Y. Han, L.A. Hemaspaandra, T. Thierauf. Threshold Computation and Cryptographic Security. SIAM Journal on Computing, vol.26, pp. 59--78, 1997.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....path on the other hand. We provide evidence that SBP does not coincide with these and other known complexity classes. In particular, in a suitable relativized world, SBP is not contained in 2 . As a consequence we get the same for BPP path . This solves an open question raised by Han et al. [HHT97]. 1 Introduction The use of randomness is one possible extension of conventional deterministic Turing machines. The origins of this idea go back to the work of de Leeuw, Moore, Shannon, and Shapiro [dLMSS56] In 1972 Gill started the investigation of probabilistic polynomial time bounded ....

.... (1 ) rej M (x) the probability gaps for the statements 3 and 2 are defined analogously. It is not difficult to see that with this modification, statement 1 describes just BPP. Moreover, we will see that statement 3 meets exactly the threshold class BPP path which was introduced by Han et al. [HHT97]. But what about statement 2 when demanding a probability gap We will see that apart from the original definition of SBP one can allow any polynomial time computable probability limit. This means that SBP can be characterized by the following equivalence: L 2 SBP if and only if there exist a ....

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Y. Han, L. A. Hemaspaandra, and T. Thierauf. Threshold computation and cryptographic security. SIAM Journal on Computing, 26(1):59--78, 1997.

.... to arbitrary constant ; 1 by forming k tuples of rectangles and taking their intersections as the new approximate majority cover with k = log(1= The de nition of approximate majority covers is similar to threshold computations on Turing machines in a class named BPP path as considered in [HHT97]. We prefer our naming to BPP path , since the class has little similarity to BPP and is not de ned in terms of paths here. It is shown in [HHT97] that BPP path contains MA[ co MA and is hence probably much more powerful than BPP . We immediately get a similar result for communication complexity ....

....The de nition of approximate majority covers is similar to threshold computations on Turing machines in a class named BPP path as considered in [HHT97] We prefer our naming to BPP path , since the class has little similarity to BPP and is not de ned in terms of paths here. It is shown in [HHT97] that BPP path contains MA[ co MA and is hence probably much more powerful than BPP . We immediately get a similar result for communication complexity using Theorems 1 and 3. Theorem 4. APP (f) O(UT s;2s (f) for all s. APP (f) O(UT s;2s ( f) for all s. APP (f) minfO(MA(f) ....

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Y. Han, L.A. Hemaspaandra, T. Thierauf. Threshold Computation and Cryptographic Security. SIAM Journal on Computing, vol.26, pp. 59-78, 1997.

....also holds in the more restricted model defined above, i.e. BalancedLeaf P (A) BalancedLeaf P (A 0 ) under all relativizations if and only if A reduces to A 0 as just sketched. What we want to point out here is that this result does not hold in the unbalanced case. Define BPP path [7] to consist of all languages A such that there exists a nondeterministic polynomial time Turing machine M and an 0 such that for all inputs x, we have: if x 2 A, then the number of accepting paths of M is greater than 1=2 times the total number of paths of M , and if x 62 A, then the ....

....0 ) BPP path . Defining A = def f0; 1g 1f0; 1g and R = f0g , obviously BalancedLeaf P (A; R) Leaf P (A; R) NP. Since there exists an oracle separating NP and BPP, we know that (A; R) does not reduce to (A 0 ; R 0 ) but on the other hand NP BPP path for all oracles (see [7]) So we see that the main result from [4] does not hold in the unbalanced case. 3. Logtime Leaf Languages for Balanced Trees Jenner, McKenzie, and Th erien showed in [14] that balanced computation trees with leaf languages from one class of the logarithmic time hierarchy characterize the ....

Y. Han, L. Hemachandra, T. Thierauf, Threshold computation and cryptographic security; Proceedings of the 4th International Symposium on Algorithms and Computation (1993), LNCS 762, pp. 230--239.

....in [HO94] An NPTM is said to be normalized if its computation tree is a complete binary tree of branching depth p(jxj) for some polynomial p. Though normalization is known not to be important in defining NP languages or even as shown by Simon [Sim75] PP languages, there is evidence [HHT] that it is critical in defining BPP languages. Here, it will be important in our definition of SelfPath; however, in our definition of SelfOutput, the same class would be defined even if a normalization requirement were added. Consider nondeterministic polynomial time machines each of whose ....

Y. Han, L. Hemachandra, and T. Thierauf. Threshold computation and cryptographic security. SIAM Journal on Computing. To appear.

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Y. Han, L. Hemaspaandra, and T. Thierauf. Threshold computation and cryptographic security. SIAM Journal on Computing, 26(1):59--78, 1997.

....1 . The length of a string x is denoted by x . denotes the set of string in # # of length exactly n. We say that a set A is a nontrivial subset of B if # A # B. usually denotes a standard, fixed, easily computable and invertible multi arity pairing function (e.g. that of [HHT97] or a standard, fixed, easily computable and invertible 2 ary pairing function (which holds will be clear from context) For any set A, #A denotes the characteristic function of A. That is, for any x A, #A (x) 0, and for any x A, #A (x) 1. A boolean predicate Q is a total function ....

Y. Han, L. Hemaspaandra, and T. Thierauf. Threshold computation and cryptographic security. SIAM Journal on Computing, 26(1):59--78, 1997.

....Our alphabet will be Sigma = f0; 1g. Let our pairing function h Delta Delta Deltai be any multi arity onto, polynomial time computable, polynomial time invertible function (that is, the ranges of different arities are disjoint, and the union over all arities covers Sigma , see, e.g. [16]) For each partial, multivalued function f , set f(x) denotes the set of values of f on input x. If f(x) is undefined, then set f(x) We will use this notation for partial single valued functions also, to avoid ambiguity regarding equality tests between potentially undefined values. For any ....

Y. Han, L. Hemaspaandra, and T. Thierauf, Threshold computation and cryptographic security, in Proceedings of the 4th International Symposium on Algorithms and Computation, Springer-Verlag Lecture Notes in Computer Science #762, Dec. 1993, pp. 230--239.

....choice. However, under our definition, on each input x every possible binary string of exactly p(jxj) bits corresponds uniquely to a computation path. Though normalization is known not to be important in defining NP languages or even as shown by Simon [Sim75] PP languages, there is evidence [HHT] that the type of normalization one uses is critical in defining BPP languages. Here, it will be important in our definition of SelfPath; however, in our definition of SelfOutput, the same class would be defined even if a normalization requirement were added. We now introduce the classes ....

....equivalent. This holds true in the unbounded error case. It is also true in the bounded error case (i.e. BPP is the same class whether one defines it using the former normalization or the latter normalization) may be different depending on whether or not there is a normalization requirement [HHT, JMT94,HVW96] Recall that SelfOutput does remain the same whether defined with or without the requirement that the underlying machines be normalized. Does SelfPath remain the same class if its normalization requirement is removed Clearly the resulting class contains SelfPath, i.e. P #P[1] ....

Y. Han, L. Hemaspaandra, and T. Thierauf. Threshold computation and cryptographic security. SIAM Journal on Computing. To appear.

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Y. Han, L.A. Hemaspaandra, T. Thierauf. Threshold Computation and Cryptographic Security. SIAM Journal on Computing, vol.26, pp. 59--78, 1997.

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