S. Toda. On the computational power of PP and \PhiP. in: Proc. 30th IEEE Symp. on Foundations of Computer Science 1989, pp. 514-519.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....main technical construction of their result is an isolation lemma that reduces an instance of SAT to an instance with a unique solution in randomized polynomial time. Valiant and Vazirani s theorem have been applied to prove several important theorems of computational complexity theory (see, e.g. [12]) 3. PRELIMINARIES In this section, we briefly survey constraint based watermarking methodology. The watermarking problem takes two inputs: the initial instance of the optimization problem (which corresponds to optimization synthesis or compilation problem) and the owner s signature in some ....

S. Toda. "On the computational power of PP and \PhiP". Proc. IEEE Foundations of Computer Science, pp. 514--519, 1989.

....description [4] Since it is not possible to directly compare #P, a class of functions, to PH, a class of languages, we take the class PP of problems asking whether more than half of the computations of a nondeterministic machine are accepting. This class is closely related to #P, Toda s theorem [12] states that PH P (2) Note that P is no simpler than #P, since knowing the full number cannot be simpler that knowing the most significant bit. Finally, we will show that some power estimation problems belong to PSPACE, the class of all decision problems that can be solved using ....

Seinosuke Toda. On the computational power of PP and \PhiP . In 30th Annual Symposium on Foundations of Computer Science, pages 514--519, Research Triangle Park, North Carolina, 30 October--1 November 1989. IEEE. 6

....by nondeterministic machines which accept if and only if the number of accepting paths is exactly equal to a given number. The power of C=P can be seen from the following facts: Facts: i) PP PH NP C=P . ii) C=P PH BP Delta C= P. Fact (i) is a consequence of Toda s theorem [18] that PP PH P PP combined with a theorem of Tor an [22] which states NP PP = NP C=P . It is significant since it states that C=P is hard for the polynomial hierarchy under nondeterministic reductions. Fact (ii) was proved by Toda and Ogiwara [20] and in a stronger form by Tarui [16] It ....

S. Toda, "On the computational power of PP and \PhiP", Proceedings 30th IEEE Symposium on Foundations of Computer Science (1989) 514-519.

....interesting open question left by the work of [6] is whether there is an oracle separating the hierarchy PP PH . More precisely, is there an oracle A such that the following is true: 8d) PP Sigma p;A d 1 6 PP Sigma p;A d ) Indeed, in light of surprising results such as Toda s theorem [12] one might wonder if it is possible that PP PH is a proper hierarchy in any relativized world. One step in this direction was provided by [5] since the result of that paper trivially implies an oracle A such that PP NP A 6 PP A . 5] also leaves open the possibility that there is an ....

....functions. Our results show that small monotone perceptrons computing these functions cannot exist. It is necessary to address a result due to Allender [1] which issues a warning as to the significance of results about monotone threshold circuits. Essentially due to the fact that Toda s theorem [12] relativizes, Yao s result (Theorem 5 above) does not carry over to the non monotone case, at least not via exponential lower bounds. Allender showed explicitly that any AC (0) circuit can be simulated by a depth 3 non monotone TC (0) circuit of sub exponential but superpolynomial size. ....

S. Toda, "On the computational power of PP and \PhiP", in Proceedings 30th IEEE Symposium on Foundations of Computer Science, IEEE Computer Society Press (1989) 514-519.

..... So, there wasn t an obvious way to amplify the Valiant Vazirani reduction from SAT to USAT. Nevertheless, the vv m reduction from SAT to USAT proved to be useful in many areas of research. For example, Richard Beigel [Bei88] used it to show that SAT is superterse unless RP = NP. Also, Toda [Tod89] used a similar reduction in his proof that PH P #P[1] This result, in turn, led to the Lund, Fortnow, Karloff and Nisan [LFKN90] result: PH IP. So, there should be little doubt in the reader s mind regarding the usefulness of the Valiant Vazirani reduction. The more pertinent questions ....

S. Toda. On the computational power of PP and \Phi P. In Proceedings of the IEEE Symposium on Foundations of Computer Science, pages 514-- 519, 1989.

