A. K. Chandra, L. Stockmeyer, and U. Vishkin. Constant Depth Reducibility. SIAM Journal on Computing, 13(2):423--439, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 they can be solved by constant depth poly size circuits of multiplication gates. Supported by NSF grant CCR 9877078. Supported in part by NSF grant CCR 9734918. Supported in part by NSF grant ....

....the number of ones in a binary string of length log n. In fact one can also determine the number of ones in a string of length n if this number is n. In TC one can multiply two n bit numbers, and add together n n bit numbers. For background on these algorithms, the reader can consult [19, 39, 14, 26, 47, 37]. Only one strict containment relation is known between the previously mentioned complexity classes. It is known that AC # TC L P. 2.3. Descriptive Complexity Classes Our goal in this paper is to show that Division and Iterated Multiplication are in the class DLOGTIME uniform TC ....

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A. K. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. SIAM J. Comput., 13:423--439, 1984.

....practically computing all the bits of xy; however, an algorithm for DMULT has the advantage of inspecting all the bits of z, the putative product. Buss [Bu92] proves that DMULT 62 AC by reducing it to counting the number of 1 s in the input (and therefore to MULT and to PARITY by results of [CSV84]) for comparison, FSS84] gives an easy reduction of MULT to PARITY to show MULT 62 AC . A simple argument [We94] shows that computing DMULT with read once programs is as hard as factoring. Given a polynomial size read once program for DMULT and any integer n, the following procedure will ....

....We may deduce similar lower bounds for other boolean functions by the standard technique of problem reduction. In order to preserve read once complexity, we will consider a very restrictive type of problem reduction. We begin with the notion of projection reductions [SV81] as defined in [CSV84]: Definition 5 A function f = ff n g n2N is projection reducible to a function g = fg n g n2N , written f proj g, if there is a mapping oe : fy 1 ; y p(n) g f0; 1; x 1 ; x n ; x 1 ; x n g such that f n (x 1 ; x n ) g p(n) oe(y 1 ) oe(y p(n) ....

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A. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. SIAM Journal of Computing, 13 (1984), pp. 423--439.

....depth measure from circuit complexity. Note that in the Circuit RAM model, the two level hashing scheme of [FKS84] has query time O(log w= log log w) since multiplication and integer division [BCH86] are in NC and any function in NC has polynomial size circuits of depth O(log w= log log w) [CSV84]. We provide a matching lower bound that also generalizes the lower bound of Theorem A. Theorem C There is no Circuit RAM solution to the static dictionary problem using space and with query time o(log w= log log w) In fact, for any word length w, there is a fixed set X f0; 1g of size n = ....

A.K. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. SIAM Journal on Computing, 13:423--439, 1984.

....the well studied depth measure from circuit complexity. In the Circuit RAM model, the two level hashing scheme of [13] has query time O(log w= log log w) since multiplication and integer division [7] are in NC and any function in NC has polynomial size circuits of depth O(logw= log log w) [9]. We provide a matching lower bound that also generalizes the lower bound of Theorem A. Theorem C There is no Circuit RAM solution to the static dictionary problem using space and with query time o(log w= log log w) In fact, for any word length w, there is a fixed set X f0; 1g of size n = ....

A.K. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. SIAM Journal on Computing, 13:423--439, 1984.

....k elements are computed with respect to the generator . By Fact (3) this can be done by an AC 0 circuit. The next task is to add the k O(log n) bit integers 1 ; 2 ; k , and reduce the sum modulo 2 m Gamma 1. The addition can be done by a logspace uniform TC 0 circuit (see [CSV84] for details) Since the sum of the k integers is at most k2 m = n O(1) reducing the sum modulo 2 m Gamma 1 can be easily accomplished by an AC 0 circuit that hardwires the conjunctive or disjunctive normal form formulas for each bit of the output. It is also clear that the circuit ....

A. Chandra, L. Stockmeyer, and U. Vishkin. Constant-depth reducibility. SIAM Journal on Computing, 13:423--439, 1984.

....section, and because it simplifies exposition of the Boolean complexity of the operations. 2) The operations listed in Facts (1) 2) and (4) can, in fact, be implemented in AC 0 [ Phi] TC 0 . Multiplying n O(1) elements, as described in Fact (5) can also be implemented in TC 0 [CSV84] Since the proof of Theorem 4 only requires these operations, our proof shows that if there is a sparse P hard set under many one reductions computable in logspace uniform TC 0 , then P = logspace uniform TC 0 . End of Remarks. Our parallel algorithm for CVP begins by computing f(hC; x; 1 ....

