J. Reif and S. Tate, "On threshold circuits and polynomial computation," SIAM J. Comput. 21, 896--908 ~1992!.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....intermediate values in the algorithm can be computed by TC circuits. The circuits will be simple to construct, and they will be composed in simple ways to compute the final results from the inputs. The most complex circuits used will be TC circuits raising an n bit number to the power n [19, 38]. Since the computation effected by each of these circuits can be described by a formula in FO(COUNT) the combined computation can be described by a combined formula in FO(COUNT) and therefore can be performed by a TC circuit. The formula for updating the a s,t upon inserting an edge ....

Reif, John H. On threshold circuits and polynomial computation. In Proceedings, Structure in Complexity Theory, Second Annual Conference (Cornell University, Ithca, NY, 16--19 June 1987), IEEE Computer Society Press, pp. 118--123.

....Hesse University of Massachusetts E mail: whesse cs.umass.edu and Eric Allender Rutgers University Piscataway, NJ 08854 8019 E mail: allender cs.rutgers.edu and David A. Mix Barrington University of Massachusetts E mail: barring cs.umass.edu It has been known since the mid 1980 s [16, 48, 49] that integer division can be performed by poly time uniform circuits of Majority gates; equivalently, the division problem lies in P uniform TC . Recently this was improved to L uniform TC [21] but it remained unknown whether division can be performed by DLOGTIMEuniform circuits. The ....

....In 1984 Beame, Cook, and Hoover [16] presented new parallel algorithms for division, powering, and iterated multiplication of integers. They showed that these problems can be solved by families of circuits with fan in two and O(log n) depth, placing them in the circuit class NC . As Reif [48, 49] observed soon after, their algorithms can also be implemented by families of threshold circuits with constant depth and polynomial size, placing these problems in the circuit class TC . Equivalently [19] these problems are reducible in a strong sense to ordinary integer multiplication ....

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J. Reif and S. Tate. On threshold circuits and polynomial computation. SIAM J. Comput., 21:896--908, 1992.

....Hesse University of Massachusetts E mail: whesse cs.umass.edu and Eric Allender Rutgers University Piscataway, NJ 08854 8019 E mail: allender cs.rutgers.edu and David A. Mix Barrington University of Massachusetts E mail: barring cs.umass.edu It has been known since the mid 1980 s [16, 48, 49] that integer division can be performed by poly time uniform circuits of Majority gates; equivalently, the division problem lies in P uniform TC . Recently this was improved to L uniform TC [21] but it remained unknown whether division can be performed by DLOGTIMEuniform circuits. The ....

....In 1984 Beame, Cook, and Hoover [16] presented new parallel algorithms for division, powering, and iterated multiplication of integers. They showed that these problems can be solved by families of circuits with fan in two and O(log n) depth, placing them in the circuit class NC . As Reif [48, 49] observed soon after, their algorithms can also be implemented by families of threshold circuits with constant depth and polynomial size, placing these problems in the circuit class TC . Equivalently [19] these problems are reducible in a strong sense to ordinary integer multiplication ....

[Article contains additional citation context not shown here]

J. H. Reif. On threshold circuits and polynomial computation. In Proceedings, Structure in Complexity Theory, Second Annual Conference, pp. 118--123, IEEE Computer Society Press.

....for updating the a s,t upon inserting an edge is s,t = a s,t a s,i r . We shall see that all of these operations upon numbers with n and n can be performed by TC circuits. The computational power of constant depth threshold circuits was investigated by Reif and Tate in [13]. They found that polynomials with size and degree bounded by n and with coe#cients and variables bounded by 2 could be computed by polynomial size constant depth threshold circuits [13, Corollary 3.4] We cannot use the result of Reif and Tate about the evaluation of polynomials directly ....

John H. Reif and Stephen R. Tate. On threshold circuits and polynomial computation. SIAM Journal on Computing, 21(5):896--908, October 1992. 16

....is in Uniform TC 0 William Hesse # Department of Computer Science University of Massachusetts Amherst, MA 01002 FAX: 001) 413 545 1249 whesse cs.umass.edu Abstract. Integer division has been known since 1986 [4, 13, 12] to be in slightly non uniform TC 0 , i.e. computable by polynomial size, constant depth threshold circuits. This has been perhaps the outstanding natural problem known to be in a standard circuit complexity class, but not known to be in its uniform version. We show that indeed division is in ....

