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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 2331 (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...
Symmetric Logspace is Closed Under Complement
 CHICAGO JOURNAL OF THEORETICAL COMPUTER SCIENCE
, 1994
"... We present a Logspace, manyone reduction from the undirected stconnectivity problem to its complement. This shows that SL = co  SL. ..."
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Cited by 26 (1 self)
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We present a Logspace, manyone reduction from the undirected stconnectivity problem to its complement. This shows that SL = co  SL.
Some Bounds on Multiparty Communication Complexity of Pointer Jumping
, 1996
"... We introduce the model of conservative oneway multiparty complexity and prove lower and upper bounds on the complexity of pointer jumping. The pointer jumping function takes as its input a directed layered graph with a starting node and k layers of n nodes, and a single edge from each node to one ..."
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Cited by 13 (1 self)
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We introduce the model of conservative oneway multiparty complexity and prove lower and upper bounds on the complexity of pointer jumping. The pointer jumping function takes as its input a directed layered graph with a starting node and k layers of n nodes, and a single edge from each node to one node from the next layer. The output is the node reached by following k edges from the starting node. In a conservative protocol Player i can see only the node reached by following the first i \Gamma 1 edges and the edges on the jth layer for each j ? i (compared to the general model where he sees edges of all layers except for the ith one). In a oneway protocol, each player communicates only once: first Player 1 writes a message on the blackboard, then Player 2, etc., until the last player gives the answer. The cost is the total number of bits written on the blackboard. Our main results are the following bounds on kparty conservative oneway communication complexity of pointer jumping wit...
Approximation From Linear Spaces And Applications To Complexity
"... . We develop an analytic framework based on linear approximation and duality and point out how a number of apparently diverse complexity related questions  on circuit and communication complexity lower bounds, as well as pseudorandomness, learnability, and general combinatorics of Boolean func ..."
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Cited by 3 (2 self)
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. We develop an analytic framework based on linear approximation and duality and point out how a number of apparently diverse complexity related questions  on circuit and communication complexity lower bounds, as well as pseudorandomness, learnability, and general combinatorics of Boolean functions  fit neatly into this framework. This isolates the analytic content of these problems from their combinatorial content and clarifies the close relationship between the analytic structure of questions. (1) We give several results that convert a statement of nonapproximability from spaces of functions to statements of approximability. We point how that crucial portions of a significant number of the known complexityrelated results can be unified and given shorter and cleaner proofs using these general theorems. (2) We give several new complexityrelated applications, including circuit complexity lower bounds, and results concerning pseudorandomness, learning, and combinator...