J. Feigenbaum, S. Kannan and N. Nisan. Lower Bounds on Random-SelfReducibility. In Proceedings of Structures 1990, 1990.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....trusted not to err, however the oracles are not restricted to answer questions of the form What is the value of f(x) Beaver Feigenbaum Kilian Rogaway] later improved this result to show that it can be done with O(jxj= log jxj) oracles. The notion of random self reducibility, as defined in [Feigenbaum Kannan Nisan] is related to private checking because it is possible to privately compute random self reducible functions. Recently [Feigenbaum Kannan Nisan] have shown that random boolean functions are not k random self reducible for any polynomial k, and that if a function is 2 random self reducible, then ....

....Rogaway] later improved this result to show that it can be done with O(jxj= log jxj) oracles. The notion of random self reducibility, as defined in [Feigenbaum Kannan Nisan] is related to private checking because it is possible to privately compute random self reducible functions. Recently [Feigenbaum Kannan Nisan] have shown that random boolean functions are not k random self reducible for any polynomial k, and that if a function is 2 random self reducible, then the function can be computed nonuniformly in nondeterministic polynomial time. Beaver Feigenbaum] Lipton] show that any function that is a ....

Feigenbaum, J., Kannan, S., Nisan, N., "Lower Bounds on RandomSelf -Reducibility", to appear in proceedings of Structure in Complexity Theory Conference, 1990.

.... University of Chicago Department of Computer Science 1100 East 58th Street Chicago, IL 60637 1 Preliminaries The concept of coherence for binary functions was introduced by Yao [Yao] as a bare bones notion of so called self reducibility (self reducibility [AL] randomself reducibility [AFK, FKN, FF], and related notions [BK, L] Loosely speaking, a function is coherent if its value at any point can be inferred in polynomial time from queries to the same function on other points. We examine the role of adaptiveness in coherent functions. We show that if a function is coherent, it is not ....

Feigenbaum, J., S. Kannan and N. Nisan, Lower bounds on random-selfreducibility, In Proceedings of the 5th Structure in Complexity Theory Conference (1990), pp. 100 -- 109.

....future answers depend on previous questions. See Beaver and Feigenbaum [2] for further details. We can also look at random self reductions as a restriction of local random reductions by requiring g 1 = g k = f . For more precise definitions and theorems about random self reductions see [1, 5, 4]. We can now state the main theorem: Theorem 2 If L has a two local random reduction with oracles g 1 and g 2 where g 1 and g 2 output a single bit then L is in NP poly co NP poly. Our proof was inspired by a weaker result by Yao [6] Any language with a two local random reduction with oracles ....

J. Feigenbaum, S. Kannan, and N. Nisan. Lower bounds on random-self-reducibility. In Proceedings of the 5th IEEE Structure in Complexity Theory Conference, pages 100--109. IEEE, New York, 1990.

....[23] considered resource bounded versions of autoreducibility in which the resource could be any Blum complexity measure. Several polynomial bounded versions of autoreducibility are particularly important in complexity theory. Motivated by previous work on uniformly random self reducibility [1, 3, 15, 26] and on efficient program checking [9, 10] Yao [29] considered sets that are autoreducible via probabilistic, polynomial time oracle Turing machines, and he called them coherent. Trakhtenbrot proved two things that are valuable in the study of coherence. First he noted that A Phi A is ....

....NP decision problems. By Theorem 4.7 we have Corollary 5.3. If s(n) is superpolynomial and space constructible, there is an uncheckable set in DSPACE(s(n) 6. Random Self Reducibility Random self reducibility and uniformly random self reducibility are important concepts in complexity theory [11, 12, 14, 15], especially in connection with program checking and secure protocols. If a set is uniformly random selfreducible in the sense of [15] then it is coherent. If a set is random self reducible in the sense of [14] then it is weakly coherent. Thus, all of our negative results about coherence yield ....

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J. Feigenbaum, S. Kannan, and N. Nisan, Lower Bounds on Random-Self-Reducibility, in Proc. of the 5th Structure in Complexity Theory Conference (1990), IEEE, 100--109.

