates at the bottom level. Since they have only one threshold gate, it ....

R. Beigel. Polynomial interpolation, threshold circuits, and the polynomial hierarchy. YALEU/DCS/TR 843, Yale University, Dept. of Computer Science, Jan. 1991.

....result is tight or whether P PP f(n) T = PP for some function f(n) 6= O(log n) Since our proof techniques are valid for computation relative to an oracle, we turn to relativized complexity for insight. There is an oracle A for which P NP A f(n) T PP A if and only if f(n) O (log n) [5]. Since NP B PP B for all B, with that same A we have P PP A f(n) T = PP A ( f(n) O (log n) This is circumstantial evidence that our collapse is the best possible, which is surprising, because we have come to expect that collapses translate upward. For example, if P NP 2 T = NP ....

....previously known lower bounds we can prove relativized separations all the way up the query hierarchies. Theorem 33. Assume that log n f(n) n O(1) and g(n) 6= O(f(n) Then there is an oracle A such that P NP A f(n) T 6 P PP A g(n) T : Proof: We take the following definitions from [5]. ffl CHUNK(A; n; f) is the set containing the lexicographically first f(n) strings, starting from 0 n , that belong to A. ffl max(B) is the lexicographically greatest element of B, if it exists; the empty string, otherwise. ffl ODD MAX ELEMENT A f is f0 n : max(CHUNK(A; n; f) ends in a ....

[Article contains additional citation context not shown here]

Richard Beigel. Polynomial interpolation, threshold circuits, and the polynomial hierarchy. YALEU/DCS/TR 843, Yale University, Dept. of Computer Science, January 1991.

....so 1 1 4 1 2 o(1) which is false. Therefore (log s) d = n Omega Gamma49 , so s = 2 n Omega Gamma4 =d) Instead of using Smolensky s idea, one could use the result from [3] that a small order perceptron cannot even approximate parity. 5. A Suggestion on Terminology Many papers [8, 9, 1, 5, 19, 6] have considered circuits whose bottom level consists of AND gates or OR gates having fanin polylog n, while gates at other levels have unbounded fanin. Perceptrons have a single threshold gate at the root and polylog fanin AND gates at the bottom level. Since they have only one threshold gate, ....

R. Beigel. Polynomial interpolation, threshold circuits, and the polynomial hierarchy. YALEU/DCS/TR 843, Yale University, Dept. of Computer Science, Jan. 1991.

....that NC NP 1 = BF NP [11] so we ask whether PP and PL are closed under Booleanformula reductions. However, it appears that BF PP is too large to be equal to PP, because of the following lower bound which we prove: P PP log 2 n= log log n T BF PP : Therefore, Beigel s oracle A from [3], which makes P NP A log n log log n T 6 PP A , also makes BF PP A 6 PP A : Thus the answer to whether PP is closed under Boolean formula reductions will require nonrelativizing techniques. While we are unable to determine whether BF PP PP, we do prove a slightly weaker upper ....

....: log n)v t Gamma1 = log n) v 1 : v t Gamma1 ) log n) t Gamma 1)v t Gamma1 (log n) t Gamma1 (t Gamma 1) log n) t Gamma 1) t Gamma1 : Since t f(n) O(logn= loglog n) we have v t = n O(1) Corollary 39. P PP log 2 n= log log n T BF PP : Theorem 40 ([3]) If f(n) 6= O(logn) there is an oracle A such that P NP A f(n) T 6 PP A . Corollary 41. There exists an oracle A such that BF PP A 6= PP A : By Theorem 40 and Corollary 41, it is impossible to prove that PP is closed under BF reductions by relativizable methods. Although it is ....

R. Beigel. Perceptrons, PP, and the polynomial hierarchy. In Proceedings of the 7th Annual Conference on Structure in Complexity Theory, pages 14--19, 1992.

No context found.

R. Beigel. Perceptrons, PP, and the Polynomial Hierarchy. Computational Complexity, 4:339--349, 1994.

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Richard Beigel. Perceptrons, pp, and the polynomial hierarchy. Computational Complexity, 4:339--349, 1994.

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R. Beigel. Perceptrons, PP, and the Polynomial Hierarchy. Computational Complexity, 4:339-- 349, 1994.

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R. Beigel. Perceptrons, PP, and the Polynomial Hierarchy. Computational Complexity, 4:339{ 349, 1994.

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Richard Beigel. Perceptrons, pp, and the polynomial hierarchy. Computational Complexity, 4:339--349, 1994.

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Beigel,R.: Perceptrons, PP, and the Polynomial Hierarchy, Proc. of IEEE Conf. SCT'92, 14--19.

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Beigel,R.: Perceptrons, PP, and the Polynomial Hierarchy. Proc. of SCT'92, 14--19.

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