Depth Lower Bounds against Circuits with Sparse Orientation

• Autorzy: Koroth S. , Sarma J.

Identyfikatory

DOI

10.3233/FI-2017-1515

• Języki publikacji: EN

Abstrakty

EN

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×ω 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 (KarchmerWigderson (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.

Słowa kluczowe

EN

monotonic functions; sparse approximation; variables; function algebras; negation

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2017
• Tom: Vol. 152, nr 2
• Strony: 123--144
• Opis fizyczny: Bibliogr. 22 poz.

Twórcy

autor

Koroth S.

• Department of Computer Science & Engineering, Indian Institute of Technology, Madras, Chennai 36, India

autor

Sarma J.

• Department of Computer Science & Engineering, Indian Institute of Technology Madras, Chennai 36, India

Bibliografia

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Uwagi

Bibliografia, poz. 4 - błąd w nazwisku autora. Powinno być: J. Håstad