M. Goldmann and M. Karpinski. Simulating threshold circuits by majority circuits. SIAM J. on Computing, 27(1):230--246, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... can be replaced with polynomial ones, by increasing the depth of a circuit by at most one layer and imposing a polynomial overhead in its size [33] An explicit construction of the underlying polynomial weight circuit whose polynomial size increase is independent of the depth can be found in [34]. Thus, polynomial weights can be assumed in multi layered perceptron networks if one extra parallel computational step can be granted. Another restriction concerns the maximum fan in of gates in the circuit, i.e. the maximum number of inputs to a single unit. Circuits with bounded fan in ....

.... of computing the basic arithmetic functions [102] 105] 128] For example, two layered perceptron networks of polynomial size and weights can be constructed for comparing two n bit binary numbers [4] or for computing the sum of two [4] or even multiple sum of n such numbers (from [124] by [34]) Further, the product and quotient of two n bit binary numbers, powering to n [124] and sorting [123] of n such numbers can be achieved with three layered feedforward networks. It is also known that the depth in these circuits (except for powering where this issue is still unresolved) cannot be ....

M. Goldmann and M. Karpinski, Simulating threshold circuits by majority circuits, SIAM Journal on Computing 27 (1) (1998) 230-246.

....P; where y(x) Q n i=1 a x i i . For our purpose it is obviously sufficient to show Theorem 6. The function f = f s has polynomial size depth 4 unweighted threshold circuits. Proof. We use the following terminology and facts about threshold circuits which are mainly based on results from [8, 9, 28]. 8 Definition 1. A Boolean function g : f0; 1g n f0; 1g is called t bounded if there are integer weights w 1 ; wn and t pairwise disjoint intervals [a k ; b k ] 1 k t of the real line such that g(x 1 ; xn ) 1 ( 9k s.t. n X i=1 w i x i 2 [a k ; b k ] The ....

M. Goldmann, M. Karpinski. Simulating threshold circuits by majority circuits. Proc. 25th ACM Symp. on Theory of Computing (STOC), 1993, 551-560.

....depth dg: Some authors, e.g. 26, 91] use the notations LT d , c LT d , from linear thresholds , for these classes. It is shown in [26] that the classes d TC 0 d , TC 0 d form a hierarchy: d TC 0 d TC 0 d d TC 0 d 1 for all d 1. For a constructive version of the proof, see [27]. Separating the levels of this hierarchy, as far as they are separate, is currently a significant research task. That class d TC 0 1 is properly contained in TC 0 1 follows from Theorem 2 above; and the proper containment of TC 0 1 in d TC 0 2 follows from the well known fact [65] that ....

Goldmann, M., Karpinski, M. Simulating threshold circuits by majority circuits. In: Proc. of the 25th Ann. ACM Symp. on Theory of Computing. ACM, New York, 1993. Pp. 551--560.

.... can be replaced with polynomial ones by increasing the depth by at most one layer while only a polynomial overhead in the network size is needed [32] The explicit construction of the underlying polynomialweight circuit whose polynomial size increase is independent of the depth can be found in [33]. Thus, polynomial weights can be assumed in multi layered perceptron networks when one more parallel computational step is granted. Another restriction concerns the maximum fan in of gates in the circuit, i.e. the maximum number of inputs to a single unit. The circuits with bounded fan in ....

.... of computing basic arithmetic functions [100, 103, 123] For example, the two layered perceptron networks of polynomial size and weights can be constructed for comparing two n bit binary numbers [4] or for computing the sum of two [4] or even multiple sum of n such numbers (from [119] by [33]) Further, the product and division of two n bit binary numbers, powering to n [119] and sorting [118] of n such numbers can be achieved with three layered feedforward networks whereas the depth in these circuits (except for powering where this issue is still unresolved) cannot be reduced when ....

M. Goldmann and M. Karpinski, Simulating threshold circuits by majority circuits, SIAM Journal on Computing 27 (1) (1998) 230-246.

....network [14] It was also suggested in [30] based on a result in [31] that multi operand addition can be computed in depth 2 and multiplication in depth 3 but no explicit construction for the networks and no complexity bounds are provided. A constructive approach can be derived if the result in [32] suggesting that a single threshold gate computing F (x) sgnf 0 1 x 1 Delta Delta Delta n x n g with arbitrary weights can be simulated by an explicit polynomial size depth 2 network is used. Such a LOGSPACE uniform construction as stated in [32] produces a network with O(log 12 ....

....can be derived if the result in [32] suggesting that a single threshold gate computing F (x) sgnf 0 1 x 1 Delta Delta Delta n x n g with arbitrary weights can be simulated by an explicit polynomial size depth 2 network is used. Such a LOGSPACE uniform construction as stated in [32] produces a network with O(log 12 W (n) wires and the weights of those wires in order of O(log 8 W (n) for a total size of O(n 20 log 20 n) The total size for such a construction was further reduced to O(n 12 log 12 n) in [33] LOGSPACE uniform constructions for depth 2 ....

