P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15(4):994--1003, November 1986.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 ....

....follows that L NP. Key Words: computer arithmetic, division, iterated multiplication, threshold circuits, uniformity, circuit complexity, complexity classes, finite model theory, Chinese remainder representation, computation in abelian groups 1. INTRODUCTION 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 ....

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P. Beame, S. Cook and J. Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15:994--1003, 1986.

....Special elliptic curves with ecient arithmetics has also been considered [11] 20] 21] In this paper we propose the application of parallel processing to speed up the exponentiation on elliptic curves. There already exists parallel algorithms for exponentiation of integers modulo a m bit integer [2, 1, 13] and also for exponentiation on the elds GF (2 ) 22] and GF (q ) 23] We present in this paper a parallel algorithm to compute scalar multiplication on elliptic curves. This algorithm would be specially ecient for anomalous binary curves, also known as Koblitz curves. The application of ....

P.W. Beame, S.A. Cook and H.J. Hoover, Log depth circuits for division and related problems, SIAM Journal on Computing, vol. 15, pp. 994-1003, 1986.

....are present on real computers. Instead we do allow them, but charge them more, using the well studied 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 ....

P.W. Beame, S.A. Cook, H.J. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15:994--1003, 1986.

....we do not disallow expensive instructions such as multiplication, we just charge them more, using 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 ....

P.W. Beame, S.A. Cook, H.J. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15:994--1003, 1986.

....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 ....

....complexity of integer division has been harder to pin down than the complexities of addition, subtraction, and multiplication. In 1986, 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 ....

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P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15(4):994--1003, 1986.

....NP. # Supported in part by NSF grant CCR 9734918. Supported in part by NSF grant CCR 9988260. # Supported by NSF grant CCR 9877078. 1 Introduction The exact complexity of division, powering, and iterated multiplication of integers has been a major open problem since Beame, Cook, and Hoover [7] showed these problems to be in P uniform TC 0 in 1984 1 . TC 0 is the set of problems solvable by threshold circuits of constant depth and polynomial size, P uniform means that these circuits can be constructed by a poly time Turing machine. In a recent breakthrough, Chiu, Davida and ....

....This predicate calculates powers modulo a prime of O(log n) bits. 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 ....

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P. Beame, S. Cook and J. Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15:994-- 1003, 1986.

....r n by dropping all but the low order n 3 bits. To raise the number ra j;i to the kth power, we will need the following lemma: Lemma 3. Finding the n O(1) th power of a number with n O(1) bits is in TC 1 uniform TC 0 Proof. A corollary to the result of Beame, Cook, and Hoover 3 [2] shows that we can multiply n numbers with n bits each with a TC 0 circuit that is constructible in logspace from the product of the rst n 2 primes [7] The product of the rst n 2 primes can be computed by a TC 1 circuit (a depth O(log n) binary tree of TC 0 circuits performing ....

Paul W. Beame, Stephen A. Cook, and H. James Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15(4):994-1003, 1986.

....other hard closure properties in [OH93] Consider, for example, bf(x) g(x)c for f 2 #P and nonzero g 2 #P. It is easy to design a #P function, say h(x) f(x)2 p(jxj) g(x) for some suitably large polynomial p, from which logarithmically depth bounded circuits can compute the division (see [BCH86]) But, combining the result of Furst, Saxe and Sipser [FSS84] with the easily provable fact that the parity function is AC 0 reducible to integer division, it is seen that no AC 0 circuit can compute the division from h above. Thus, studying the closure properties of #P in context AC 0 ....

P. Beame, S. Cook and H. Hoover, Log depth circuits for division and related problems, SIAM J. Comput. 15 (1986), 994--1003. 15

....breakthrough will be presented at ICALP 2001 by Bill Hesse, a student at the University of Massachusetts. He will be receiving the best paper award for Track A at ICALP 2001 (combined with the best student paper award) All of these results build on the earlier work of Beame, Cook, and Hoover ([9]) In the following sections, I will provide the necessary background about the complexity classes I ll be discussing, and then I ll present the history of these breakthroughs, and the main ideas involved. In a closing section, I ll discuss some of the applications that these advances have already ....

....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 fact that, for small enough ....

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P. Beame, S. Cook and J. Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15:994--1003, 1986.

