Ravi Boppana. Amplification of Probabilistic Boolean Formulas. In Advances in Computing Research, 5:4, pages 27--45, 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....1 ; z q ) where q = poly(n) h has size polynomial in q, and such that h(z 1 ; z q ) is a ( amplifier. Proof: The formula h is constructed out of two different amplifiers. The first is the ( amplifier A(y 1 ; y ) described in [7] see also [13] and [6]) where = 4n log n log log n . This amplifier is simply a CNF formula with 2n= log n disjoint clauses, each of which contains 2n= log log n variables. The second amplifier is the ( 4 ) n ) amplifier B(y 1 ; y c log n ) Majority(y 1 ; y c log n ) Since B depends ....

R. Boppana. Amplification of probabilistic boolean formulas. Advances in Computing Research, 5(4):27--45, 1989.

....ffi) jCj, so at most ln jCj= ln( 1 Gammaffi ) ln jCj equivalence queries are required to isolate the target concept. In order to generate ffi good hypotheses, we use amplifiers. The concept of amplification was first investigated by Moore and Shannon [MS56] Valiant [V84a] and Boppana [B89]. Definition: Let 0 p p q q 1. A (boolean) function G(y 1 ; ym ) is a (p; q) p ; q ) amplifier if: a) When y 1 ; ym are each independently set to 1 with probability at least q, Pr[G(y 1 ; ym ) 1] q 6 (b) When y 1 ; ym are each ....

....any x such that fl (x;1) 1 Gamma ffi. Therefore, the probability that there exists an x 2 f0; 1g which, when returned as a counterexample to the equivalence query G(f 1 ; fm ) eliminates less than a ffi fraction of the elements of C, is less than 2 Delta 2 . 2 Lemma 2 [B89, K93]: a) The function MAJORITY(y 1 ; y 48n ) is a ( 2 ) amplifier. b) Define A(y 1 ; ym ) as a (2n= log n) ary of (2n= log n) ary s of distinct variables. Thus, the number of inputs to the formula is m = log . Then A(y 1 ; ym ) is a 2 ; 1 ....

Ravi Boppana. Amplification of Probabilistic Boolean Formulas. In Advances in Computing Research, 5:4, pages 27--45, 1989.

....q = poly(n) h has size polynomial in q, and such that h(z 1 ; z q ) is a ( amplifier. Proof: The formula h is constructed out of two different amplifiers. The first is the ( 2 ; 1 Gamma2 ) amplifier A(y 1 ; y ) described in [5] see also [8] and [4]) where = 4n log n log log n . This amplifier is simply a CNF formula with 2n= log n disjoint clauses, each of which contains 2n= log log n variables. The second amplifier is the ( amplifier B(y 1 ; y c log n ) Majority(y 1 ; y c log n ) Since B ....

R. Boppana. Amplification of probabilistic boolean formulas. Advances in Computing Research, 5(4):27--45, 1989.

....up to a polylogarithmic factor. At the heart of our approach lie various links between the shrinkage properties of a function and its behavior under random restrictions assigning all variables. These links allow us to apply to our problem the strong machinery developed by Valiant [10] and Boppana [2]. More generally, our proofs are assembled from several independent pieces. It seems that many of these auxiliary statements have a scope of application much broader than the original task they were designed for. We hope that at least some of them will be useful for attacking the general case. ....

....q and to 0 with probability (1 Gamma q) It may be helpful for the reader to think of p as being small and q as being around 1=2. We will denote ae (f) by f . Note that, unlike f p , f is always a constant. Let A f (q) P [f j 1] The following remarkable result of Boppana [2] lies at the heart of our approach: Proposition 3.1. Boppana) If f is a read once monotone function and q 2 (0; 1) then f (q) L(f) where H(q) Gammaq log q Gamma (1 Gamma q) log(1 Gamma q) Our first lemma is an easy exercise in mathematical calculus: Lemma 3.2. Let f be as in ....

[Article contains additional citation context not shown here]

R. B. Boppana. Amplification of probabilistic Boolean formulas. In S. Micali, editor, Advances in Computer Research, vol. 5: Randomness and Computation, pages 27--46. JAI Press, Greenwich, CT, 1989.