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M. Goldmann and M. Karpinski, "Simulating threshold circuits by majority circuits," SIAM Journal on Computing, vol. 27, pp. 230--246, Feb. 1998.

....Element y = sgn( Gammat P n i=1 w i x i ) Many results from the theory of threshold circuits could be applied to the implementation of circuits on silicon. Results such as the relationship between the maximal size allowed for the weights and the power of the resulting element or circuit [5] [8], not to mention efficient designs for XOR, ADD, MULT IPLY and other useful functions, see [11] 13] 15] Our research has three distinct goals: 1. The implementation aspect. To design and implement efficient threshold elements on silicon. 2. The theoretical aspect. To leverage the work done ....

M. Goldmann and M. Karpinski. Simulating threshold circuits by majority circuits. In Proc. 25th ACM STOC, pp. 551--560, 1993.

.... with by defining a class within the set of linear threshold functions : the class of functions with small (i.e. polynomialy growing) weights [Siu 91] Most of the recent research focuses on the power of circuits with small weights, relative to circuits with arbitrary weights [Goldmann 92] Goldman 93] Rather than dealing with circuits we are interested in studying a single threshold gate. The main contribution of the present paper is to further refine the division of small versus arbitrary weights. We separate the set of functions with small weights into classes indexed by d, the degree of ....

M. Goldmann and M. Karpinski. Simulating threshold circuits by majority circuits. In Proc. 25th ACM STOC, pp. 551--560, 1993.

....more complicated. It is known that exponential weights suffice. Hofmeister has recently described a simple construction that replaces exponentially bounded weighted threshold circuits by polynomially bounded at the expense of increasing the depth by 1 [H96] which simplifies previous work [GHR92,GK93]. However, this simpler construction still does not seem to directly translate into fault tolerant circuit designs. Finally, in the weakly reliable model threshold circuits can be made arbitrarily fault tolerant using redundancy of moderate size. This indicates a fundamental difference between the ....

M. Goldmann, M. Karpinski, Simulating Threshold Circuits by Majority Circuits, Proc. 25. SToC, 1993, 551 -- 560.

....November 13, 1998 Abstract We present an explicit construction of a circuit for the COMPARISON function in c LT 2 , the class of polynomial size linear threshold circuits of depth two with polynomially growing weights. Goldmann and Karpinski proved that LT 1 ae c LT 2 in [4]. Hofmeister presented a simplified version of the same result in [6] We have further simplified the results of these two papers by limiting ourselves to the simulation of COMPARISON. Our construction has size O(n 4 log n) a significant improvement on the general bound of O(n 12 log 11 n) ....

....the property that the weights are integers bounded by a polynomial in n, i.e. jw i j n c for some constant c 0. Siu and Bruck proved that LT d ae d LT 2d 1 [11] Goldmann and Karpinski improved the bound to LT d ae d LT d 1 by showing that LT 1 ae d LT 2 and generalizing to arbitrary depth [4]. Hofmeister presented a simplified version of the proof that LT 1 ae d LT 2 [6] The idea is to use two operations in order to reduce the weights: divide them by powers of two and divide them modulo a prime. The resulting small weight gates are connected into a circuit that produces the correct ....

[Article contains additional citation context not shown here]

M. Goldmann and M. Karpinski. Simulating threshold circuits by majority circuits. In SIAM J. Discrete Math, Vol. 7, No. 1, pp. 230--246, 1998.

.... with by defining a class within the set of linear threshold functions : the class of functions with small (i.e. polynomialy growing) weights [Siu 91] Most of the recent research focuses on the power of circuits with small weights, relative to circuits with arbitrary weights [Goldmann 92] Goldman 93] Rather than dealing with circuits we are interested in studying a single threshold gate. The main contribution of the present paper is to further refine the division of small versus arbitrary weights. We separate the set of functions with small weights into classes indexed by d, the degree of ....

M. Goldmann and M. Karpinski. Simulating threshold circuits by majority circuits. In Proc. 25th ACM STOC, pages pp. 551--560, 1993.

....weights bounded polynomially in the fan in [6, 15, 18] Let d LT d denote the set of Boolean functions computable by a depth d polynomial size circuit composed of LTUs with small weights. The strongest relationship between LT d and d LT d has been obtained recently by Goldmann and Karpinski [5], who proved that LT d ae d LT d 1 8d 1. The class of linear threshold Boolean functions with integer parameters w i bounded by a constant, constitutes naturally the next stage in this simplification of the LTUs. The simplest situation, where each w i is either 1; 0 or Gamma1, corresponds to ....

M. Goldmann and M. Karpinski, Simulating threshold circuits by majority circuits, in Proceedings of the Twenty Fifth Annual ACM Symposium on Theory of Computating, 1993, pp. 551--560.

....relations. Then, we provide functions that demonstrate the separation between classes. 2.1 Inclusions Most inclusion relations follow from the definitions : c LT LT LTM and c LT d LTM LTM . Only one requires a proof : LTM c LT 2 To show the above statement we use a result from [Goldman 93] a single LT gate with arbitrary weights can be realized by an c LT 2 circuit. Furthermore the non linearity in the second layer can be removed without affecting the output of the circuit (a property called 1 approximability , Hofmeister 96] So, given f 2 LT , f(X) P p i 1 w i f i (X) ....

