Results 1  10
of
43
Monotone Complexity
, 1990
"... We give a general complexity classification scheme for monotone computation, including monotone spacebounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple ..."
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Cited by 2837 (11 self)
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We give a general complexity classification scheme for monotone computation, including monotone spacebounded and Turing machine models not previously considered. We propose monotone complexity classes including mAC i , mNC i , mLOGCFL, mBWBP , mL, mNL, mP , mBPP and mNP . We define a simple notion of monotone reducibility and exhibit complete problems. This provides a framework for stating existing results and asking new questions. We show that mNL (monotone nondeterministic logspace) is not closed under complementation, in contrast to Immerman's and Szelepcs 'enyi's nonmonotone result [Imm88, Sze87] that NL = coNL; this is a simple extension of the monotone circuit depth lower bound of Karchmer and Wigderson [KW90] for stconnectivity. We also consider mBWBP (monotone bounded width branching programs) and study the question of whether mBWBP is properly contained in mNC 1 , motivated by Barrington's result [Bar89] that BWBP = NC 1 . Although we cannot answer t...
On Uniformity within NC¹
 JOURNAL OF COMPUTER AND SYSTEM SCIENCES
, 1990
"... In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defin ..."
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Cited by 127 (19 self)
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In order to study circuit complexity classes within NC¹ in a uniform setting, we need a uniformity condition which is more restrictive than those in common use. Two such conditions, stricter than NC¹ uniformity [Ru81,Co85], have appeared in recent research: Immerman's families of circuits defined by firstorder formulas [Im87a,Im87b] and a uniformity corresponding to Buss' deterministic logtime reductions [Bu87]. We show that these two notions are equivalent, leading to a natural notion of uniformity for lowlevel circuit complexity classes. We show that recent results on the structure of NC¹ [Ba89] still hold true in this very uniform setting. Finally, we investigate a parallel notion of uniformity, still more restrictive, based on the regular languages. Here we give characterizations of subclasses of the regular languages based on their logical expressibility, extending recent work of Straubing, Th'erien, and Thomas [STT88]. A preliminary version of this work appeared as [BIS88].
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 61 (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 Permanent Requires Large Uniform Threshold Circuits
, 1999
"... We show that the permanent cannot be computed by uniform constantdepth threshold circuits of size T (n) for any function T such that for all k, T (k) (n) = o(2 n ). More generally, we show that any problem that is hard for the complexity class C=P requires circuits of this size (on the unif ..."
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Cited by 36 (9 self)
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We show that the permanent cannot be computed by uniform constantdepth threshold circuits of size T (n) for any function T such that for all k, T (k) (n) = o(2 n ). More generally, we show that any problem that is hard for the complexity class C=P requires circuits of this size (on the uniform constantdepth threshold circuit model). In particular, this lower bound applies to any problem that is hard for the complexity classes PP or #P.
Circuit Complexity before the Dawn of the New Millennium
, 1997
"... The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bound ..."
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Cited by 35 (4 self)
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The 1980's saw rapid and exciting development of techniques for proving lower bounds in circuit complexity. This pace has slowed recently, and there has even been work indicating that quite different proof techniques must be employed to advance beyond the current frontier of circuit lower bounds. Although this has engendered pessimism in some quarters, there have in fact been many positive developments in the past few years showing that significant progress is possible on many fronts. This paper is a (necessarily incomplete) survey of the state of circuit complexity as we await the dawn of the new millennium.
A Uniform Circuit Lower Bound for the Permanent
 SIAM Journal on Computing
, 1994
"... We show that uniform families of ACC circuits of subexponential size cannot compute the permanent function. This also implies similar lower bounds for certain sets in PP. This is one of the very few examples of a lower bound in circuit complexity whose proof hinges on the uniformity condition; it is ..."
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Cited by 32 (10 self)
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We show that uniform families of ACC circuits of subexponential size cannot compute the permanent function. This also implies similar lower bounds for certain sets in PP. This is one of the very few examples of a lower bound in circuit complexity whose proof hinges on the uniformity condition; it is still unknown if there is any set in Ntime #2 n O#1# # that does not have nonuniform ACC circuits.
Searching Constant Width Mazes Captures the AC° Hierarchy
 In Proceedings of the 15th Annual Symposium on Theoretical Aspects of Computer Science
, 1997
"... We show that searching a width /' maze is complete for II, i.e., for the /"th level of the AC hierarchy. Equivalently, stconnectivity for width /' grid graphs is complete for II. As an application, we show that there is a data structure solving dynamic stconnectivity for constan ..."
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Cited by 28 (6 self)
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We show that searching a width /' maze is complete for II, i.e., for the /"th level of the AC hierarchy. Equivalently, stconnectivity for width /' grid graphs is complete for II. As an application, we show that there is a data structure solving dynamic stconnectivity for constant width grid graphs with time bound O (log log n) per operation on a random access machine. The dynamic algorithm is derived from the parallel one in an indirect way using algebraic tools.
Logspace and Logtime Leaf Languages
, 1996
"... The computation tree of a nondeterministic machine M with input x gives rise to a leaf string formed by concatenating the outcomes of all the computations in the tree in lexicographical order. We may characterize problems by considering, for a particular "leaf language" Y , the set of all ..."
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Cited by 25 (2 self)
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The computation tree of a nondeterministic machine M with input x gives rise to a leaf string formed by concatenating the outcomes of all the computations in the tree in lexicographical order. We may characterize problems by considering, for a particular "leaf language" Y , the set of all x for which the leaf string of M is contained in Y . In this way, in the context of polynomial time computation, leaf languages were shown to capture many complexity classes. In this paper, we study the expressibility of the leaf language mechanism in the contexts of logarithmic space and of logarithmic time computation. We show that logspace leaf languages yield a much finer classification scheme for complexity classes than polynomial time leaf languages, capturing also many classes within P. In contrast, logtime leaf languages basically behave like logtime reducibilities. Both cases are more subtle to handle than the polynomial time case. We also raise the issue of balanced versus nonbalanced comp...
Superlinear lower bounds for boundedwidth branching programs
 J. Comp. and System Sci
, 1995
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On Monotone Planar Circuits
, 1999
"... In this paper we show several results about monotone planar circuits. We show that monotone planar circuits of bounded width, with access to negated input variables, compute exactly the functions in nonuniform AC 0 . This provides a striking contrast to the nonplanar case, where exactly NC 1 i ..."
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Cited by 19 (2 self)
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In this paper we show several results about monotone planar circuits. We show that monotone planar circuits of bounded width, with access to negated input variables, compute exactly the functions in nonuniform AC 0 . This provides a striking contrast to the nonplanar case, where exactly NC 1 is computed. We show that the circuit value problem for monotone planar circuits, with inputs on the outer face only, can be solved in LOGDCFL ` SC, improving a LOGCFL upper bound due to Dymond and Cook. We show that for monotone planar circuits, with inputs on the outer face only, excessive depth compared to width is useless; any function computed by a monotone planar circuit of width w with inputs on the outer face can be computed by a monotone planar circuit of width O(w) and depth w O(1) . Finally, we show that monotone planar readonce circuits, with inputs on the outer face only, can be efficiently learned using membership queries. 1 Introduction In this paper, we prove a number of ...