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The Polynomial Method in Circuit Complexity
 In Proceedings of the 8th IEEE Structure in Complexity Theory Conference
, 1993
"... The representation of functions as lowdegree polynomials over various rings has provided many insights in the theory of smalldepth circuits. We survey some of the closure properties, upper bounds, and lower bounds obtained via this approach. 1. Introduction There is a long history of using polyno ..."
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Cited by 70 (4 self)
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The representation of functions as lowdegree polynomials over various rings has provided many insights in the theory of smalldepth circuits. We survey some of the closure properties, upper bounds, and lower bounds obtained via this approach. 1. Introduction There is a long history of using polynomials in order to prove complexity bounds. Minsky and Papert [39] used polynomials to prove early lower bounds on the order of perceptrons. Razborov [46] and Smolensky [49] used them to prove lower bounds on the size of ANDOR circuits. Other lower bounds via polynomials are due to [50, 4, 10, 51, 9, 55]. Paturi and Saks [44] discovered that rational functions could be used for lower bounds on the size of threshold circuits. Toda [53] used polynomials to prove upper bounds on the power of the polynomial hierarchy. This led to a series of upper bounds on the power of the polynomial hierarchy [54, 52], AC 0 [2, 3, 52, 19], and ACC [58, 20, 30, 37], and related classes [21, 42]. Beigel and Gi...
Representing Boolean Functions As Polynomials Modulo Composite Numbers
 Computational Complexity
, 1994
"... . Define the MODm degree of a boolean function F to be the smallest degree of any polynomial P , over the ring of integers modulo m, such that for all 01 assignments ~x, F (~x) = 0 iff P (~x) = 0. We obtain the unexpected result that the MODm degree of the OR of N variables is O( r p N ), wher ..."
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Cited by 56 (6 self)
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. Define the MODm degree of a boolean function F to be the smallest degree of any polynomial P , over the ring of integers modulo m, such that for all 01 assignments ~x, F (~x) = 0 iff P (~x) = 0. We obtain the unexpected result that the MODm degree of the OR of N variables is O( r p N ), where r is the number of distinct prime factors of m. This is optimal in the case of representation by symmetric polynomials. The MOD n function is 0 if the number of input ones is a multiple of n and is one otherwise. We show that the MODm degree of both the MOD n and :MOD n functions is N\Omega\Gamma1/ exactly when there is a prime dividing n but not m. The MODm degree of the MODm function is 1; we show that the MODm degree of :MODm is N\Omega\Gamma30 if m is not a power of a prime, O(1) otherwise. A corollary is that there exists an oracle relative to which the MODmP classes (such as \PhiP) have this structure: MODmP is closed under complementation and union iff m is a prime power, and...
The Complexity of Computation on the Parallel Random Access Machine
, 1993
"... PRAMs also approximate the situation where communication to and from shared memory is much more expensive than local operations, for example, where each processor is located on a separate chip and access to shared memory is through a combining network. Not surprisingly, abstract PRAMs can be much m ..."
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Cited by 32 (4 self)
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PRAMs also approximate the situation where communication to and from shared memory is much more expensive than local operations, for example, where each processor is located on a separate chip and access to shared memory is through a combining network. Not surprisingly, abstract PRAMs can be much more powerful than restricted instruction set PRAMs. THEOREM 21.16 Any function of n variables can be computed by an abstract EROW PRAM in O(log n) steps using n= log 2 n processors and n=2 log 2 n shared memory cells. PROOF Each processor begins by reading log 2 n input values and combining them into one large value. The information known by processors are combined in a binarytreelike fashion. In each round, the remaining processors are grouped into pairs. In each pair, one processor communicates the information it knows about the input to the other processor and then leaves the computation. After dlog 2 ne rounds, one processor knows all n input values. Then this processor computes th...
Feasible TimeOptimal Algorithms for Boolean Functions on ExclusiveWrite PRAMs
, 1994
"... It was shown some years ago that the computation time for many important Boolean functions of n arguments on concurrentread exclusivewrite parallel randomaccess machines (CREW PRAMs) of unlimited size is at least '(n) 0:72 log 2 n. On the other hand, it is known that every Boolean function of n ..."
