S. Muroga. Threshold Logic and Its Applications. Wiley-Interscience, 1971.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... On binary inputs it computes a Boolean threshold function f : f0; 1g f0; 1g defined by f(x 1 ; xn ) 1 iff w 1 x 1 Delta Delta Delta wnxn t: It is well known that integer weights are sufficient to represent any Boolean threshold function, see e.g. Minsky and Papert, 1988; Muroga, 1971; Parberry, 1994 ] As a useful tool for the investigation of the learning complexity of Perceptron like rules we introduce the weight complexity of a threshold function. It is defined as the smallest natural number such that the function can be represented by a weight vector where the weights ....

.... 1=2 [ Schmitt, 1994 ] For bipolar inputs f Gamma1; 1g the more succinct bound Gamman is known (see also [ Hastad, 1994; Parberry, 1994 ] By different methods several functions of weight complexity at least 2 Omega Gamma n) have been constructed [ Minsky and Papert, 1988; Muroga, 1971; Parberry, 1994 ] By a more involved construction [ Hastad, 1994 ] has recently defined a function that requires weights at least as large as 2 Omega Gamma n log n) see also [ Parberry, 1994 ] Using a counting argument [ Hampson and Volper, 1986, Section 3.3.3 ] proved that the average ....

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Saburo Muroga. Threshold Logic and Its Applications. John Wiley & Sons, New York, 1971.

....sgn a i x i = sgn a i x i Proposition 3.3. Every integer polynomial p(x 1 ; x n ) of degree d which is not an identically zero on f Gamma1; 1g , differs from zero on at least 2 n Gammad points from f Gamma1; 1g . The proof of Proposition 3. 2 can be found e.g. in [21]; Proposition 3.3 is folklore. Let f n 2 F n . If f n is weakly represented by a polynomial p of degree at most n=20, we firstly apply Proposition 3.2 to the vector of coefficients of p. The length N of this vector is P n=20 n(H(1=20) o(1) where H(ffl) is the entropy function. We find ....

S. Muroga. Threshold logic and its applications. Wiley-Interscience, 1971. 24

....enumeration of minimal V cuts for G 1 and G 2 . # 2. 3 2 Monotonic functions A monotone function f : 2 R is called 2 monotonic, if there exists a permutation # S V of the ground set V such that f(X # v u ) # f(X) whenever u V , v X and v precedes u in their # order (see e.g. [57]) For instance, any non negative modular function f(X) v#X w(v) is 2 monotonic with respect to the permutation that puts the non negative singleton weights w(v) in non increasing order (but not all 2 monotonic functions are modular) Theorem 3 ( 10] cf. 22] Let for a system of ....

M. Muroga (1971). Threshold logic and its applications. Wiley-Interscience, New York.

....5.12 depends on the results of Propositions 5.10 and 5.11, while Proposition 5.15 builds on 5.13 and 5.14. The expression of these results also depends on some notions of minimal and irreducible boolean representations of functions which are stated immediately below. These are reproduced from [Mur71], but equivalent statements should be found in any work on switching theory. 41 Definition 5.5 [Mur71, Defn 2.1.8] If there exists a disjunctive form for a function f ffl BN such that the literal u i does not appear in any term of this form, then f is said to be positive in u i , u i is a ....

S Muroga. Threshold Logic and its Applications. John Wiley & Sons, 1971.

.... is a realvalued threshold, such that for all a 2 f0; 1g , f(a) 1 if i=1 w i a(x i ) and f(a) 0 otherwise. Such inequalities form a representation class for Boolean halfspace functions, which we call the class of halfspaces over the Boolean domain, or just halfspaces. It is known [48] that any Boolean halfspace function can be represented by an inequality i=1 w i x i such that the weights w i and threshold are integers satisfying jw i j; j j 2 Gamman (n 1) n 1) 2 . Note that the weights and thresholds in these inequalities can be encoded with a polynomial number ....

S. Muroga, Threshold logic and its applications. Wiley-Interscience, 1971.

....of pyramids is a pyramid cover for a function f if every minterm in foN is contained in at least one pyramid and no minterm in the fOFF set is contained in any pyramid. This concept of cover can be For a more detailed description of the relation between t u eshold gates and logic functions, see [6]. further extended to a more general one that leads to implementations where the second level gate is also a general threshold gate instead of an or gate. A bag of pyramids, B, is an M cover (M 1) for a function f iff all minterms in the fo:v set are covered by at least M pyramids and all ....

S. Muroga "Threshold Logic and its Applications", Wiley-Interscience, 1971.

....then modified to optimize one or more criteria. Therefore, a strong motivation exists to apply some of the techniques used in traditional circuit design to the definition of architectures for general networks. While algorithms for the synthesis of threshold gate networks have been proposed before [Muroga, 1971] they are only applicable for functions of a very small number of variables and, in most cases, find solutions very far from the optimum. The approach presented in this paper is different and introduces some new concepts like pyramids and M covers. We will show that a substantial subset of the ....

....pyramid, k h 1) where k is the number of literals in c and is the cube obtained by complementing all such literals. It is trivial to show that p = 3 andp u = 0, 1 . This fact, together with a trivial generalization of the De Morgan laws leads to the following interesting result [Muroga, 1971]: Lemma 1: Ifa function f can be implemented by anetwork with H threshold gates and the inputs are available in both negated and non negated form, then function y can also be implemented by a network of H threshold gates. 2.3 Covers and M covers We are interested in the synthesis of a two level ....