....is very cheap compared with a multiple product or a modular exponentiation. As described in the Introduction and in Section 2.2, we are also 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 ....

P. W. Beame, S. A. Cook and H. J. Hoover, Log depth circuits for division and related problems, SIAM J. Comput., vol. 15, 1986, pp. 994-1003.

....5 D(n) The proof of theorem 3.2 requires that we build up families of Threshold Circuits for the basic problems of multiplication, iterated sum, and iterated product. The most costly problem we encounter is iterated product, and this is solved using techniques introduced for integer division [2, 8, 18]. As a consequence of Theorem 3.2, we have Corollary 3.3. Suppose an analytic function f(x) has a convergent Taylor Series Expansion of form f(x) 1 X n=0 c n (x Gamma x 0 ) n over an interval jx Gamma x 0 j ffl where 0 ffl 1, and the coefficients are rationals c n = an bn ....

.... logarithm of all input values (f Gamma1 i (x) above) performing the iterated sum of these values modulo p i Gamma 1, then raising the generator to the resulting power in Z p i (this is just f i (x) above) This is a fairly common method of performing iterated product (see, for example, [2]) The only part we haven t examined here is the iterated sum. By Lemma 5.1, we can calculate the exact iterated sum of m ffi numbers, each of log p i bits, in size O(m ffi ffi 2 log p i ) and depth O( 1 ffi ) With an m ffi log p i bit approximation to (1= p i Gamma 1) we can reduce ....

P. W. Beame, S. A. Cook, and H. J. Hoover, Log depth circuits for division and related problems, SIAM J. Comput., 15 (1986), pp. 994--1003.

....uniformity notion under which NC 1 becomes the class of regular languages. If our non uniformity resource is between DLOGTIME and NC 1 itself, we get a robust class, equal to ALOGTIME. And if we allow polynomial time to build our circuits, we can then do integer division and related problems [BCH] which (as far as we know) we couldn t do before. In general our techniques for proving lower bounds on circuit complexity are combinatorial and algebraic and apply to the totally non uniform versions of the circuit classes. A notable exception is the result by Allender and Gore [AG] that the ....

P. Beame, S. Cook, H.J. Hoover, "Log depth circuits for division and related problems," SIAM J. Comput.15:4 (1986), 994-1003.

....the entire process can be performed with depth O( log n) 2 ) and size O(n 3 ) There are alternative methods for performing iterated multiplication achieving various combinations of depth and size. In particular, it was recently proved by Chiu, Davida, and Litow [9] improving on results in [4]) that a product such as we have above can be computed by O(log n) depth boolean circuits of polynomial size. While the size is likely to exceed the O(n 3 ) bound obtained above, the result has an interesting consequence regarding simulations of logarithmic depth quantum circuits: if ....

P. Beame, S. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15(4):994--1003, 1986.

....basic problem build Boolean input Boolean output threshold circuits, to compute useful Boolean functions efficiently. Threshold circuits have been shown to be surprisingly powerful [1] For example, integer division can be implemented by a polynomial size threshold circuit of constant depth, [3], 20] In other words, if one is to implement a threshold circuit to compute the division of two n bit integers, one needs polynomially many, in n, threshold elements. On the other hand, using the traditional logic circuits, composed of AND ,OR and NOT gates, requires exponentially many gates. ....

P.W. Beame, S.A. Cook and H.J. Hoover. Log depth circuits for division and related problems. Proc. 25th IEEE Symposium on Foundations of Computer Science, pages 1 -- 6, 1984.

....leads to a polynomial increase in the number of wires. Hence, depth d polynomial size majority circuits are equivalent to depth d polynomial size, small weight threshold circuits. Threshold circuits have been shown to be surprisingly powerful. It is implicit in work by Beame, Cook, and Hoover [4] that integer division can be carried out by polynomial size threshold circuits of constant depth. Allender [1] inspired by Toda [28] shows that any function in AC 0 can be computed by depth 3 majority circuits of quasi polynomial size. Yao [33] extends this to all of ACC 0 (see also [5] ....