....main result in [AFK] that no NP hard function is random selfreducible in this sense, unless the polynomial time hierarchy collapses at the third level. Random self reductions that produce several, correlated random instances y 1 , y k were defined formally by Feigenbaum, Kannan, and Nisan [FKN]; however, they only considered reductions that produce y i s that are uniformly distributed over f0; 1g jxj . Their main result is that self reductions that map x to two instances y 1 and y 2 , each of which is uniformly distributed over f0; 1g jxj , do not exist for NP hard functions, ....

.... For example, there exists a function that is nonadaptively k(n) uniformly random selfreducible but not in FPSPACE poly, for any unbounded k(n) This should be contrasted with Feigenbaum, Kannan, and Nisan s theorem that all nonadaptively 2 uniformlyrandom self reducible sets are in NP=poly (cf. [FKN]) These results first appeared in our Technical Memorandum [FFLS] 2 Definitions and Notation Throughout this paper, x is the input to a randomized reduction, n is the size of x, and r is a random variable distributed uniformly on f0; 1g w(n) where w is some polynomially 4 bounded ....

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J. Feigenbaum, S. Kannan, and N. Nisan. Lower bounds on random-selfreducibility. In Proceedings of the 5th IEEE Structure in Complexity Theory Conference, 100--109, 1990. --17--

....precise notation, terminology, and definitions for the concepts discussed above. Our main results are given in Section 4. Section 5 contains a brief discussion of very recent related results, and Section 6 contains open problems. The results given here first appeared in our Technical Memorandum [16]. In what follows, some details of proofs have been omitted because of space limitations; they will appear in the full paper. 2 Preliminaries We first fix notation for the following concepts, with which we assume familiarity on the part of the reader. The class of total functions computable in ....

J. Feigenbaum, S. Kannan, and N. Nisan. Lower Bounds on Random-Self-Reducibility, AT&T Bell Laboratories Technical Memorandum, December 4, 1989.

....[23] considered resource bounded versions of autoreducibility in which the resource could be any Blum complexity measure. Several polynomial bounded versions of autoreducibility are particularly important in complexity theory. Motivated by previous work on uniformly random self reducibility [1, 3, 15, 26] and on efficient program checking [9, 10] Yao [29] considered sets that are autoreducible via probabilistic, polynomial time oracle Turing machines, and he called them coherent. Trakhtenbrot proved two things that are valuable in the study of coherence. First he noted that A Phi A is ....

....By Theorem 4.7 we have 12 Beigel Feigenbaum Corollary 5.3. If s(n) is superpolynomial and space constructible, there is an uncheckable set in DSPACE(s(n) 6. Random Self Reducibility Random self reducibility and uniformly random self reducibility are important concepts in complexity theory [11, 12, 14, 15], especially in connection with program checking and secure protocols. If a set is uniformly random selfreducible in the sense of [15] then it is coherent. If a set is random self reducible in the sense of [14] then it is weakly coherent. Thus, all of our negative results about coherence yield ....

[Article contains additional citation context not shown here]

J. Feigenbaum, S. Kannan, and N. Nisan, Lower Bounds on Random-Self-Reducibility, in Proc. of the 5th Structure in Complexity Theory Conference (1990), IEEE, 100--109.

....1.6 There is a (strongly) incoherent set in DSPACE(n log n ) The unconditional consequences for checking and random self reducible sets are immediate. In particular we get a set in DSPACE(n log n ) that is not uniformly random self reducible, improving Feigenbaum, Kannan and Nisan [FKN]. 2 Search Versus Decision in NP In this section we present a simple construction of a language in NP for which search does not reduce to decision, assuming that EE 6= NEE. In later sections we will extend the argument to interactive proofs and program checking. Let us begin with a more precise ....

....are given to any of queries 1 through i Gamma 1. A set A is uniformly random self reducible (abbreviated ursr) if it has a random self reduction in which each random variable q A i (x; r) is uniformly distributed over f0; 1g jxj . Uniformly random self reducible sets are treated at length in [FKN]. Yao [Ya] showed that if L has a checker then it is coherent, and thus reduced the problem of proving negative results on checking to that of constructing incoherent languages. Similarly, it is not hard to see the following Theorem 5.1 If search reduces to decision for L then L is ....