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Cited by 13 (3 self)
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It was shown some years ago that the computation time for many important Boolean functions of n arguments on concurrentread exclusivewrite parallel randomaccess machines (CREW PRAMs) of unlimited size is at least '(n) 0:72 log 2 n. On the other hand, it is known that every Boolean function of n arguments can be computed in '(n) + 1 steps on a CREW PRAM with n \Delta 2 n\Gamma1 processors and memory cells. In the case of the OR of n bits, n processors and cells are sufficient. In this paper it is shown that for many important functions there are CREW PRAM algorithms that almost meet the lower bound in that they take '(n) + o(log n) steps, but use only a small number of processors and memory cells (in most cases, n). In addition, the cells only have to store binary words of bounded length (in most cases, length 1). We call such algorithms "feasible". The functions concerned include: the PARITY function and, more generally, all symmetric functions; a large class of Boolean formulas...
Lower Bounds on Representing Boolean Functions as Polynomials in Z_m
 SIAM Journal on Discrete Mathematics
, 1996
"... . Define the MODmdegree of a boolean function F to be the smallest degree of any polynomial P , over the ring of integers modulo m, such that for all 01 assignments ~x , F (~x) = 0 iff P (~x) = 0. By exploring the periodic property of the binomial coefficients modulo m, two new lower bounds on th ..."
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Cited by 9 (2 self)
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. Define the MODmdegree of a boolean function F to be the smallest degree of any polynomial P , over the ring of integers modulo m, such that for all 01 assignments ~x , F (~x) = 0 iff P (~x) = 0. By exploring the periodic property of the binomial coefficients modulo m, two new lower bounds on the MODm degree of the MOD l and :MODm functions are proved, where m is any composite integer and l has a prime factor not dividing m. Both bounds improve from sublinear to \Omega\Gamma n). With the periodic property, a simple proof of a lower bound on the MODm degree with symmetric multilinear polynomial of the OR function is given. It is also proved that the majority function has a lower bound n 2 and the MidBit function has a lower bound p n. Key words. boolean function complexity, circuit complexity, computational complexity AMS subject classifications. 68Q05, 68Q15, 68Q25, 94C10 1. Introduction. Proving lower bounds on explicitly given boolean functions is a notoriously difficul...
Multiple Threshold Neural Logic
 In Advances in Neural Information Processing, Volume 10: NIPS’1997
, 1996
"... We introduce a new Boolean computing element related to the Boolean version of a neural element. Instead of the sign function in the Boolean neural element (also known as an LT element), it computes an arbitrary (with polynomialy many transitions) Boolean function of the weighted sum of its inputs. ..."
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Cited by 8 (1 self)
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We introduce a new Boolean computing element related to the Boolean version of a neural element. Instead of the sign function in the Boolean neural element (also known as an LT element), it computes an arbitrary (with polynomialy many transitions) Boolean function of the weighted sum of its inputs. We call the new computing element an LTM element, which stands for Linear Threshold with Multiple transitions. The paper consists of the following main contributions related to our study of LTM circuits: (i) the characterization of the computing power of LTM relative to LT circuits, (ii) a proof that the area of the VLSI layout is reduced from O(n 2 ) in LT circuits to O(n) in LTM circuits, for n inputs symmetric Boolean functions, and (iii) the creation of efficient designs of LTM circuits for the addition of a multiple number of integers and the product of two integers. In particular, we show how to compute the addition of m integers with a single layer of LTM elements. Category : The...
Lower Bounds for Randomized Exclusive Write PRAMs
 in Proc. 7th ACM Symp. on Parallel Algorithms and Architectures, (ACM
, 1995
"... In this paper we study the question: How useful is randomization in speeding up Exclusive Write PRAM computations? Our results give further evidence that randomization is of limited use in these types of computations. First we examine a compaction problem on both the CREW and EREW PRAM models, and w ..."