.... : 7 Conclusions and Open Problems Our results entail the first explicit constructions for the optimal depth, polynomial size majority circuits for the number of basic functions including, among others, 17 powering (depth 3) integer multiplication and integer division (depth 3) see [26] and [4]. More generally, our results entail the uniformity of the classes of majority circuits simulating the corresponding classes of threshold circuits. We look at the following functions. ADDITION: given two n bit numbers, compute their sum. MULTIPLE ADDITION: given n n bit numbers, compute their ....

P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. In Proc. 25th IEEE Symposium on Foundations of Computer Science, pages 1--6, 1984.

....complexity than the other operations has been an open theoretical and practical problem. If division could be computed within O(log n) time, according to a common measure of parallel complexity known as NC 1 , then the iterated product problem could be solved sequentially in logarithmic space [1]. Since the iterated product problem has been hypothesized as a candidate for a problem within PTIME log space, research into the parallel time complexity of division is also closely tied to low level space complexity. Approaches to parallel division with better than O(log 2 n) time, e.g. 10, ....

....[1] Since the iterated product problem has been hypothesized as a candidate for a problem within PTIME log space, research into the parallel time complexity of division is also closely tied to low level space complexity. Approaches to parallel division with better than O(log 2 n) time, e.g. [10, 1, 12, 3, 2 8], have all used alternate number representation systems, such as discrete Fourier transforms, or Chinese remaindering. These alternate systems represented numbers as small, independent units. This allowed for greater parallelism than ordinary binary arithmetic since the circuits were not limited ....

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P. Beame, S. Cook, and H. Hoover. Log depth circuits for division and related problems. SIAM Journal of Computing, 15:994--1003, 1986.

....Therefore, each a i ; b i equals to a quotient of a suitable pair of (t 1 t 2 Gamma 1) Theta (t 1 t 2 Gamma 1) minors of this linear system. Then maxfja i j; jb i jg (Np 2t 2 d 2t) 2t (N t 4dt 2 ) 2t . The lemma is proved. 2 One can construct in NC the integer t 4dt 2 ( BCH 86] then by Lemma 2 an integer larger than 2 M=2t and again using [BCH 86] an integer larger than 2 M . Then the algorithm constructs an integer N 0 36 Delta 2 3M Delta d 5 . Finally, the algorithm yields the number N = q(q(N 0 ) We claim that N is big enough (see [GK 91] namely, ....

....1 t 2 Gamma 1) Theta (t 1 t 2 Gamma 1) minors of this linear system. Then maxfja i j; jb i jg (Np 2t 2 d 2t) 2t (N t 4dt 2 ) 2t . The lemma is proved. 2 One can construct in NC the integer t 4dt 2 ( BCH 86] then by Lemma 2 an integer larger than 2 M=2t and again using [BCH 86] an integer larger than 2 M . Then the algorithm constructs an integer N 0 36 Delta 2 3M Delta d 5 . Finally, the algorithm yields the number N = q(q(N 0 ) We claim that N is big enough (see [GK 91] namely, divide with the remainder f = eg rem(f; g) then for each integer N 1 N ....

Beame, P. W., Cook, S. A., Hoover, H. J., LOG Depth Circuit for Division and Related Problems, SIAM J. Comput. 15 (1986), pp. 994-1003.

....The smallest natural circuit class that is not known to be strictly contained in NP is TC 0 , the set of functions computable by constant depth polynomialsize circuits containing threshold gates. Threshold gates are quite powerful and many fairly complicated functions (like division, implicit in [6]) are in TC 0 . It also seems like the techniques used for proving lower bounds for usual constant depth circuits (with or without modular gates) are not sufficient to prove lower bounds for threshold circuits. The best known results about smalldepth threshold circuits are those by Hajnal et ....

P. W. Beame, S. A. Cook, and H.J. Hoover. Log depth circuits for division and related problems. Proceedings 25'th Annual Symposium on Foundations of Computer Science, pages 1--6, 1984.

.... 0 n : n is prime ## dspace(log log n) to show that 0 n : n is prime ## DSPACE(log log n) it would follow that L #= NP. 1 Introduction The exact complexity of division, powering, and iterated multiplication of integers has been a major open problem since Beame, Cook, and Hoover [7] # Rutgers University, allender cs.rutgers.edu U. of Massachusetts, barring cs.umass.edu 1 showed these problems to be in P uniform TC 0 in 1986 1 . TC 0 is the set of problems solvable by threshold circuits of constant depth and polynomial size, P uniform means that these circuits ....

