C. K. Chow, "On the characterization of threshold functions," in Proc. IEEE Symp. Switching Theory and Logic Design, pp. 34--38, 1961.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....i is just the number of agreements between the value of x i and the value of f(x) less the number of disagreements divided by the total number of assignments of values to variables. R i is known as a first order spectral coefficient of f(X) 3] and is related to the Chow parameters of the function [2]. For example, if f(X) x i , then R i =1, corresponding to complete correlation and if f(X) x i , then R i = 1, corresponding to complete anti correlation. All spectral coefficients of a completely specified Boolean function can be computed by scanning the nodes beginning at the root node and ....

C. K. Chow, "On the characterization of threshold functions," in Proc. IEEE Symp. Switching Theory and Logic Design, pp. 34--38, 1961.

....but can be naturally reformulated as an RFA problem. Consider the class of linearly separable half spaces over f0; 1g n (perceptrons) It is well known that the rst order Fourier coecients of a perceptron (also called the Chow parameters of the perceptron) uniquely determine the perceptron (see [10], or [9] for a more general result) Is it possible to eciently compute a good weights based approximation of the perceptron from good approximations of these coecients This question can be naturally formulated as a 1 RFA learning problem, as follows. It can be shown that when learning from a ....

Chow, C. (1961). On the characterization of threshold functions. In Proc. Symp. on Switching Circuit Theory and Logical Design, pages 34-38.

....Coventry CV4 7AL, U.K. pwg dcs.warwick.ac.uk Abstract. A boolean perceptron is a linear threshold function over the discrete boolean domain f0; 1g n . That is, it maps any binary vector to 0 or 1 depending on whether the vector s components satisfy some linear inequality. In 1961, Chow [9] showed that any boolean perceptron is determined by the average or center of gravity of its true vectors (those that are mapped to 1) Moreover, this average distinguishes the function from any other boolean function, not just other boolean perceptrons. We address an associated ....

....and a real valued threshold t, and it maps a binary vector x to the output value 1 provided that w:x t, otherwise it maps x to 0. In this paper we consider the problem of estimating a perceptron from an approximate value of the mean, or center of gravity of its satisfying assignments. Chow [9] originally showed that the exact value of the average of the satisfying assignments of a boolean perceptron determines it, in that there are no other boolean functions of any kind for which the average satisfying assignment is the same. Bruck [8] also gives a more general result. The question ....

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C.K. Chow (1961). On the characterization of threshold functions. Proc. Symp. on Switching Circuit Theory and Logical Design, 34-38.

....used to detect the parity function are the 0 th and 1 st ordered Walsh coefficients and a single i th ordered Walsh coefficient, where i may range from 2 to n. The 0 th and 1 st ordered Walsh coefficients are usually referred to as the Chow parameters in recognition of the work in [5] although the original definition differs slightly from that originally defined by Chow. The realization portion of this process uses n 1 spectral coefficients corresponding to constituent functions that are not used in any of the common transformation matrices. This problem has great practical ....

.... of the commonly used transformation matrices always include constituent functions that correspond to the constant function (either f c0 = 0, or, f c0 = 1) and functions that are equal to each primary input (f ci = x i for i = 1 : n) This set of spectral coefficients form the Chow parameters [5]. As an example, the following transformation matrix could be used to compute the Chow parameters of a three input function: 1 x 1 x 2 x 3 2 6 6 4 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 3 7 7 5 Notice that the constituent functions corresponding to each ....

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C. K. Chow. On the Characterization of Threshold Functions. IEEE Special Publication S.134, pages 34--38, 1961.

....Many applications have been proposed and developed using spectral methods for logic circuits. Some of these include logic synthesis [13] 20] 22] 26] 31] 33] 39] decomposition and partitioning techniques [2] 22] 24] 40] 41] testing [10] 18] 29] 34] function classification [5] [12] 20] and others. The application of spectral based methodologies to digital logic analysis has been studied and developed since the mid 1970 s in an attempt to use the vast amount of results that have been very effective in areas such as signal processing and systems analysis. 4.1 Function ....

....in this area since the logic diagram may be more readily available than a BDD description of the circuit under analysis. 4.3 Function classification using spectral coefficients The use of spectral coefficients has been proven to uniquely specify threshold functions. The work by Chow in 1961 [5] resulted in a formal proof that n 1 spectral coefficients are sufficient to define a specific threshold function. In terms of the definitions used in this paper, the constituent functions that are used to compute these coefficients are f c (x) 0 and f c (x) x i , where each x i is a primary ....

C. K. Chow. On the Characterization of Threshold Functions. IEEE Special Publication S.134, pages 34--38, 1961.

....circuits into disjoint sets where any one member may be used to provide the output of another member by possible inversion and permutation of some inputs, and perhaps, inverting the output. The theory of NPN equivalence and its relation to certain spectral coefficients has been studied extensively [8] [9] 10] Therefore, the ability to efficiently determine spectral coefficients from a structural representation of a circuit will allow spectral based Boolean matching techniques to be easily included in this class of existing logic synthesis systems. Another area of application of the use of ....