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J. Feigenbaum, S. Kannan and N. Nisan. Lower Bounds on Random-Self-Reducibility, Proceedings of the 5th Structures, IEEE (1990).

....in C=poly. For example, there is a nonadaptively k(n) u niformly random self reducible function that is not in FPSPACE poly, for any unbounded k(n) This should be contrasted with Feigenbaum, Kannan, and Nisan s theorem that all nonadaptively 2 uniformly randomself reducible sets are in NP=poly [13]. 2. Definitions and Notation Throughout this paper, x is the input to a randomized reduction, n is the size of x, and r is a random variable distributed uniformly on f0; 1g w(n) where w is some polynomially bounded function of n. The parameter k(n) is also a polynomially bounded function of ....

....(abbreviated k(n) ursr) and stress that the word uniform describes the distribution of random queries, i.e. it is not meant to distinguish between reduction procedures that take advice and those that do not. Uniformly random self reductions are the type of rsr s studied in [13]. Let BPE be the class of languages recognizable in bounded error, probabilistic exponential time, i.e. the union of BPTIME(2 cn ) for all constants c. Similarly, let BPEE and BPEEE denote the languages recognizable in boundederror, probabilistic double exponential and triple exponential time. ....

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J. Feigenbaum, S. Kannan, and N. Nisan, Lower bounds on random-self-reducibility. In Proceedings of the 5th Structure in Complexity Theory Conference (1990), IEEE Computer Society, 100--109.

....main result in [1] that no NP hard function is random self reducible in this sense, unless the polynomial time hierarchy collapses at the third level. Random self reductions that produce several, correlated random instances y 1 , y k were defined formally by Feigenbaum, Kannan, and Nisan [15]; however, they only considered reductions that produce y i s that are uniformly distributed over f0; 1g jxj . Their main result is that self reductions that map x to two instances y 1 and y 2 , each of which is uniformly distributed over f0; 1g jxj , do not exist for NP hard functions, ....

....f can be locally randomly reduced to a related function g; see [7, 8] for a thorough discussion. In this paper, we continue the study of random self reductions from a complexitytheoretic point of view. We further generalize the formal definition of random selfreducibility that is studied in [15]. Specifically, we look at reductions that map a given instance x to a sequence of random instances y 1 , y k , with the property that the induced distribution on each y i depends only on the length of x. We consider both nonadaptive k random self reductions, in which the k random ....

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J. Feigenbaum, S. Kannan, and N. Nisan, Lower bounds on random-self-reducibility, in Proceedings of the 5th Structure in Complexity Theory Conference, IEEE Computer Society, Los Alamitos, 1990, pp. 100--109.

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J. Feigenbaum, S. Kannan and N. Nisan. Lower Bounds on Random-SelfReducibility. In Proceedings of Structures 1990, 1990.

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J. Feigenbaum, S. Kannan and N. Nisan. Lower Bounds on Random-SelfReducibility. In Proceedings of Structures 1990, 1990.

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J. Feigenbaum, S. Kannan and N. Nisan. Lower Bounds on Random-SelfReducibility. In Proceedings of Structures 1990, 1990.

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J. Feigenbaum, S. Kannan and N. Nisan. Lower Bounds on Random-SelfReducibility. In Proceedings of Structures 1990, 1990.

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J. Feigenbaum, S. Kannan, and N. Nisan. Lower bounds on random-self-reducibility. In Proceedings of the 5th IEEE Conference on Structure in Complexity Theory, pages 100--109, 1990.

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J. Feigenbaum, S. Kannan and N. Nisan. Lower Bounds on Random-SelfReducibility. In Proceedings of Structures 1990, 1990.

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Feigenbaum, J., Kannan, S., Nisan, N., "Lower Bounds on Random Self-Reducibility", proceedings of the 5 th Structures in Complexity Theory Conference, 1990.

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