....to deal with depth 3 circuits with small weights and small bottom fanin. These lower bounds agree very well with our intuition, as do the results about monotone threshold circuits by Yao [21] extended in [7] The first surprise was presented by Allender [1] who, inspired by the results of Toda [18], proved that depth 3 threshold circuits of subexponential size could do all of AC 0 . Yao [22] extended this to ACC 0 which consists of all functions computable by polynomial size constant depth circuits over the basis f; mod mg for an arbitrary fixed m (note that this last class could ....

S. Toda, On the computational power of PP and \PhiP , in Proceedings of the 30th IEEE Symposium on Foundations of Computer Science, 1989, 514--519.

....the polynomial hierarchy and PSPACE hold also for the quasilinear hierarchy (QLH) and QLSPACE. Section 3 shows that the randomized reduction from NP to parity given by Valiant and Vazirani [VV86] and used by Toda [Tod91] previously proved by constructions which run in quadratic time (see [VV86, Tod89, CRS93, Gup93] can be made to run in time qlin . Our construction also markedly improves the number of random bits needed and the success probability, and uses error correcting codes in an interesting manner first noted in [NN90] Section 4 studies what may be the major difference between ....

S. Toda. On the computational power of PP and \Phi P. In Proc. 30th FOCS, pages 514--519, 1989.

....property that whenever OE x has a unique satisfying assignment, it always answers 1 , and whenever OE x 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 ....

....L 2 NP) to an oracle for Unique SAT which meets the above promise condition. Given L 2 NP, let R(x; y) be a witness predicate for L; i.e. such that for all x, x 2 L ( 9y 2 f 0; 1 g p(n) R(x; y) where p is a polynomial. For the case L = SAT, VV86] took p(n) n. Here we follow Toda [Tod89] and write p as short for p(n) The 2 machine M first flips p 2 dlog(p 1)e coins to form p many vectors w 1 ; w p 2 f 0; 1 g p and an integer k, 0 k p. Then M makes p many nondeterministic moves to form y, and accepts iff R(x; y) 8i : 1 i k) y Delta w k = 0; 3) ....

S. Toda. On the computational power of PP and \Phi P. In The Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, pages 514--519, 1989.

....polynomial size, small weight threshold circuits. Threshold circuits have been shown to be surprisingly powerful. It is implicit in work by Beame, Cook, and Hoover [4] that integer division can be carried out by polynomial size threshold circuits of constant depth. Allender [1] inspired by Toda [28]) shows that any function in AC 0 can be computed by depth 3 majority circuits of quasi polynomial size. Yao [33] extends this to all of ACC 0 (see also [5] There are some strong lower bounds for majority circuits of very small depth. Hajnal et al. 11] prove exponetial lower bounds on the ....

S. Toda. On the computational power of PP and \PhiP . In Proc. 30th IEEE Symposium on Foundations of Computer Science, pages 514--519, 1989.

....about the negation of f k , f k and the corresponding distributions p k and q k . It is interesting to note that while the functions f k cannot be computed by depth k Gamma 1 monotone threshold circuits of size 2 n 1 2k , by a result of Allender [2] which is based on work by Toda [20]) they can be computed by depth 3 general threshold circuits of size 2 O( log n) k ) This might be taken as another piece of evidence that monotonicity is a severe restriction, and that new techniques probably have to be developed to attack general threshold circuits of small depth. A ....

S. Toda. On the computational power of pp and \Phip. Proceedings 30th Annual IEEE Symposium on Foundations of Computer Science, pages 514-- 519, 1989.

.... These circuits capture essential aspects in neural net computations [RMtPrg86] Hop82] They have been shown to be equivalent to constant depth arithmetic circuits over finite fields [Rei87] SFB89] and were recently related to simulating the polynomial hierarchy by counting oracles [Tod89] All89] The fundamental question of whether the inclusion TC 0 NC 1 is proper, surfaced naturally after AC 0 6= TC 0 was resolved ( FSS84] Ajt83] and their improvements) and after the results about constant depth circuits with prime modulo gates were proved ( Raz87] Smo87] ....

