Results 1  10
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117
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 2350 (12 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...
Boundedwidth polynomialsize branching programs recognize exactly those languages
 in NC’, in “Proceedings, 18th ACM STOC
, 1986
"... We show that any language recognized by an NC ’ circuit (fanin 2, depth O(log n)) can be recognized by a width5 polynomialsize branching program. As any boundedwidth polynomialsize branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such prog ..."
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Cited by 209 (13 self)
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We show that any language recognized by an NC ’ circuit (fanin 2, depth O(log n)) can be recognized by a width5 polynomialsize branching program. As any boundedwidth polynomialsize branching program can be simulated by an NC ’ circuit, we have that the class of languages recognized by such programs is exactly nonuniform NC’. Further, following
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
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 defined by ..."
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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].
The Boolean formula value problem is in ALOGTIME
 in Proceedings of the 19th Annual ACM Symposium on Theory of Computing
, 1987
"... The Boolean formula value problem is in alternating log time and, more generally, parenthesis contextfree languages are in alternating log time. The evaluation of reverse Polish notation Boolean formulas is also in alternating log time. These results are optimal since the Boolean formula value ..."
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Cited by 66 (7 self)
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The Boolean formula value problem is in alternating log time and, more generally, parenthesis contextfree languages are in alternating log time. The evaluation of reverse Polish notation Boolean formulas is also in alternating log time. These results are optimal since the Boolean formula value problem is complete for alternating log time under deterministic log time reductions. Consequently, it is also complete for alternating log time under AC reductions.
A timespace tradeoff for sorting on a general sequential model of computation
 SIAM Journal on Computing
, 1982
"... Abstract. In a general sequential model of computation, no restrictions are placed on theway in which the computation may proceed, except that parallel operations are not allowed. We show that in such an unrestricted environment TIME.SPACE fl(N2/logN) in order to sort N integers, each in the range [ ..."
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Cited by 55 (6 self)
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Abstract. In a general sequential model of computation, no restrictions are placed on theway in which the computation may proceed, except that parallel operations are not allowed. We show that in such an unrestricted environment TIME.SPACE fl(N2/logN) in order to sort N integers, each in the range [,N]. Key words, timespace tradeoffs, conputational complexity, sorting, time lower bounds, space lower bounds
On the complexity of space bounded interactive proofs
 In 30th Annual Symposium on Foundations of Computer Science
, 1989
"... Some of the most exciting developments in complexity theory in recent years concern the complexity of interactive proof systems, defined by Goldwasser, Micali and Rackoff (1985) and independently by Babai (1985). In this paper, we survey results on the complexity of space bounded interactive proof s ..."
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Cited by 54 (5 self)
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Some of the most exciting developments in complexity theory in recent years concern the complexity of interactive proof systems, defined by Goldwasser, Micali and Rackoff (1985) and independently by Babai (1985). In this paper, we survey results on the complexity of space bounded interactive proof systems
Descriptive and Computational Complexity
 COMPUTATIONAL COMPLEXITY THEORY, PROC. SYMP. APPLIED MATH
, 1989
"... Computational complexity began with the natural physical notions of time and space. Given a property, S, an important issue is the computational complexity of checking whether or not an input satisfies S. For a long time, the notion of complexity referred to the time or space used in the computatio ..."
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Cited by 47 (0 self)
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Computational complexity began with the natural physical notions of time and space. Given a property, S, an important issue is the computational complexity of checking whether or not an input satisfies S. For a long time, the notion of complexity referred to the time or space used in the computation. A mathematician might ask, "What is the complexity of expressing the property S?" It should not be surprising that these two questions  that of checking and that of expressing  are related. However it is startling how closely tied they are when the second question refers to expressing the property in firstorder logic. Many complexity classes originally defined in terms of time or space resources have precise definitions as classes in firstorder logic. In 1974 Fagin gave a characterization of nondeterministic polynomial time (NP) as the set of properties expressible in secondorder existential logic
An Optimal Parallel Algorithm for Formula Evaluation
, 1992
"... A new approach to Buss’s NC¹ algorithm [Proc. 19thACM Symposium on Theory of Computing, Association for Computing Machinery, New York, 1987, pp. 123131] for evaluation of Boolean formulas is presented. This problem is shown to be complete for NC¹ over AC¬ reductions. This approach is then used to s ..."
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Cited by 43 (6 self)
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A new approach to Buss’s NC¹ algorithm [Proc. 19thACM Symposium on Theory of Computing, Association for Computing Machinery, New York, 1987, pp. 123131] for evaluation of Boolean formulas is presented. This problem is shown to be complete for NC¹ over AC¬ reductions. This approach is then used to solve the more general problem of evaluating arithmetic formulas by using arithmetic circuits.
The Complexity of Iterated Multiplication
 INFORMATION AND COMPUTATION
, 1995
"... For a monoid G, the iterated multiplication problem is the computation of the product of n elements from G. By refining known completeness arguments, we show that as G varies over a natural series of important groups and monoids, the iterated multiplication problems are complete for most natural, lo ..."
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Cited by 40 (4 self)
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For a monoid G, the iterated multiplication problem is the computation of the product of n elements from G. By refining known completeness arguments, we show that as G varies over a natural series of important groups and monoids, the iterated multiplication problems are complete for most natural, lowlevel complexity classes. The completeness is with respect to "firstorder projections"  lowlevel reductions that do not obscure the algebraic nature of these problems.