A.Bernasconi, B.Codenotti, J.Simon, On the Fourier Analysis of Boolean Functions, Preprint, 1996  Home/Search   Document Details and Download   Summary   Related Articles   Check

....= 4; F(1; 1; 0) Gamma4: Substituting these values of F (w) w 2 B 3 ; into (20) we have: Delta f (ffl) 1 16 max u12B;u22B;u32B j( Gamma1) u2 ( Gamma1) u3 )8ffl ( Gamma1) u1 Phiu 3 Gamma ( Gamma1) u1 Phiu 2 )16ffl 2 j = ffl: Example 3.3. The majority function ([1]) f(x) 1; W (x) k; 0; W (x) k; where x 2 B 2k 1 ; k 0. It is known ( 1] that for the majority function F (w) 8 : 0; W (w) 2s; Gamma1) s 2 ( 2s s ) 2k Gamma2s k Gammas ) k s ) W (w) 2s 1; 9 for s = 0; k. It leads to the relation Delta f (ffl) Gamma ....

....we have: Delta f (ffl) 1 16 max u12B;u22B;u32B j( Gamma1) u2 ( Gamma1) u3 )8ffl ( Gamma1) u1 Phiu 3 Gamma ( Gamma1) u1 Phiu 2 )16ffl 2 j = ffl: Example 3.3. The majority function ( 1] f(x) 1; W (x) k; 0; W (x) k; where x 2 B 2k 1 ; k 0. It is known ([1]) that for the majority function F (w) 8 : 0; W (w) 2s; Gamma1) s 2 ( 2s s ) 2k Gamma2s k Gammas ) k s ) W (w) 2s 1; 9 for s = 0; k. It leads to the relation Delta f (ffl) Gamma 2k k Delta (2k 1) 2 2k ffl o(ffl 2 ) and hence Delta f (ffl) ffl M ....

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A.Bernasconi, B.Codenotti, J.Simon, On the Fourier Analysis of Boolean Functions, Preprint, 1996

....coe#cient u 0 is taken to be 1. We provide estimates for the average sensitivity and the size of the Fourier coe#cient of highest order for these functions. These measures have important consequences for the computational complexity of functions, and therefore they have often received study, see [8, 13, 24, 26]. Then, using our estimates, we derive lower bounds on the decision tree size, on the average decision tree depth, on the formula size and on the degree of certain polynomial representations for g and h. Although, as we mentioned, our techniques are similar to those used for the analogues of the ....

.... # implies #(#) # K(#)n. The following bound on the formula size in terms of average sensitivity was derived in [9, 10] Lemma 7 Let # be a Boolean function depending on n variables. Then L(#) # 1 1 (E(#) 2 s(#) 2 . 10 Notice that this bound has essentially been mentioned also in [8, 13]. Finally we mention the following statement which is a slightly relaxed version of the result of [31] Lemma 8 Let p be a prime, and let d not be a power of p. Then the Boolean function Mod d (x) is not in AC 0 [p] 4 Squarefree and Co prime Polynomials over IF 2 4.1 A preliminary identity We ....

A. Bernasconi, B. Codenotti and J. Simon, `On the Fourier analysis of Boolean functions', Preprint , 1996, 1--24.

....based on the Abstract Harmonic Analysis on the hypercube have been shown to represent a very useful tool for obtaining lower complexity bounds. Various links between Fourier coefficients of Boolean functions and their complexity characteristics have been studied in a number of works, see [1, 2, 3, 4, 6, 8, 13, 19, 20, 22, 23]. In particular, these Institut fur Informatik, Technische Universitat Munchen, D 80290 Munchen, Germany. bernasco informatik.tu muenchen.de y Fachbereich fur Informatik, Universitat Trier, D 54286 Trier, Germany. damm uni trier.de z School of MPCE, Macquarie University, Sydney, NSW 2109, ....

....= n by considering u = v = p Gamma 1) 2 where p is a prime number with 2 m 1 p 2 m 1 Gamma 1. It is also known that the average sensitivity oe av (f) 2 Gamman X x2Bn n X i=1 fi fi fif (x) Gamma f(x (i) fi fi fi can be expressed via the Fourier coefficients of f , see [1, 4, 6]. Applying our results, one can derive the estimate oe av (f) c n 1=2 log Gamma2 n for the functions g and h given by (1) and (2) where c 0 is an absolute constant. It would be interesting to obtain a linear lower bound on the average sensitivity of g and h. There are also close ....

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A. Bernasconi, B. Codenotti and J. Simon, `On the Fourier analysis of Boolean functions', Preprint (1996), 1--24.

....based on the Abstract Harmonic Analysis on the hypercube have been shown to represent a very useful tool for obtaining lower complexity bounds. Various links between Fourier coefficients of Boolean functions and their complexity characteristics have been studied in a number of works, see [2 5, 8, 14, 20, 21, 24, 25]. In particular, these spectral techniques have been successfully applied to the parity function and to threshold functions. However, a limitation of such approach to the study of Boolean function complexity is that, besides the results for parity and threshold functions, spectral methods have ....

.... that the CREW PRAM complexity of g is at least 0:5 log n O(1) see [23] It is also known that the average sensitivity can be expressed via the Fourier coefficients of f and related to the formula size complexity of f and to the degree of the polynomial approximation of f over the reals, see [2, 5, 8, 22]. Applying our results, one can derive the estimate oe av (f) c n 1=2 log Gamma2 n for the function f given by (1) where c 0 is an absolute constant. However, using a more direct approach, it is shown in [7] that in fact oe av (f) 4 9 2 n o(n) This bound implies several other ....

A. Bernasconi, B. Codenotti and J. Simon, `On the Fourier analysis of Boolean functions', Preprint (1996), 1--24.

....with an unlimited number of allpowerful processors, such that simultaneous reads of a single memory cell by several processors are permitted, but simultaneous writes are not. The average sensitivity is a finer characteristic of Boolean functions which has been studied in a number of papers, see [2, 5, 14]. Our main result consists in a linear lower bound on the average sensitivity of testing square free numbers. More precisely, we consider the function g which decides whether a given (n 1) bit odd integer is square free, that is the function for which g(x 1 ; x n ) 1; if 2x 1 is ....

....A = f Gamma1 (0) and B = f Gamma1 (1) We obtain the desired lower bound by observing that X u2A; v2B q uv = 2 n Gamma1 s(f) jAj = 2 n (1 Gamma p) and jBj = 2 n p. ut Notice that this bound on the formula size in terms of average sensitivity was essentially mentioned also in [2, 5]. Lemma 2. Let f be a Boolean function. Then D(f) s(f) Proof. Let a be an input assignment. The inequality D a (T ) oe a (f) holds since otherwise some untested variable still could decide about the function value. Hence, E[D a (T ) E[oe a (f ) s(f) where E denotes the expectation with ....

A. Bernasconi, B. Codenotti and J. Simon, `On the Fourier analysis of Boolean functions', Preprint (1996), 1-24.

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