Muroga, Saburo "Threshold Logic and its Applications", Wiley-Interscience, 1971.

....and have been studied extensively. See, e.g. 11] and its references. It is easy to see that every threshold gate can be realized with integer weights. Various researchers proved that there is always a realization with integer weights satisfying jw i j 2 Gamman (n 1) n 1) 2 : See, e.g. [13] for a proof. There are several simple constructions of threshold gates of n inputs that require some weights of size 2 Omega Gamma n) Hastad [10] constructed threshold gates that require larger weights, thus showing that the above mentioned upper bound is nearly tight. The precise statement ....

S. Muroga, Threshold Logic and its Applications, Wiley-Interscience, New York, 1971.

.... of functions f such that both f and its complement f are Horn [12] also called disguised double Horn functions) CND , the class of nested differences of concepts [21] where each concept is described by a single term; C 2M CR 1 , the intersection of the classes of 2 monotonic functions [32] and read once functions, i.e. functions definable by a formula in which each variable occurs at most once [18, 25, 39, 37] C TH CR 1 , the intersection of threshold functions (also called linearly separable functions) 32] and read once functions; and CLR 1 , the class of linear ....

.... 2M CR 1 , the intersection of the classes of 2 monotonic functions [32] and read once functions, i.e. functions definable by a formula in which each variable occurs at most once [18, 25, 39, 37] C TH CR 1 , the intersection of threshold functions (also called linearly separable functions) [32] and read once functions; and CLR 1 , the class of linear read once functions [12] i.e. functions represented by a read once formula such that each binary connective involves at least one literal. Observe that the inclusion C 1 DL C TH CR 1 follows from the result that C 1 DL C TH [5, ....

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S. Muroga. Threshold Logic and its Applications. Wiley-Interscience, New York, 1971.

....as the key component. Figure 5 shows the logic diagram we have implemented. It consists only of a 16 input sorter, and a threshold gate, realized based on the ideas sketched in the previous Section. The output of the threshold gate implements the parity function following the Muroga s method [8] as: 3) The F Block circuit using the MOS sorter has been designed and laid out in a 0.8m double poly CMOS process. Figure 6 plots the simulated waveforms for the parity output of the extracted F Block. The inputs correspond to a sequence of input patterns with an increasing number of ones: ....

S. Muroga, Threshold Logic and Its Applications, John Wiley & Sons, 1971.

....it, apart from peripheral circuitry necessary for data scheduling, is a combinational functional block (F Block) with 16 inputs and nine outputs. Eight of the outputs correspond to threshold functions and . The ninth is the parity function. A logic diagram for the F Block, based on Muroga s method [7] for implementing the parity function is shown in Figure 5, which uses a 16 input sorter, T n n I On nV DD C u ( C tot x1 x 2 x n T1 n T 2 n T n n V 1 second stage I O1 I O2 I On M 1 M 2 M 3 I 1 R R R R V 0 M 4 V FG first stage Figure 3: Two stage ....

S. Muroga, Threshold Logic and Its Applications, John Wiley & Sons, 1971. 88 ()

....and continuous time to point out alternative sources of efficient computation. 1 Introduction The computational potential and limits of neural networks have been studied for more than a decade in order to understand what is, either ultimately or efficiently, computable by particular models [15, 28, 48, 50, 53, 54, 55, 56, 58, 61, 67, 72]. This interest is motivated partly by the quest to formally justify heuristics used in practical neurocomputing, and partly by the realization that despite their formal simplicity, neural networks are computationally quite powerful, and thus may serve as a useful reference model for investigating ....

S. Muroga, Threshold Logic and its Applications, Wiley--Interscience, New York (1971).

....and limits of conventional computers are by now well understood in terms of classical models of computation such as Turing machines. Analogously, many fundamental results have been achieved in the past decade concerning the capabilities of neural networks for general computation [26] 45] [86], 90] 94] 95] 96] 105] 110] 128] 136] In particular, the computational and descriptive powers of neural nets have been investigated by comparing their various architectures with each other and with more traditional computational models and descriptive tools such as nite automata, ....

S. Muroga, Threshold Logic and its Applications (Wiley{ Interscience, New York, 1971).

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S. Muroga. Threshold Logic and Its Applications. Wiley-Interscience, 1971.

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S. Muroga. Threshold Logic and its Application. Wiley, New York, 1971.

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S. Muroga. Threshold Logic and its Applications. Wiley, New York, 1971.

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S Muroga. Threshold Logic and its Applications. John Wiley & Sons, 1971.

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Muroga, Saburo "Threshold Logic and its Applications", Wiley-Interscience, 1971.

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S. Muroga. Threshold logic and its applications. Wiley-Interscience, 1971.

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S. Muroga, Threshold Logic and Its Applications, Wiley, New York, 1971.

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S. Muroga, Threshold Logic and its Applications. New York, NY: John Wiley, 1971.

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S. Muroga, "Threshold Logic and its Applications ", Wiley and Sons Inc., 1971;

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Muroga, S. Threshold Logic and Its Applications. Wiley-Interscience, New York, 1971.

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S. Muroga, Threshold Logic and its Applications, Wiley, New York, 1971.

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Muroga, S. (1971). Threshold Logic and Its Application. WileyInterscience.

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