Boolean Function Complexity

Jukna, Stasys

doi:10.1007/978-3-642-24508-4

Cited by 224 publications

(59 citation statements)

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“…However, we recall that achieving a exp(ω(n 1/(d−1) )) lower bound against depth-d circuits for an explicit function, even for worst-case computation, is a well-known and major open problem in complexity theory (see e.g. Chapter §11 of [Juk12] and [Val83,GW13,Vio13]). In particular, an extension of the 2 Ω(n/polylog(n)) -type lower bound of [OW07] to depth 3, even for worst-case computation, would constitute a significant breakthrough.…”

Section: Previous Workmentioning

confidence: 99%

An Average-Case Depth Hierarchy Theorem for Boolean Circuits

Håstad

Rossman

Servedio

et al. 2017

J. ACM

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We prove an average-case depth hierarchy theorem for Boolean circuits over the standard basis of AND, OR, and NOT gates. Our hierarchy theorem says that for every d ≥ 2, there is an explicit n-variable Boolean function f , computed by a linear-size depth-d formula, which is such that any depth-(d − 1) circuit that agrees with f on (1/2 + o n (1)) fraction of all inputs must have size exp (n Ω(1/d) A key ingredient in our proof is a notion of random projections which generalize random restrictions.

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Section: Previous Workmentioning

confidence: 99%

An Average-Case Depth Hierarchy Theorem for Boolean Circuits

Håstad

Rossman

Servedio

et al. 2017

J. ACM

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“…To represent a Boolean function Binary Decision Diagram (BDD) can be used [6]. It is a rooted Directed Acyclic Graph (DAG), where every non-leaf node have exactly two child nodes.…”

Section: The Quidd Data Structure 21 Binary Decision Diagrammentioning

confidence: 99%

An Idea to Improve QuIDD Based Quantum Simulations

Friedl

Kabódi

2015

Period. Polytech. Elec. Eng. Comp. Sci.

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“…To complete the discussion, we refer the reader to Razborov [1991] and Part 5 in Jukna [2012], for a detailed survey on BP lower bounds. Specifically, we note that good lower bounds are known against read-once BP models (see Wegener [1987], Zák [1984], and Jukna and Zák [2001]), although not for the problem that is of interest in this article.…”

Section: Introductionmentioning

confidence: 99%

Pebbling, Entropy, and Branching Program Size Lower Bounds

Komarath

Sarma

2015

ACM Trans. Comput. Theory

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We contribute to the program of proving lower bounds on the size of branching programs solving the Tree Evaluation Problem introduced by Cook et al. [2012]. Proving a superpolynomial lower bound for the size of nondeterministic thrifty branching programs would be an important step toward separating NL from P using the tree evaluation problem. First, we show that Read-Once Nondeterministic Thrifty BPs are equivalent to whole black-white pebbling algorithms, thus showing a tight lower bound (ignoring polynomial factors) for this model. We then introduce a weaker restriction of nondeterministic thrifty branching programs called Bitwise Independence. The best known [Cook et al. 2012] nondeterministic thrifty branching programs (of size O(k h/2+1 )) for the tree evaluation problem are Bitwise Independent. As our main result, we show that any Bitwise Independent Nondeterministic Thrifty Branching Program solving BT 2 (h, k) must have at least k 2 h/2 states. Prior to this work, lower bounds were known for nondeterministic thrifty branching programs only for fixed heights h = 2, 3, 4 [Cook et al. 2012]. We prove our results by associating a fractional blackwhite pebbling strategy with any bitwise independent nondeterministic thrifty branching program solving the Tree Evaluation Problem. Such a connection was not known previously, even for fixed heights.Our main technique is the entropy method introduced by Jukna and Zák [2001] originally in the context of proving lower bounds for read-once branching programs. We also show that the previous lower bounds known [Cook et al. 2012] for deterministic branching programs for the Tree Evaluation Problem can be obtained using this approach. Using this method, we also show tight lower bounds for any k-way deterministic branching program solving the Tree Evaluation Problem when the instances are restricted to have the same group operation in all internal nodes.

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Boolean Function Complexity

Abstract: The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Cited by 224 publications

References 0 publications

An Average-Case Depth Hierarchy Theorem for Boolean Circuits

An Average-Case Depth Hierarchy Theorem for Boolean Circuits

An Idea to Improve QuIDD Based Quantum Simulations

Pebbling, Entropy, and Branching Program Size Lower Bounds

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