....the construction and proof exploits the fact that a (2; 2) regular OR AND tree has a critical value of OE = 3 Gamma p 5) 2, and that if y Gamma1 = OE ffl then y OE cffl for a constant c 1. The analogous rate of divergence is true if y Gamma1 = OE Gamma ffl. In further work, [3] and [5] provide beautiful proofs that the construction size of [13] is optimal, this time using amplification analysis to prove a lower bound. Our work has a similar spirit to the amplification work, but it differs in several ways. We generalize to allow the number of children of each node to ....

R. Boppana, "Amplification of probabilistic Boolean formulas", Advances in Computer Research, Vol. 5: Randomness and Computation, JAI Press, Greenwich, CI, 1989, pp. 27--45.

.... of Khrapchenko [16] Pippenger [26] Paterson [21] and Peterson [25] Relevant results on formula size are the construction of concise B 2 formulae for MOD k with k = 3; 5; 7 by Van Leijenhorst [34] the construction of monotone formulae for the majority function by Valiant [33] see also [3]) and for general threshold functions by Boppana [3] and Friedman [11] Depth and the logarithm of formula size are closely related. It is known for example that log LB 2 (f) DB 2 (f) 2:47 log LB 2 (f) in fact even that DU 2 (f) 2:47 log LB 2 (f) and that log LU 2 (f) DU 2 (f) 1:81 log LU ....

.... [21] and Peterson [25] Relevant results on formula size are the construction of concise B 2 formulae for MOD k with k = 3; 5; 7 by Van Leijenhorst [34] the construction of monotone formulae for the majority function by Valiant [33] see also [3] and for general threshold functions by Boppana [3] and Friedman [11] Depth and the logarithm of formula size are closely related. It is known for example that log LB 2 (f) DB 2 (f) 2:47 log LB 2 (f) in fact even that DU 2 (f) 2:47 log LB 2 (f) and that log LU 2 (f) DU 2 (f) 1:81 log LU 2 (f) see [5] 28] 31] These relations are ....

Boppana R.B., Amplification of probabilistic Boolean formulas. Advances in Computing Research, Vol. 5 on Randomness and Computation, Editor S. Micali, JAI Press, Greenwich, Conn., pp. 27-45.

....for the remaining concepts then, starting with the concept class C, at most ln jCj= ln( 1 1 Gammaffi ) 1 ffi ln jCj equivalence queries are required to isolate the target concept. The concept of amplification was first investigated by Moore and Shannon [MS56] Valiant [V84a] and Boppana [B89]. Definition: Let 0 p 0 p q q 0 1. A (boolean) function G(y 1 ; y m ) is a (p; q) p 0 ; q 0 ) amplifier if: a) When y 1 ; y m are each independently set to 1 with probability at least q, Pr[G(y 1 ; y m ) 1] q 0 ; b) When y 1 ; y m are ....

....C (x;1) 1 Gamma ffi. Therefore, the probability that there exists an x 2 f0; 1g n which, when returned as a counterexample to the equivalence query G(f 1 ; f m ) eliminates at most a ffi fraction of the elements of C, is less than 2 n Delta 2 Gamma2n = 2 Gamman . 2 Lemma 2 [B89, K93]: a) The function MAJORITY(y 1 ; y 48n ) is a ( 1 4 ; 3 4 ) 2 Gamma2n ; 1 Gamma 2 Gamma2n ) amplifier. b) Define A(y 1 ; y m ) as a (2n= log n) ary of (2n= log n) ary s of distinct variables. Thus, the number of inputs to the formula is m = 4n 2 log 2 n ....

Ravi Boppana. Amplification of Probabilistic Boolean Formulas. In Advances in Computing Research, 5:4, pages 27--45, 1989.

....appears at most once. The amplification function A f (p) for a function f : f0; 1g n f0; 1g is defined as the probability that the output of f is 1 when each of the n inputs to f is 1 independently with probability p. Amplification functions were first studied by Valiant [24] and Boppana [4, 5] in obtaining bounds on monotone formula size for the majority function. The method used by our algorithms is of central interest. For several classes of formulas, we show that the behavior of the amplification function is unstable near the fixed point; that is, the value of A f (p) varies greatly ....