....and Berman [14] says that the set of unary strings of prime length is not in dspace(o(log n) The new translational lemma shows that proving an analogous result for DSPACE(log log n) would separate the classes L and NP. In Section 2 we review the history and context of these numeric problems. 1 [7] claimed only P uniform NC 1 , but it was observed later in [22] that their algorithm is implementable in TC 0 . 2 In Section 3 we outline the new proof of Chiu, Davida, and Litow [9] that the necessary CRR operations for division can be carried out in log space, and we show that this ....

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P. Beame, S. Cook and J. Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15:994--1003, 1986.

....size. The situation for division has been different until now. It has long been known that log uniform, polynomial size, O(log 2 n) depth circuits exist for division. See [15] The question of the existence of log n depth circuits was affirmatively resolved by Beame, Cook and Hoover in [1]. Unfortunately, their division circuits are not log uniform, although they can be produced in P. x EECS Dept. University of Wisconsin Milwaukee, Milwaukee, WI, USA EECS Dept. University of Wisconsin Milwaukee, Milwaukee, WI, USA davida cs.uwm.edu k School of Information Technology, ....

....size variables. We use the expression NC1 in terms of c to mean that a problem can be computed by circuits of c O(1) size, O(log c) depth and that the circuits can be computed in O(log c) space. The basic theory of NC and in particular, NC1 and its associated uniformity issues can be found in [3, 13, 5, 1]. We will need the following facts from this theory. Iterated 2 product is the computation of the binary notations of all of the prefix products x 1 Delta Delta Delta x i for i = 1; n where x 1 ; x n are n bit integers in binary notation. Powering is the computation of the ....

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P. Beame, S. Cook, and H. Hoover. Log depth circuits for division and related problems. SIAM J. Comp., 15,4:994--1003, 1986.

....in [12] and information about the parallel complexity class NC may be found in [11, 4] It is also know that division can be done in the same time and size bounds, but slightly more than logspace is needed to build the requisite Boolean circuits. It is open whether or not division is in NC1. See [2, 5, 7] for more information about division. It is natural to ask about other arithmetic functions. Perhaps the most important function after the basic operations is GCD (greatest common divisor. Unfortunately, very little is known about the parallel complexity of GCD. In this situation it makes sense ....

P. Beame, S. Cook, and H. Hoover. Log depth circuits for division and related problems. SIAM J. Comp., 15,4:994--1003, 1986.

...., computing the product of two n bit numbers x and y . By 2 i j Gamma2 x i y j we mean 2 i j Gamma2 if both x i and y j are true, and 0 otherwise. Lastly, we will describe our TC 0 formula for computing the iterated product of m numbers. This formula is basically the original formula of [BCH], and articulated as a TC 0 formula in [M] The iterative product, PROD[z 1 ; z m ] gives the product of z 1 ; z m , where each z i is of length n , and we assume that m;n are both bounded by N . The basic idea is to compute the product modulo small primes using iterated addition, ....

Beame, P., Cook, S., and Hoover, J. " Log depth circuits for division and related problems," SIAM J. Comput, vol. 15 (1986), pp. 996-1003.

....other hard closure properties in [OH93] Consider, for example, bf(x) g(x)c for f 2 #P and nonzero g 2 #P. It is easy to design a #P function, say h(x) f(x)2 p(jxj) g(x) for some suitably large polynomial p, from which logarithmically depthbounded circuits can compute the division (see [BCH86]) But, combining the result of Furst, Saxe and Sipser [FSS84] with the easily provable fact that the parity function is AC 0 reducible to integer division, it is seen that no AC 0 circuit can compute the division from h above. Thus, studying the closure properties of #P in context AC 0 #P ....

P. Beame, S. Cook and H. Hoover, Log depth circuits for division and related problems, SIAM J. Comput. 15 (1986), 994-1003.

