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Depth Lower Bounds against Circuits with Sparse Orientation
Published in IOS Press
2017
Volume: 152
   
Issue: 2
Pages: 123 - 144
Abstract
We study depth lower bounds against non-monotone circuits, parametrized by a new measure of non-monotonicity: the orientation of a function f is the characteristic vector of the minimum sized set of negated variables needed in any DeMorgan circuit (circuits where negations appear only at the leaves) computing f. We prove trade-off results between the depth and the weight/structure of the orientation vectors in any circuit C computing the CLIQUE function on an n vertex graph. We prove that if C is of depth d and each gate computes a Boolean function with orientation of weight at most w (in terms of the inputs to C), then d × w must be Ω(n). In particular, if the weights are o(n log k n), then C must be of depth ω(log k n). We prove a barrier for our general technique. However, using specific properties of the CLIQUE function (used in Amano Maruoka (2005)) and the Karchmer-Wigderson framework (Karchmer Wigderson (1988)), we go beyond the limitations and obtain lower bounds when the weight restrictions are less stringent. We then study the depth lower bounds when the structure of the orientation vector is restricted. Asymptotic improvements to our results (in the restricted setting) separates NP from NC. As our main tool, we generalize Karchmer-Wigderson games (Karchmer Wigderson (1988)) for monotone functions to work for non-monotone circuits parametrized by the weight/structure of the orientation. We also prove structural results about orientation and prove connections between number of negations and weight of orientations required to compute a function. © IOS Press and the authors. All rights reserved.
About the journal
JournalFundamenta Informaticae
PublisherIOS Press
ISSN01692968
Open AccessYes
Concepts (12)
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    Boolean functions
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    Economic and social effects
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    Timing circuits
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    CHARACTERISTIC VECTORS
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    CLIQUE FUNCTION
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    MONOTONE CIRCUIT
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    MONOTONE FUNCTIONS
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    NON-MONOTONICITY
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    ORIENTATION VECTOR
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    SPECIFIC PROPERTIES
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    WEIGHT RESTRICTION
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    Electric network analysis