A. Chandra, L. Stockmeyer, and U. Vishkin. Constant-depth reducibility. SIAM Journal on Computing, 13:423--439, 1984. 16

....and Hoover presented a FO reduction from POWERING to DIVISION in [7] This suffices to also reduce the binary MULTIPLICATION problem to DIVISION, because the expression XY = X Y ) 2 X 2 Y 2 ] 2 reduces MULTIPLICATION to POWERING. MAJORITY is reducible to MULTIPLICATION as shown in [9]. It remains to solve the POW predicate a i # b (mod m) with these tools. We can use POWERING to compute a i , find q = #a i m# using DIVISION directly, compute a i qm using MULTIPLICATION, and compare the result to b. 2 To be completely formal, the statement of this problem ....

A. K. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. SIAM J. Comput., 13:423--439, 1984.

.... parallel programming models form quite a menagerie: ffl Semaphores, CCRs, Monitors [BA90, AS83] ffl CSP, Occam [Hoa85] 17 ffl Data Flow model [AA82] ffl RPCs, Client Server models [BN84, MW91] ffl Petri Nets [Rei85] ffl Alternating Turing Machines [CKS81, Ruz80] ffl Boolean circuits [CSV84, SV84] ffl Systolic Arrays [Kun82] ffl Associative Processors [Pot92, SKA92] ffl PRAM (several varieties: EREW, CREW, several kinds of CRCW) FW78, Gol82, SS79] ffl V RAM (data parallel) model [Ble90, HS86] Each model sprang from a different research community in response to completely ....

A.K. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. SIAM Journal of Computing, 13:423--439, 1984. 127

....constant depth circuits of AND, OR and NOT gates of polynomial size (called AC 0 circuits) can add and subtract binary numbers. The class ACC extends AC 0 by allowing modular counting gates. The class TC 0 , consisting of constant depth threshold circuits, can compute iterated multiplication [16]. Thus there are many problems of practical importance that can be solved in constant parallel time. Here we address the following question: If we extend these models to the quantum setting, F. Green, S. Homer, C. Moore, and C. Pollett 37 do we obtain more powerful models of computation More ....

A. Chandra, L. Stockmeyer, and U. Vishkin, \Constant depth reducibility," in SIAM J. Comp., 13, (1984), pp. 423-439.

....a typical property of densely interconnected neural networks, which thereby may gain more ef ciency. For example, any classical circuit of size s, depth d, and with maximum fan in 2 can be implemented by a threshold circuit with unbounded fan in and O(s 2 ) gates whose depth is O(d= log log s) [16]. This yields a speed up factor of O(log log n) in polynomial size feedforward neural networks. The exponential size (6) in the above discussed universal networks is not practically realizable for large input lengths. On the other hand, no naturally interesting function (e.g. from the complexity ....

A.K. Chandra, L.J. Stockmeyer, and U. Vishkin, Constant depth reducibility, SIAM Journal on Computing 13 (2) (1984) 423-439.

....well known adders built out of AND OR bounded fan in BGs (we use delay instead of depth, and gates instead of size, as in most of the original articles) are shown in Table 1. here n is the number of bits needed to represent one input) Delay (depth) Gates (size) Method Reference 4 O (n 2 ) Chandra et al. 1984) also in Wegener (1987) 3 O (n 2 ) Wegener (1990) also in Parberry (1994) 2n 1 5n 3 School method 4logn 35n 6 Carry lookahead Chang et al. 1992) 4logn 14n logn 10 Carry lookahead Brent and Kung (1982) 3logn 3nlogn 12.5n 8 Conditional sum Kelliher et al. 1992) 2 e)logn (2 e)nlogn 5n ....

Chandra, A.K., Stockmeyer, L.J., & Vishkin, U. (1984). Constant Depth Reducibility. SIAM Journal on Computing, 13(2), 423-539.

....fan in is a typical property of densely interconnected neural networks which may gain more eciency. For example, a classical circuit of size s, depth d, and with the maximum fan in 2 can be implemented by a threshold circuit with unbounded fan in and O(s 2 ) gates whose depth is O(d= log log s) [16]. This yields O(log log n) factor speed up in polynomial size feedforward neural networks of unbounded fan in. The exponential size (2.6) in the above discussed universal networks is far from being practically realizable for larger input lengths. On the other hand, there is no speci c function ....