.... that correspond to the constant function (either f c0 = 0, or, f c0 = 1) and functions that are equal to each primary input (f ci = x i for i = 1 : n) The particular subset of spectral coefficients that correspond to these n 1 constituent functions are referred to as the Chow parameters [8]. As an example, the following transformation matrix could be used to compute the Chow parameters of a three input function: 1 x 1 x 2 x 3 2 6 6 6 6 4 1 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 1 1 0 0 1 1 0 1 0 1 0 1 0 1 3 7 7 7 7 5 4 Notice that the constituent functions ....

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C. K. Chow. On the characterization of threshold functions. IEEE Special Publication S.134, pages 34--38, 1961.

....; x n ) The original two valued data is fully recoverable from S by application of the inverse transform [T ] Gamma1 . The exploitation of the spectral data provides a sound mathematical basis for many applications in digital logic design. Some of them include Boolean function classification [1], 4] 15] logic network synthesis [4] 5] 6] 7] 8] 9] 11] 16] fault diagnosis [12] 13] 14] and others. Spectral based methodologies have also proved very effective in other digital areas such as signal processing [3] and transmission of information [2] In many applications ....

....application of this decomposition. This view of the inverse transform gives rise to the approach to deciding spectrality, presented here. We employ spectra generated from the f 1; Gamma1g encoding of two valued data by applying the Hadamard transform matrix, defined inductively by: T 0 = [1] T n = T n Gamma1 T n Gamma1 T n Gamma1 GammaT n Gamma1 # Let S be the spectrum of a Boolean function f(x 1 ; x n ) and S 0 and S 1 be the spectra of the subfunctions f(x 1 ; x n Gamma1 ; 0) and f(x 1 ; x n Gamma1 ; 1) involved in a general Shannon ....

C. K. Chow, On the characterization of threshold functions, IEE Special Pub. S. 134 (1961), 34-38.

....can be naturally reformulated as an RFA problem. Consider the class of linearly separable half spaces over f0; 1g n (perceptrons) It is well known that the first order Fourier coefficients of a perceptron (also called the Chow parameters of the perceptron) uniquely determine the perceptron (see [17], or [15] for a more general result) Is it possible to efficiently compute a good weights based approximation of the perceptron from good approximations of these coefficients This question can be naturally formulated as a 1 RFA learning problem, as follows. It can be shown that when learning ....

C.K. Chow. On the characterization of threshold functions. In Proc. Symp. on Switching Circuit Theory and Logical Design, pages 34--38, 1961.

.... coefficient may be computed in a reasonable amount of CPU time, an entire spectral description of a circuit requires an exponential number of spectral coefficients (2 n ) for this reason a subset of the Walsh coefficients, the first order spectral coefficients referred to as the Chow parameters [4] are computed only. Depending on the Chow parameter magnitudes, a series of heuristic rules are invoked that govern the OBDD decomposition logic realization step. Table 2 contains data regarding the synthesis of some common benchmark circuits and a comparison using the misII tool from Berkeley ....

C. K. Chow. On the Characterization of Threshold Functions. IEEE Special Publication S.134, pages 34--38, 1961.

....of depth two or more do not carry a simple geometric interpretation in R S . The inputs to gates in the second level are themselves threshold functions, hence the linear combination computed at the second level can be a non linear function of the inputs. Lacking a geometric view, researchers [6, 4] have used indirect approaches, applying spectral analysis techniques to analyze threshold gates. These techniques, apart from their complexity, restricted the input functions of the gates to be of very special types: input variables, or parities of the input variables, thus not applying even to ....

....gY 6= C fY . This implies, for example, that any set of S input functions can give rise to at most (2 n 1) S different threshold functions. The special case of the uniqueness property where the functions f 1 ; f S are the input variables (or a constant function) had been proven in [6] (see also [15] The proof used spectral analysis tools such as Parseval s theorem and relied on the mutual orthogonality of the input functions (namely, C x i ;x j = 0 for all i 6= j) Another special case where the input functions are parities of the input variables was proven in [4] ....

C. K. Chow. On The Characterization of Threshold Functions. Proceedings of the 6th Symposium on Switching Circuit Theory and Logical Design, pages 34--38, 1961.

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C. K. Chow, "On the characterization of threshold functions," in Proc. IEEE Symp. Switching Theory and Logic Design, pp. 34--38, 1961.

No context found.

C. K. Chow. On the Characterization of Threshold Functions. IEEE Special Publication S.134, pages 34--38, 1961.

No context found.

C. K. Chow. On the characterization of threshold functions. IEEE Special Publication S.134, pages 34--38, 1961.

No context found.

C. K. Chow, "On the characterization of threshold functions," in Proc. IEEE Symp. Switching Theory and Logic Design, pp. 34--38, 1961.

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