S. Toda. On the computational power of PP and \PhiP. In Proc. 30th IEEE Symp. on Foundations of Computer Science, pages 514--519, 1989.

.... Delta Delta (3) of predicates obtained from the base class P by finite quantification. Note the analogy with the Kleene s arithmetic hierarchy from classical recursion theory. At the base of the hierarchy we find the classes P and NP = 9P that were encountered in the previous section. Toda [17] has shown the following. Theorem 1 Every predicate in PH is polynomial time Turing reducible to a function in #P. In other words, the class #P essentially contains the entire polynomial hierarchy. The power of the counting operator # to simulate arbitrary finite alternations of the operators 9 ....

Seinosuke Toda, On the computational power of PP and \Phi P, Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, Computer Society Press (1989), 514--519.

....to Ryser, which runs in time O(n 2 2 n ) Rys63] In terms of computational complexity theory, in 1979, Valiant [Val79] proved the seminal result that computing the permanent of integer matrices is complete for the counting complexity class #P, and is therefore NP hard. A decade later, Toda [Tod89] demonstrated the surprising power of #P; Toda s theorem, Supported in part by NSF grant CCR 9634665, and by a Guggenheim Fellowship. Supported in part by an NSF CAREER award CCR 9734164. together with Valiant s result, implies that the permanent is hard for the entire polynomial time ....

S. Toda. On the computational power of PP and \PhiP. In Proceedings of the 30th FOCS, pages 514--519, 1989.

....of such proofs implies unprovability of the principle in bounded arithmetic. 7 Many important open problems in bounded arithmetic deal with counting, see [16, 2] While the questions from [16] about explicit counting in the hierarchy of bounded predicates were to a large extent answered in [24] (assuming that the polynomial time hierarchy does not collapse) questions about the relations between various simple principles of counting (such the pigeonhole principle and the parity principle considered in [2] remain open. Let me give an example of such a problem formulated for ....

S. Toda: "On the computational power of PP and \PhiP ", Proc. 30th IEEE Symp. on the Found. of Computer Science, (1989), pp. 514-519.

....d polynomial size, small weight threshold circuits. Threshold circuits have been shown to be surprisingly powerful. It is implicit in work by Beame, Cook, and Hoover [4] that integer division can be carried out by polynomial size threshold circuits of constant depth. Allender [1] inspired by Toda [29]) shows that any function in AC 0 can be computed by depth 3 majority circuits of quasi polynomial size. Yao [34] extends this to all of ACC 0 (see also [5] There are some strong lower bounds for majority circuits of very small depth. Hajnal et al. 11] prove exponetial lower bounds on the ....

S. Toda. On the computational power of PP and \PhiP . In Proc. 30th IEEE Symposium on Foundations of Computer Science, pages 514--519, 1989.

....d polynomial size, small weight threshold circuits. Threshold circuits have been shown to be surprisingly powerful. It is implicit in work by Beame, Cook and Hoover [4] that integer division can be carried out by polynomial size threshold circuits of constant depth. Allender [1] inspired by Toda [23]) shows that any function in can be computed by depth three majority circuits of quasi polynomial size. Yao [27] extends this to all of (see also [5] There are some strong lower bounds for majority circuits of very small depth. Hajnal [10] prove exponetial lower bounds on the size of depth two ....

S. Toda. On the computational power of PP and \PhiP . In Proc. 30 IEEE Symposium on Foundations of Computer Science, pages 514--519, 1989.

....that for some y, the r th bit of the (i; j) th entry of a product of matrices is 1. The question is then how to write this existential statement as an arithmetic expression. Note also that this is intimately connected with a similar kind of bit separation that is important in Toda s theorem, [Tod89]. A solution to this problem would give a simplified and generalized version of that theorem. See [GKT] for related work. 2. Q (Sn ) 6 fop Q (S 5 ) This is true iff NC 1 is strictly contained in L. 3. Q (Z) fop Q (S 5 ) This is true iff Q (Z) 2 NC 1 . 4. Q (S 5 ) fop Q (Z) ....