.... 2 n) The first improvement of this was given by Reif in [4] 5] Using a clever construction involving discrete Fourier transforms he was able to construct polynomial size circuits of depth O(log n log log n) By using an elegant method relying on modular arithmetic Beame, Cook and Hoover [1] were able to construct circuits of depth O(log n) This is of course optimal but the construction had two imperfections. The circuits were of size Theta(n 5 ) and only P uniform while Reif s circuits were small and logspace uniform. The result of this paper is remove one of these imperfections ....

....The result of this paper is remove one of these imperfections in that we for any ffl 0 construct circuits for division of depth O(log n) and size O(n 1 ffl ) Unfortunately our circuits are still only P uniform. Our construction basically follows the same ideas as the construction in [1], but we introduce a number of improvements to make the construction more efficient. The most important of these improvements is to introduce a more efficient way of implementing Chinese remaindering. The other key idea is to use one extra level of modular arithmetic to reduce the sizes of the ....

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Beame P.W., Cook S.A. and Hoover H.J. "Log Depth Circuits for Division and Related Problems" Proceedings 25'th Annual Symposium on Foundations of Computer Science, 1984, pp 1-6.

.... has 4 been considered at least once in the past by Allender and Gore [2] although in a completely different context) The reason amounts to a connection with the problem of the uniformity of Boolean circuits for integer division, an interesting issue that has received a good deal of attention [7, 28, 23, 19]. See the end of Section 5 for more details. Finally, it is obvious that our objects of study are intimately related to questions about the rudimentary languages, a well studied topic [32, 8, 20, 35, 27, 1] We point out that the rudimentary languages, and the techniques related to them, have been ....

....and still a constant number of alternations. That is, on the original inputs, the evaluation of can be done in LH as required. ut As mentioned in the introduction, Theorem 3 sets the link to an important problem related to the uniformity of circuits for integer division. Beame, Cook, and Hoover [7] showed that the problem of dividing two numbers can be computed by P uniform bounded fan in, logarithmic depth circuits (NC 1 ) The result was improved by Reif [28] see also [19] who showed that the problem could be computed by P uniform unbounded fan in, bounded depth circuits with majority ....

P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal of Computing, 15(4):994--1003, 1986.

....While a variety of problems, such as integer multiplication, can be solved in log time uniform TC 0 (the method of [CSV84] can be made log time uniform following [BIS88] there are several which are only known to be in P uniform TC 0 . These include integer division and related problems [BCH86], and various operations in large finite fields [Re87] There seems to be a sort of orthogonality in these classes, where new computing power can be added either by allowing more powerful gates or by weakening the uniformity condition. 5. New Atomic Predicates The atomic predicates in our ....

P. W. Beame, S. A. Cook, and H. J. Hoover, "Log-depth circuits for division and related problems", SIAM J. Comp. 15 (1986), 994-1003. 19

....and 2n Gamma 1 internal nodes. Thus, the entire process can be performed with depth O( log n) 2 ) and size O(n 3 ) There are alternative methods for performing iterated multiplication achieving various combinations of depth and size. In particular, it was proved by Beame, Cook and Hoover [4] that a product such as we have above can be computed by O(log n) depth boolean circuits of size O(n 5 (log n) 2 ) While O(n 5 log n) qubits may seem a high price to pay in order to save a factor of O(log n) in the circuit depth, the result has an interesting consequence regarding ....

P. Beame, S. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15(4):994--1003, 1986.

....integer by another, iterated multiplication of n n bit integers, and powering of an n bit integer by a log n bit exponent. How do these problems compare, for example, with multiplication or iterated addition of integers Can they be carried out in deterministic log space Beame, Cook, and Hoover [9] showed that these three problems could be solved by threshold circuits of constant depth and polynomial size. In logical terms, this means that each problem may be defined by a first order formula (with no iterated quantifier block) using majority quantifiers (see [7, 5] The main open problem ....

....that these three problems could be solved by threshold circuits of constant depth and polynomial size. In logical terms, this means that each problem may be defined by a first order formula (with no iterated quantifier block) using majority quantifiers (see [7, 5] The main open problem left by [9] was the uniformity of the threshold circuits. In logical terms, this refers to the numerical predicates that are needed to express the formula in addition to addition and multiplication of input position numbers. In the construction of [9] the necessary numerical predicate is computable in P, ....

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P. Beame, S. Cook and J. Hoover, Log depth circuits for division and related problems, SIAM Journal on Computing, 15(4):994--1003, 1986.