....class of all read once Boolean formulas constructed from the usual basis fand; or; notg. He shows that an arbitrarily good approximation of such formulas can be inferred in polynomial time against any product distribution. 2 Preliminaries Given a Boolean function f : f0; 1g n f0; 1g, Boppana [4, 5] defines its amplification function A f as follows: A f (p) Pr[f(X 1 ; X n ) 1] where X 1 ; X n are independent Bernoulli variables that are each 1 with probability p. The quantity A f (p) is called the amplification of f at p. Valiant [24] uses properties of the ....

Ravi B. Boppana. Amplification of probabilistic Boolean formulas. In 26th Annual Symposium on Foundations of Computer Science, pages 20--29, October 1985.

....amplification function. The amplification function A f (p) for a function f : f0; 1g n f0; 1g is defined as the probability that the output of f is 1 when each of the n inputs to f is 1 independently with probability p. Amplification functions were first studied by Valiant [20] and Boppana [3, 4] in obtaining bounds on monotone formula size for the majority function. The method used by our algorithms is of central interest. For several classes of formulas, we show that the behavior of the amplification function is unstable near the fixed point; that is, the value of A f (p) varies greatly ....

....any product distribution. For example, his result demonstrates that the class of read once Boolean formulas over the usual basis can be learned in polynomial time against the uniform distribution in the sense of Valiant. 2 Preliminaries Given a Boolean function f : f0; 1g n f0; 1g, Boppana [3, 4] defines its amplification function A f as follows: A f (p) Pr[f(X 1 ; X n ) 1] where X 1 ; X n are independent Bernoulli variables that are each 1 with probability p. The quantity A f (p) is called the amplification of f at p. Valiant [20] uses properties of the ....

Ravi B. Boppana. Amplification of probabilistic Boolean formulas. In 26th Annual Symposium on Foundations of Computer Science, pages 20--29, October 1985.

....i jCj, so at most ln jCj= ln( 1 1 Gammaffi ) 1 ffi ln jCj equivalence queries are required to isolate the target concept. In order to generate ffi good hypotheses, we use amplifiers. The concept of amplification was first investigated by Moore and Shannon [MS56] Valiant [V84a] and Boppana [B89]. Definition: Let 0 p 0 p q q 0 1. A (boolean) function G(y 1 ; ym ) is a (p; q) p 0 ; q 0 ) amplifier if: a) When y 1 ; ym are each independently set to 1 with probability at least q, Pr[G(y 1 ; ym ) 1] q 0 ; b) When y 1 ; ym are each ....

....C (x;1) 1 Gamma ffi. Therefore, the probability that there exists an x 2 f0; 1g n which, when returned as a counterexample to the equivalence query G(f 1 ; fm ) eliminates less than a ffi fraction of the elements of C, is less than 2 n Delta 2 Gamma2n = 2 Gamman . 2 Lemma 2 [B89, K93]: a) The function MAJORITY(y 1 ; y 48n ) is a ( 1 4 ; 3 4 ) 2 Gamma2n ; 1 Gamma 2 Gamma2n ) amplifier. b) Define A(y 1 ; ym ) as a (2n= log n) ary of (2n= log n) ary s of distinct variables. Thus, the number of inputs to the formula is m = 4n 2 log 2 n . ....

Ravi Boppana. Amplification of Probabilistic Boolean Formulas. In Advances in Computing Research, 5:4, pages 27--45, 1989.

....[12] to construct short monotone formulae for the majority function. The exponent 1= log 2 ( p 5 Gamma 1) 3:27 is the exponent in his first stage which amplifies ( Gamma 1 n ; 1 n ) to ( 1 4 ; 3 4 ) where = p 5 Gamma1 2 , and which was shown to be optimal by Boppana [2]. Let fr n g be a sequence of bits. Define two sequences of read once formulae fT n;0 g and fT n;1 g as follows : T 1;0 = x 1 x 2 T 1;1 = x 1 x 2 T n;i = T n Gamma1;1 Gammai NAND T n Gamma1;r n for n 2 and i = 0; 1. In defining T n;i for n 2, disjoint sets of variables are to be used for ....

R.B. Boppana, Amplification of probabilistic Boolean formulas. In Advances in Computer Research, Vol.5: Randomness and Computation (JAI Press, Greenwich, Conn., to appear).