Seinosuke Toda (1989), On the Computational Power of PP and \PhiP,30th IEEE FOCS Symp., 514-519.

....of fan in (log n) c at the leaves. This result improves the known upper bounds for the class ACC. 1 Introduction The complexity classes PP (probabilistic polynomial time [Gi 77] and Phi P (parity P, PaZa 83, GoPa 86] have received much attention since the well known result by Toda [Tod 89] proving that the polynomial time hierarchy (PH) is Turing reducible to PP. These classes are closely related to the class of counting functions #P [Va 79] that count the number of accepting paths on nondeterministic Turing machines. Observe that sets in PP and Phi P can be respectively decided ....

....Clearly ACC contains AC 0 and is contained in TC 0 . Since the PARITY function cannot be computed in AC 0 the first inclusion is proper. Barrington [Ba 89] conjectured that the second inclusion is also proper i.e. TC 0 6aeACC, but no proof of this fact has been obtained. Using Toda s result [Tod 89] and building on some work on AC 0 by Allender and Hertrampf [Al 89] AlHe 90] Yao [Yao 90] proved the first non trivial upper bound for ACC. He showed that every language in ACC is recognized by a family of depth 2 probabilistic circuits of size 2 (log n) c with a symmetric gate at the root ....

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S. Toda, On the computational power of PP and \Phi P. In Proceedings of the 30th Symposium on Foundations of Computer Science 1989, 514-519.

....x and random input r can be regarded as the opinion of voter r about whether x is in L. From this point of view PP is the class of all languages L such that membership of x in L can be determined via election with 2 poly(jxj) voters, every voter being polynomial time bounded. Second, as shown in [12], the class PP proved to be suprisingly powerful: polynomial hierarchy PH is Turing reducible to PP. Third, PP is closed under polynomial truth table reductions (see [4] and [5] Thus the class PP has a rather regular structure. 2 Definitions We consider languages over the binary alphabet B = ....

S. Toda. "On the computational power of PP and \PhiP", Proc. of 30th Symp. on Found. of Comp. Sci., 1989, pp. 514--519.

....tick uaimzti.mathematik.uni mainz.de ABSTRACT We introduce the class MP of languages L which can be solved in polynomial time with an oracle that returns one bit of a #P function value f(x) That one bit suffices for any L in the polynomial hierarchy follows from the proof of S. Toda s theorem [Tod89, Tod91] that PH P #P , so PH BP[ Phi P] C Phi P MP P #P[1] We show that the middle bit of f(x) is as powerful as any other bit, and that a wide range of bits around the middle have the same power. By contrast, the O(log n) many least significant bits are equivalent to Phi P [BGH90] ....

....bit. We denote by MP the class of languages that can be decided with the help of any one selected bit. MP is a natural generalization of both PP and Phi P which seems to be easier than the much studied class P #P . One reason for the importance of MP is that S. Toda s celebrated proof [Tod89, Tod91] that the polynomial hierarchy PH is contained in P #P actually shows the following (where we have written C[ Phi P] for Toda s P Delta Phi P) Theorem 1. after Toda] PH C[ Phi P] MP. In section 2 we give basic definitions and observe that the power of MP is essentially captured by ....

S. Toda. On the computational power of PP and \Phi P. In The Proceedings of the 30th Annual IEEE Symposium on Foundations of Computer Science, pages 514--519, 1989.

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S. Toda. On the computational power of PP and \PhiP. In Proceedings of the 30th IEEE Symposium on Foundations of Computer Science, pages 514--519. IEEE Computer Society Press, 1989.

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S. Toda. On the computational power of PP and \PhiP. in: Proc. 30th IEEE Symp. on Foundations of Computer Science 1989, pp. 514-519.

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S.Toda, On the computational power of PP and \PhiP , 30

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S. Toda "On the computational power of PP and \PhiP ", Proc. 30th IEEE Symp. on the Foundations of Computer Science", pp. 514-519, 1989

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S. Toda. On the computational power of PP and \Phi P. In Proceedings of the IEEE Symposium on Foundations of Computer Science, pages 514--519, 1989.

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