....Euler function, inversion can be considered as a special case of the more general question of modular exponentiation. Both these problems can also be considered over finite fields and other algebraic domains. For inversion, exponentiation and gcd, several parallel algorithms are in the literature [1, 2, 3, 9, 10, 11, 12, 13, 14, 15, 18, 20, 21, 23, 28, 30]. The question of obtaining a general parallel algorithm running in poly logarithmic time (log n) O(1) for n bit integers M is wide open [11, 12] Some lower bounds on the depth of arithmetic circuits are known [11, 15] On the other hand, some examples indicate that for this kind of problem the ....

....functions, and its generalization has not been worked out yet. Open Question 6.1. Extend Theorem 4.1 to arbitrary moduli M . Moduli of the form M = p m , where p is a small prime number, are of special interest because Hensel s lifting allows to design e#cient parallel algorithms for them [2, 11, 15]. Theorem 5.1 and its proof demonstrate how to deal with such moduli and what kind of result should be expected. Each Boolean function f(X 1 , X n ) can be uniquely represented as a multilinear polynomial of degree n over F 2 of the form f(X 1 , X n ) X 0#k#d X 1#i1 . ik ....

Paul W. Beame, Stephen A. Cook and H. James Hoover, `Log depth circuits for division and related problems', SIAM J. Comp., 15 (1986) 994--1003.

....threshold circuits are also appealing theoretically since they provide a surprisingly strong bounded depth computational model. Indeed, it has been shown that basic operations like addition, multiplication, division and sorting can be performed by bounded depth polynomial size threshold circuits [7, 17, 22, 5, 2, 6, 24, 27, 13, 3]. On the other hand, unbounded fan in bounded depth polynomial size circuits over the standard basis (even when supplemented with mod p gates for prime p) cannot compute majority [5, 21, 25] Therefore, separating the class of functions computable by bounded depth polynomial size threshold ....

Beame, P., Cook, S.A. and Hoover, H.J. (1986), Log Depth Circuits for Division and Related Problems, SIAM Journal on Computing, 15, pp. 994--1003.

....There are a number of problems that are known to have P uniform circuits but not DLOGTIME uniform circuits. For example, it is an open question whether integer division can be done by a DLOGTIME uniform family of NC 1 circuits, while it is known 67 that a P uniform such family exists (see [BCH86] This connection with uniform circuits was pointed out without proof by Gurevich, Immerman, and Shelah [GIS94] They isolated the class of finite structures of the form B n = f0; n Gamma 1g; Bn ; BIT Bn g) where Bn is the standard linear order, and BIT Bn is the binary ....

P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal of Computing, 15(4):994--1003, 1986.

....upper bound on the complexity of K, viz. K uniform NC 2 . The main technical work goes into showing that the b 1 de nable functions of C 0 2 [div] are closed under short iterated products. This is done by formalizing the reduction of iterated multiplication to division of Beame et al. [1], and involves formalizing some basic number theory in C 0 2 [div] for small numbers, e.g. the Chinese Remainder Theorem and the structure theorem for the multiplicative groups (Z=qZ) for prime powers q. We then study C 0 2 [div] with respect to interpolation. We say that a theory T of ....

....out in detail, it is straightforward though tedious. But the TC 0 circuits occurring in the proofs obtained by the simulation are all DLogTime uniform, whereas the proofs of the Die Hellman tautologies in [2] make essential use of the P uniform, but probably not DLogTime uniform circuits of [1]. Thus it is highly unlikely that C 0 2 itself could prove the arithmetic form of these tautologies, and thus could be shown not to enjoy feasible b 1 interpolation by this method. On the other hand, all the functions used in the proofs in [2] are in the class K, so our rst Main Theorem ....

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P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal of Computing, 15:994-1003, 1986.

..... However, the exact relations are not known. For example, the best known simulation of L uniform NC 1 by DLOGTIME uniform circuits requires already depth log 2 n (see Ruzzo [11] On the other hand there are problems computable in P uniform NC 1 , but not known to be computable in NC 1 [3]. The question whether or not, e.g. L uniform NC 1 = P uniform NC 1 does not seem to be equivalent to the question whether L = P. This is because polynomial time uniform circuits can not make use of all the computational power of the complexity class P. But what features of P computations can ....