.... Beame, Cook, and Hoover showed that iterated multiplication, and thus division, could be performed by Boolean circuits of logarithmic depth (NC 1 circuits) 4] In 1987, Reif showed that these circuits could be implemented as constant depth circuits containing threshold gates (TC 0 circuits) [12, 13]. Since then, the remaining issue has been the complexity of constructing these circuits. Division is the only prominent natural problem whose computation uses non uniform circuits, circuits which require a non trivial amount of computation for their construction. The division problem discussed in ....

[Article contains additional citation context not shown here]

J. H. Reif and S. R. Tate. On threshold circuits and polynomial computation. SIAM Journal on Computing, 21(5):896--908, 1992.

....is in Uniform TC 0 William Hesse # Department of Computer Science University of Massachusetts Amherst, MA 01002 FAX: 001) 413 545 1249 whesse cs.umass.edu Abstract. Integer division has been known since 1986 [4, 13, 12] to be in slightly non uniform TC 0 , i.e. computable by polynomial size, constant depth threshold circuits. This has been perhaps the outstanding natural problem known to be in a standard circuit complexity class, but not known to be in its uniform version. We show that indeed division is in ....

.... Beame, Cook, and Hoover showed that iterated multiplication, and thus division, could be performed by Boolean circuits of logarithmic depth (NC 1 circuits) 4] In 1987, Reif showed that these circuits could be implemented as constant depth circuits containing threshold gates (TC 0 circuits) [12, 13]. Since then, the remaining issue has been the complexity of constructing these circuits. Division is the only prominent natural problem whose computation uses non uniform circuits, circuits which require a non trivial amount of computation for their construction. The division problem discussed in ....

[Article contains additional citation context not shown here]

J. H. Reif. On threshold circuits and polynomial computation. In Proceedings, Structure in Complexity Theory, Second Annual Conference, pages 118--123, IEEE Computer Society Press.

....We then consider various algorithms for powering modulo a small prime, and their consequences for the uniformity of division circuits. Following an argument of Chiu [10] powering modulo a small prime can be achieved by 1 [7] claimed only P uniform NC 1 , but it was observed later by Reif [23] that their algorithm is implementable in TC 0 . fully uniform circuits of logarithmic depth and fan in two ( Ruzzo uniform NC 1 ) and hence division is itself in uniform NC 1 , in fact in NC 1 uniform TC 0 . But we also show that powering modulo a small prime lies in another natural ....

J. Reif and S. Tate. On threshold circuits and polynomial computation. SIAM J. Comput., 21:896--908, 1992.

.... Computing the updates using a TC 1 uniform TC 0 circuit The formula for updating the a s;t upon inserting an edge is a 0 s;t = a s;t n 2 X k=0 a s;i r(ra j;i ) k a j;t mod r n : 5) The computational power of constant depth threshold circuits was investigated by Reif and Tate in [10]. They found that polynomials with size and degree bounded by n O(1) and with coecients and variables bounded by 2 n O(1) could be computed by polynomial size constant depth threshold circuits (Corollary 3.4 in that paper) We cannot use the result of Reif and Tate about the evaluation of ....

John H. Reif and Stephen R. Tate. On threshold circuits and polynomial computation. SIAM Journal on Computing, 21(5):896-908, October 1992.

....and [123] respectively. Moreover, the multiple product of n n bit binary numbers is performable in depth 4 [124] It thus follows that any analytic function can be implemented to high precision by a perceptron network of polynomial size and weights, using only a small constant number of layers [103]. There are also trade o results known among di erent complexity measures including size, depth, and connectivity in threshold circuits. For example, one can achieve smaller polynomial size feedforward networks for some of the arithmetic tasks discussed above if the permissible depth is ....