A.K. Chandra, L.J. Stockmeyer, and U. Vishkin, Constant depth reducibility, SIAM Journal on Computing 13 (2) (1984) 423-439.

....a) holds. Then the 1;b 1 formula A s t is equivalent to the 1;b 1 formula 8 m T able(s H ; t H ; m ; T s ) T s 1; m ) x) Now the existence of is proved from the counting axiom by formalizing in D 0 k a reduction of vector summation to counting such as the one in [6], and the uniqueness follows from extensionality. The de nitions for the other function symbols can be found in [15, 16] To de ne A H for a formula A, rst (s t) H is de ned as s H H t H , where H expresses the lexicographic ordering of predicates. Then (s = t) H is (s t) H ....

....by openLIND and open COMP respectively, where the latter is provable in C 0 k 1 by Lemma 3. The translation of 1;b 0 AC is provable by use of BB b 0 . Finally the translation of the counting axiom can be proved in C 0 k 1 by use of the reduction of counting to multiplication in [6]. Finally, we show that the translations H and L are inverse to each other. There are very easy translations from L k 1 to itself and 2 from L k to itself such that the following holds. Theorem 19. 1. C 0 k 1 A A HL for every L k 1 formula A. 2. D 0 k B B LH2 , for ....

A. C. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. SIAM Journal of Computing, 13:423-439, 1984.

.... arrays (F n ) n#N of Boolean functions F n : 0, 1 n # 0,1 are computable by arrays (N n ) n#N of neural nets of depth O(1) and size O(n O(1) with linear threshold gates, no matter whether one uses as weights arbitrary reals, rationals, integers, or elements of 1, 0, 1 ; see [Mu] [CSV], HMPST] GHR] MT] The resulting class of arrays (F n ) n#N of Boolean functions is called (nonuniform) TC 0 (see [HMPST] J] In comparison, very little is known about upper bounds for the computational power and the learning complexity of feedforward neural nets whose gates g employ ....

.... operations (multiple addition, multiplication, division) can be carried out on a circuit of small constant depth with O(a O(1) w O(1) MAJORITYgates, hence also on a network architecture of depth O(1) and size O(a O(1) w O(1) with Heaviside gates and weights from 1, 0, 1 (see [CSV], PS] HMPST] GHR] SR] SBKH] Thus one can simulate for inputs from 0, 1 n any depth d network architecture N as in Theorem 2.1 with arbitrary parameter assignments # # R w by a network architecture of depth O(d) and size O(a O(1) w O(1) with Heaviside gates and weights ....

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A. K. Chandra, L. Stockmeyer, and U. Vishkin, Constant depth reducibility, SIAM J. Comput., 13 (1984), pp. 423--439.

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A. K. Chandra, L. Stockmeyer, and U. Vishkin. Constant Depth Reducibility. SIAM Journal on Computing, 13(2):423--439, 1984.

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A. K. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. SIAM J. Comput., 13:423--439, 1984.

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A.K. Chandra, L. Stockmeyer and U. Vishkin, Constant depth reducibility, SIAM J. Computing 13 (1984), 423-439.

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A. Chandra, L. Stockmeyer, and U. Vishkin. Constant-depth reducibility. SIAM J. Comput., 13:423--439, 1984.

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A. K. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. SIAM Journal on Computing, 13:423--439, 1984.

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A. K. Chandra, L. Stockmeyer, and U. Vishkin, Constant depth reducibility, SIAM Journal on Computing, 13 (1984), pp. 423--439. 4

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A. Chandra, L. Stockmeyer, and U. Vishkin. Constant depth reducibility. SIAM Journal on Computing, 13:423--439, 1984.

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Ashok K. Chandra, Larry Stockmeyer, and Uzi Vishkin. Constant depth reducibility. SIAM J. Comput., 13(2):423--439, 1984.

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Ashok Chandra, Larry Stockmeyer, and Uzi Vishkin. Constant depth reducibility. SIAM Journal on Computing, 13(2):423--439, 1984.

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A. Chandra, L. Stockmeyer and U. Vishkin, Constant Depth Reducibility, SIAM J. on Computing 13, 2 (1984), pp. 423-439. 30

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A. Chandra, L. Stockmeyer and U. Vishkin, Constant Depth Reducibility, SIAM J. on Computing 13, 2 (1984), pp. 423-439.

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