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Cited by 7 (0 self)
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In this paper we study the question: How useful is randomization in speeding up Exclusive Write PRAM computations? Our results give further evidence that randomization is of limited use in these types of computations. First we examine a compaction problem on both the CREW and EREW PRAM models, and we present randomized lower bounds which match the best deterministic lower bounds known. (For the CREW PRAM model, the lower bound is asymptotically optimal.) These are the first nontrivial randomized lower bounds known for the compaction problem on these models. We show that our lower bounds also apply to the problem of approximate compaction. Next we examine the problem of computing boolean functions on the CREW PRAM model, and we present a randomized lower bound which improves on the previous best randomized lower bound for many boolean functions, including the OR function. (The previous lower bounds for these functions were asymptotically optimal, but we improve the constant multiplicat...
An Improved Lower Bound for the QRQW PRAM
 In Proc. 7th IEEE Symp. on Para. and Distr. Proc
, 1996
"... The queueread, queuewrite (QRQW) parallel random access machine (PRAM) model is a shared memory model which allows concurrent reading and writing with a time cost proportional to the contention. This is designed to model currently available parallel machines more accurately than either the CRCW PR ..."
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Cited by 3 (2 self)
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The queueread, queuewrite (QRQW) parallel random access machine (PRAM) model is a shared memory model which allows concurrent reading and writing with a time cost proportional to the contention. This is designed to model currently available parallel machines more accurately than either the CRCW PRAM or EREW PRAM models. Here we present a lower bound for the problem of Linear Approximate Compaction (LAC) on the QRQW PRAM. The input to LAC consists of at most m marked items in an array of size n, and the output consists of the marked items in an array of size O(m). There is an O( p log n) expected time randomized algorithm for LAC on the QRQW PRAM. We prove a lower bound of \Omega\Gamma/66 log n) expected time for any randomized algorithm for LAC, an improvement over the previous best bound of \Omega\Gamma/11 log log n). Our bound applies regardless of the number of processors and memory cells of the QRQW PRAM. 1 Introduction The PRAM model of computation has been the most widely us...
Complexity of Boolean Functions on PRAMs  Lower Bound Techniques
"... Determining time necessary for computing important functions on parallel machines is one of the most important problems in complexity theory for parallel algorithms. Recently, a substantial progress has been made in this area. In this survey paper, we discuss the results that have been obtained for ..."
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Determining time necessary for computing important functions on parallel machines is one of the most important problems in complexity theory for parallel algorithms. Recently, a substantial progress has been made in this area. In this survey paper, we discuss the results that have been obtained for three types of parallel random access machines (PRAMs): CREW, ROBUST and EREW. 1 Introduction Parallel random access machine (PRAM) is a most abstract model of parallel computers, where interprocessor communication is realized using a shared memory. Each processor of a PRAM can access any cell of a shared memory in one computation step. This is certainly an unrealistic assumption, but it makes the analysis of the parallel algorithms much easier, and we can can concentrate ourselves on inherent complexity of a given problem. This is a reason, why most parallel algorithms have been described in terms of PRAMs. Each PRAM consists of a collection of processors and common memory cells. Each comp...
A Note On The Polynomial Representation Of Boolean Functions Over GF(2)
, 1998
"... We investigate the representation of Boolean functions as polynomials over the eld GF (2), and prove an interesting characterization theorem: the degree of a Boolean function over GF (2) is equal to the size of its largest subfunction that takes the value 1 on an odd number of input strings. We t ..."
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We investigate the representation of Boolean functions as polynomials over the eld GF (2), and prove an interesting characterization theorem: the degree of a Boolean function over GF (2) is equal to the size of its largest subfunction that takes the value 1 on an odd number of input strings. We then present some properties of odd functions, i.e., functions that take the value 1 on an odd number of strings, and analyze the connections between the problem of the existence of odd functions with very low maximal sensitivity and the long standing open problem of the relationship between the maximal sensitivity and the block sensitivity of Boolean functions. 1. Introduction Over the last decade, several authors have employed polynomial methods in circuit complexity [2]. A key step in this approach is showing that Boolean functions computable by certain kinds of circuits can be represented by polynomials of a given type, e.g., lowdegree polynomials. Every Boolean function can be r...