J.H. Reif and S.R. Tate, On threshold circuits and polynomial computations, SIAM Journal on Computing 21 (5) (1992) 896{ 908.

....circuit to produce the correct output input x, and put a 0 in that entry otherwise. 4 When these lectures were presented, division was known to be hard for TC 0 , but it was still unknown whether division was in uniform TC 0 . At the time, division was known only to lie in nonuniform TC 0 [RT92] In the mean time, this has been resolved [Hes01] 24 Eric Allender and Catherine McCartin 2 m random sequences 2 n inputs x 000 . 0 001. 1 000 . 1 . 110. 1 . 111 . 1 010. 1 Each row has # 2 m (1 1 2 2n ) 1 s (for good random ....

J. Reif and S. Tate. On threshold circuits and polynomial computation. SIAM Journal on Computing, 21:896--908, 1992.

....We now know that division is also complete for TC 0 under AC 0 T reductions. It had been known for a while that multiplication reduces to division, and thus division was known to be hard for TC 0 . In fact, division had been known to be in P uniform TC 0 ever since it was observed in [27, 28] that the algorithm of [9] can be implemented in P uniform TC 0 . The breakthrough of [18] is that division is in DLOGTIME uniform TC 0 . 3 Background on Division All of the recent work on division builds on the work of Beame, Cook, and Hoover [9] Beame, Cook, and Hoover make use of the ....

J. Reif and S. Tate. On threshold circuits and polynomial computation. SIAM J. Comput., 21:896--908, 1992.

....We now know that division is also complete for TC 0 under AC 0 T reductions. It had been known for a while that multiplication reduces to division, and thus division was known to be hard for TC 0 . In fact, division had been known to be in P uniform TC 0 ever since it was observed in [27, 28] that the algorithm of [9] can be implemented in P uniform TC 0 . The breakthrough of [18] is that division is in DLOGTIME uniform TC 0 . 3 Background on Division All of the recent work on division builds on the work of Beame, Cook, and Hoover [9] Beame, Cook, and Hoover make use of the ....

J. Reif, On threshold circuits and polynomial computation. Proc. 2nd IEEE Structure in Complexity Theory Conference, 1987, pp. 118--123.

....interested in nding the parallel time complexity of the pseudo random functions. In order to do so, let us rst recall the result of Beame, Cook and Hoover [2] who showed that division and related operations including multiple product are computable in NC 1 . Based on this result Reif and Tate [62, 63] showed that these operations are also computable in TC 0 . The exact depth 6 In the case that Q is much smaller than P we have that the rst multiple product is much cheaper than the second 20 required for these operations was considered in [68, 69] where it was shown that multiple sum is in ....

J. Reif and S. Tate, On threshold circuits and polynomial computation, SIAM J. Comput., vol. 5, 1992, pp. 896-908.

....interested in nding the parallel time complexity of the pseudo random functions. In order to do so, let us rst recall the result of Beame, Cook and Hoover [2] who showed that division and related operations including multiple product are computable in NC 1 . Based on this result Reif and Tate [62, 63] showed that these operations are also computable in TC 0 . The exact depth 6 In the case that Q is much smaller than P we have that the rst multiple product is much cheaper than the second 20 required for these operations was considered in [68, 69] where it was shown that multiple sum is in ....

J. Reif, On threshold circuits and polynomial computation, Proc. of the 2nd Conference on Structure in Complexity Theory 1987, pp. 118-123.

....are thus better motivated. Furthermore, proofs using logical uniformity are arguably simpler two examples of this are the uniformity [AG, B92] of the upper bounds on the power of ACC 0 [Ya, BT, GKRST] and the relationship between threshold circuits and algebraic circuits over GF (2 n ) [R, BFS, FVB]. In this paper we show that the descriptive complexity framework can deal with all three dimensions and with general operations, in virtually any possible combination. Specifically: ffl In Section 2 we review the descriptive complexity framework and define quantifiers for arbitrary operations. ....