....ADD = 2 d LTM . 2 IP k 2 c LT 2 but IP k = 2 LTM Proof : Let IP (X; Y ) P n 1 x i y i . Define the function IP k (X; Y ) 1 iff IP k, else IP k = 0. We claim that IP k = 2 LTM . Indeed, if IP k was in LTM then it could be implemented by a layer of c LT gates followed by a weighted sum [Goldman 93] We could then combine the circuits for k = 1: n to implement IP2 (Inner Product mod 2) in c LT 2 which is known to be false [Hajnal 94] 2 What remains to be shown in order to complete the classification picture is c LT = LT d LTM . We conjecture that this is true and we are in the process ....

M. Goldmann and M. Karpinski. Simulating threshold circuits by majority circuits. In Proc. 25th ACM STOC, pages pp. 551--560, 1993.

....weights the analysis seems to be significantly more complicated. Hofmeister has recently described a simple construction that replaces exponentially bounded weighted threshold circuits by polynomially bounded at the expense of increasing the depth by 1 [H96] which simplifies previous work [GHR92,GK93]. However, this construction does not seem to translate directly into fault tolerant circuit designs. Finally, in the weak model threshold circuits can be made arbitrarily fault tolerant by moderate redundancy. This indicates a fundamental difference between both fault models, which has not been ....

M. Goldmann, M. Karpinski, Simulating Threshold Circuits by Majority Circuits, Proc. 25. ACM Symp. on the Theory of Computing, STOC'93, 1993, 551 -- 560.

....depth dg: Some authors, e.g. 26, 91] use the notations LT d , c LT d , from linear thresholds , for these classes. It is shown in [26] that the classes c TC 0 d , TC 0 d form a hierarchy: c TC 0 d TC 0 d c TC 0 d 1 for all d 1. For a constructive version of the proof, see [27]. Separating the levels of this hierarchy, as far as they are separate, is currently a significant research task. That class c TC 0 1 is properly contained in TC 0 1 follows from Theorem 2 above; and the proper containment of TC 0 1 in c TC 0 2 follows from the wellknown fact [65] that the ....

Goldmann, M., Karpinski, M. Simulating threshold circuits by majority circuits. In: Proc. of the 25th Ann. ACM Symp. on Theory of Computing. ACM, New York, 1993. Pp. 551--560.

....threshold, etc. or alternatively if AC 0 circuitry is considered cheap and only applications of MAJORITY are considered expensive [MT93, Mac95] Many separations are known among the various low levels; a good survey of these separations and inclusions is found in [Raz92] See also [GHR92, GK93, Hof96] and the articles in [RSO94] The state of the art in this direction still only yields superpolynomial bounds for restricted classes of depth two or depth three circuits: Threshold of MODm [KP94] extending [Gol95] An alternate proof is presented in [ES] Depth Three MAJORITY ....

M. Goldmann and M. Karpinski. Simulating threshold circuits by majority circuits. In ACM Symposium on Theory of Computing (STOC), pages 551-- 560, 1993.

....recent paper [13] an alternate geometric approach is used to show that IP 2 62 d LT 2 ; in fact related results in [13] imply that IPm 62 d LT 2 . We should also note that the computational power of threshold circuits where all weights are polynomially bounded integers has recently been explored [27, 8, 9]. For example, it is shown in [8, 9] that LT 1 ae d LT 2 , and in general LT d d LT d 1 . The method of correlation and other related techniques have been used in [8] to obtain these and other related separation results. We have therefore seen that if f is to be computed by a threshold gate with ....

....approach is used to show that IP 2 62 d LT 2 ; in fact related results in [13] imply that IPm 62 d LT 2 . We should also note that the computational power of threshold circuits where all weights are polynomially bounded integers has recently been explored [27, 8, 9] For example, it is shown in [8, 9] that LT 1 ae d LT 2 , and in general LT d d LT d 1 . The method of correlation and other related techniques have been used in [8] to obtain these and other related separation results. We have therefore seen that if f is to be computed by a threshold gate with input functions f 1 ; f S ....

M. Goldmann and M. Karpinski. Simulating threshold circuits by majority circuits. In Proceedings of the 25th Annual ACM Symposium on Theory of Computing, pages 551--560, 1993.

No context found.

M. Goldmann and M. Karpinski. Simulating threshold circuits by majority circuits. SIAM J. on Computing, 27(1):230--246, 1998.

No context found.

Goldmann, M. and Karpinski, M. (1993). Simulating threshold circuits by majority circuits. Proc. of 25th Symposium on Theory of Computing (STOC), 551--560.

No context found.

M. Goldmann and M. Karpinski, Simulating Threshold Circuits by Majority Circuits, In Proc. of the 25th annual ACM Symposium on the Theory of Computing, pp. 551-560, 1993.

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