....so called blind Turing machines introduced by Hartmanis and Mahaney [8] see also Jenner and Kirsig [9] 5.2 Trade offs between Depth and Uniformity Uniformity of circuits is a resource. e.g. there are problems known to be solvable in P uniform NC 1 but not known to be solvable in NC 1 [3]. The question arises what increase in the other resources of the circuits (size, depth) have to be paid in order to make circuits more uniform. Let DLOGTIME uniform Size Depth(n O(1) d(n) denote the class of languages that can be recognized by DLOGTIME uniform polynomial size circuits of ....

P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15:994--1003, 1986.

...., computing the product of two n bit numbers x and y . By 2 i j Gamma2 x i y j we mean 2 i j Gamma2 if both x i and y j are true, and 0 otherwise. Lastly, we will describe our TC 0 formula for computing the iterated product of m numbers. This formula is basically the original formula of [BCH], and articulated as a TC 0 formula in [M] The iterative product, PROD[z 1 ; z m ] gives the product of z 1 ; z m , where each z i is of length n , and we assume that m;n are both bounded by N . The basic idea is to compute the product modulo small primes using iterated addition, and ....

Beame, P., Cook, S., and Hoover, J. " Log depth circuits for division and related problems," SIAM J. Comput, vol. 15 (1986), pp. 996-1003.

....and (there exist integer matrices X;Y such that MX Gamma pY = I) This can be encoded as a many one reduction to IFSLE. This reduction works as long as p is at most polynomially large. Thus a P uniform NC 1 reduction can use Chinese Remaindering to compute the exact value of the determinant [BCH86]. This shows that #L is P uniform NC 1 reducible to IFSLE. In contrast, we do not know of any correspondingly efficient way to reduce computation of the determinant (or other #L hard problems) to the problem FSLE. 5 Open Questions The most obvious open question is: Is C=L closed under ....

P. Beame, S. Cook, and H. J. Hoover, Log depth circuits for division and related problems. SIAM Journal on Computing 15 (1996), 994--1003.

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P. W. Beame, H. J. Hoover, and S. A. Cook. Log depth circuits for division and related problems. SIAM Journal on Computing, 15(4):994--1003, November 1986.

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P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15(4):994--1003, November 1986.

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P. W. Beame, S. A. Cook, and H. J. Hoover, Log depth circuits for division and related problems, SIAM J. Comput., 15 (1986), pp. 994--1003.

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P. Beame, S. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15(4):994--1003, 1986.

No context found.

P. Beame, S. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15(4):994--1003, 1986.

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P. Beame, S. Cook, and H. Hoover. Log depth circuits for division and related problems. In Annual IEEE Symposium on Foundations of Computer Science, pages l--6, 1984.

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P. Beame, S. Cook and J. Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15:994--1003, 1986.

No context found.

P. Beame, S. Cook, and H. Hoover. Log depth circuits for division and related problems. SIAM Journal on Computing, 15:994--1003, 1986.

No context found.

P. Beame, S. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM J. Comput., 15:994--1003, 1986.

No context found.

P. W. Beame, S. A. Cook, and H. J. Hoover. Log depth circuits for division and related problems. SIAM Journal of Computing, 15(4):994--1003, 1986.

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Paul W. Beame, Stephen A. Cook, and H. James Hoover. Log depth circuits for division and related problems. SIAM J. Comput. , 15(4):994--1003, 1986.

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P.W. Beame, S.A. Cook, H.J. Hoover. Log depth circuits for division and related problems. S.I.A.M. Journal on Computing, 15(4), 1986, pp. 994-1003.

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Beame, P., Cook, S.A., and Hoover, J. (1986). Log depth circuits for division and related problems. SIAM Journal on Computing 15, 994--1003. 28

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P.W. Beam, S.A. Cook, and H.J. Hoover, Log Depth Circuits for Division and Related Problems, in Proc. of the 25th IEEE Symp. on the Foundations of Computer Science (1984), pp. 1-6.

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P. Beame, S. Cook, H. J. Hoover, Log depth circuits for division and related problems. SIAM Journal on Computing, 15:994--1003, 1986.

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