J. Reif, "On threshold circuits and polynomial computation," 2nd Structure in Complexity Theory Conf. (1987), 118-123.

.... and [118] respectively) Moreover, the multiple product of n n bit binary numbers is performable in depth 4 [119] It follows that any analytic function can be implemented with a large precision by perceptron networks of polynomial size and weights within only a small constant number of layers [101]. There are also trade o results known among di erent complexity measures including size, depth, and connectivity in threshold circuits. For example, one can achieve smaller polynomial size feedforward networks in some cases of the arithmetic circuits above if the optimal depth is increased by ....

J.H. Reif and S.R. Tate, On threshold circuits and polynomial computations, SIAM Journal on Computing 21 (5) (1992) 896-908.

.... and addition on two n bit numbers can be computed by such networks in depth 2 [52, 57] the product of two n bit numbers can be computed in depth 4 [70] and analytic functions with a convergent rational power series (e.g. sin, cos, exp, log, sqrt) can all be approximated in bounded depth [63] (cf. also [69] It is therefore of great interest to consider also the following fixed depth sublevels of the class TC 0 individually: TC 0 d = ffunctions computable by neural circuits of polynomial size and depth dg; c TC 0 d = ffunctions computable by threshold circuits of polynomial ....

Reif, J. H., Tate, S. R. On threshold circuits and polynomial computation. SIAM J. Comput. 21 (1992), 896--908.

....A Omega Gamma4 =d) where d is the given depth of the circuit (Lemma 3.1) Thus sine and oe are not equivalent with respect to error e(s; d) 2 Gammas . On the other hand, constant depth and size polynomial in log A suffice to approximate sine(Ax) provided the input x is given in binary (Reif, 1987). Therefore, the complexity of extracting bits from the given analog input is high as well. 3 1.2 Error e(s; d) s Gammad . Now we are facing a far less demanding error requirement (if depth is small) But, as we will see in section 1.3, linear splines and the standard sigmoid are still not ....

....Finally, let fl(x) exp(x) This time fl( Gammax) is fast converging. Remark 3.2 (a) Of course, the equivalence of spline circuits and foeg circuits also holds for binary input. Since threshold circuits can add and multiply m m bit numbers in constant depth and size polynomial in m (Reif, 1987), threshold circuits can efficiently approximate polynomials and splines. Thus, we obtain that oe circuits of depth d, size s and Lipschitz bound L can be simulated by circuits of binary thresholds. The depth of the simulating threshold circuit will increase by a constant factor and its size will ....

[Article contains additional citation context not shown here]

REIF, J. H. (1987), On threshold circuits and polynomial computation, in "Proceedings of the 2nd Annual Structure in Complexity theory", pp. 118-123.

....it had been shown that multiplication and iterated sum can be computed by polynomial size, constant depth threshold circuits [46] Furthermore, when we consider P uniform circuit families, it is possible to compute iterated product and integer 6 CHAPTER 1. INTRODUCTION division in constant depth [52]. In chapter 3, we show how to reduce the size of constant depth circuits for iterated product and division. Specifically, we construct a constant depth circuit family for integer division with size O(n 1 ffl ) for any constant ffl 0, and then show how to use these results to give simulations ....

J. H. Reif. On Threshold Circuits and Polynomial Computation. In 2nd Conference on Structure in Complexity Theory, pages 118--123, 1987.

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J. Reif and S. Tate, "On threshold circuits and polynomial computation," SIAM J. Comput. 21, 896--908 ~1992!.

No context found.

J. Reif and S. Tate. On threshold circuits and polynomial computation. SIAM J. Comput., 21:896--908, 1992.

No context found.

J. Reif. On threshold circuits and polynomial computation. In Proceedings, Structure in Complexity Theory, Second Annual Conference, pp. 118--123, IEEE Computer Society Press.

No context found.

J. Reif and S. Tate. On threshold circuits and polynomial computation. SIAM J. Comput., 21:896--908, 1992.

No context found.

J. Reif and S. Tate. On threshold circuits and polynomial computation. SIAM J. Comput., 21:896--908, 1992.

No context found.

J. H. Reif. On threshold circuits and polynomial computation. In 2nd IEEE Structure in Complexity Theory, pages 118--123, 1987.

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