....a ( 1 4 ; 3 4 ) 2 Gamma2n ; 1 Gamma 2 Gamma2n ) amplifier. Proof: The formula h is constructed out of two different amplifiers. The first is the ( 1 n ; 1 Gamma 1 n ) 2 Gamma2n ; 1 Gamma 2 Gamma2n ) amplifier A(y 1 ; y ) described in [7] see also [13] and [6]) where = 4n 2 log n log log n . This amplifier is simply a CNF formula with 2n= log n disjoint clauses, each of which contains 2n= log log n variables. The second amplifier is the ( 1 4 ; 3 4 ) 1 n ; 1 Gamma 1 n ) amplifier B(y 1 ; y c log n ) Majority(y 1 ; ....

R. Boppana. Amplification of probabilistic boolean formulas. Advances in Computing Research, 5(4):27--45, 1989.

....the construction and proof exploits the fact that a (2; 2) regular OR AND tree has a critical value of OE = 3 Gamma p 5) 2, and that if y Gamma1 = OE ffl then y OE cffl for a constant c 1. Analogously, if y Gamma1 = OE Gamma ffl, then y OE Gamma cffl. In further work, [3] and [5] provide beautiful proofs that the construction size of [13] is optimal, this time using amplification analysis to prove a lower bound. Our work has a similar spirit to the amplification work, but it differs in several ways. We generalize to allow the number of children of each node to ....

R. Boppana, "Amplification of probabilistic Boolean formulas", Advances in Computer Research, Vol. 5: Randomness and Computation, JAI Press, Greenwich, CI, 1989, pp. 27-- 45.

.... Gamma = ff. It is the purpose of this paper to give a tighter analysis of the shrinkage of read once formulae, leading in particular to an alternative proof of the result Gamma = ff. Both the work of Hastad, Razborov and Yao [9] and the current work use results of Valiant [14] and Boppana [1] concerning the amplification properties of read once formulae. In this work we also use some extensions of Boppana s results obtained by Dubiner and Zwick [3] 4] 2 Preliminaries Definition 2.1 (Formulae) A (de Morgan) formula in the n variables x 1 ; x n is defined recursively as ....

.... ) 2; 1 with probability (1 Gamma ) 2; x i with probability . We denote by E (f) E[L(f ) the expected formula size of f . If f is a read once formula, then E (f) is just the expected number of variables on which f depends. The following results are easily verified. Lemma 2. 4 ([1]) Let f be a read once formula. 1. If f = f 1 f 2 then f(x) f 1 (x)f 2 (x) 2. If f = f 1 f 2 then f(x) 1 Gamma (1 Gamma f 1 (x) 1 Gamma f 2 (x) Lemma 2.5 ( 9] Let f be a read once formula. 1. If f = f 1 f 2 then E (f) f 1 ( 1 2 )E (f 2 ) f 2 ( 1 2 )E (f 1 ) 2. If ....

[Article contains additional citation context not shown here]

R.B. Boppana, "Amplification of probabilistic Boolean formulas", in Advances in Computer Research, Vol.5: Randomness and Computation, JAI Press, Greenwich, Conn., 1989.

....the construction and proof exploits the fact that a (2; 2) regular And Or tree has a critical value of OE = 3 Gamma p 5) 2, and that if y Gamma1 = OE ffl then y OE cffl for a constant c 1. Analogously, if y Gamma1 = OE Gamma ffl, then y OE Gamma cffl. In further work, [4] and [6] provide beautiful proofs that the construction size of [17] is optimal, this time using amplification analysis to prove a lower bound. This type of analysis has also been noted as a possible attack for specific random graph problems, including the greedy matching algorithm of [9] and the ....

R. Boppana, "Amplification of probabilistic Boolean formulas", Advances in Computer Research, Vol. 5: Randomness and Computation, JAI Press, Greenwich, CI, 1989, pp. 27--45.

No context found.

Ravi Boppana. Amplification of Probabilistic Boolean Formulas. In Advances in Computing Research, 5:4, pages 27--45, 1989.

No context found.

R. Boppana. Amplification of probabilistic Boolean formulas. In 26th Annual Symposium on Foundations of Computer Science, pages 20--29, Oct 1985.

No context found.

R. B. Boppana. Amplification of Probabilistic Boolean Formulas. In Advances in Computing Research - Randomness and Computation 5, pp. 27-46, JAI Press (1989).

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