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Boolean Function Complexity: Advances and Frontiers
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TLDR
In this article, a comprehensive description of basic lower bound arguments, covering many of the gems of this complexity Waterloo that have been discovered over the past several decades, right up to results from the last year or two, is given.Abstract:
Boolean circuit complexity is the combinatorics of computer science and involves many intriguing problems that are easy to state and explain, even for the layman. This book is a comprehensive description of basic lower bound arguments, covering many of the gems of this complexity Waterloo that have been discovered over the past several decades, right up to results from the last year or two. Many open problems, marked as Research Problems, are mentioned along the way. The problems are mainly of combinatorial flavor but their solutions could have great consequences in circuit complexity and computer science. The book will be of interest to graduate students and researchers in the fields of computer science and discrete mathematics.read more
Citations
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Journal ArticleDOI
On the Resolution of the Sensitivity Conjecture
TL;DR: The ideas that inspired the proof of the Sensitivity Conjecture are explored by an exposition of four papers that had the most impact on the Conjectures.
Posted Content
An NP-Complete Problem in Grid Coloring
TL;DR: Three results are presented that support the conjecture that the problem of grid coloring is difficult in general and yield statements from Ramsey Theory which are of size polynomial in their parameters and require exponential size in various proof systems.
Journal ArticleDOI
On the CNF-complexity of bipartite graphs containing no squares
TL;DR: In this paper, a bipartite graph with average degree d which can be expressed as a conjunctive normal form using C log d clauses was found, which negatively resolves Research Problem 1.33 of Jukna.
Proceedings ArticleDOI
Shrinkage Under Random Projections, and Cubic Formula Lower Bounds for AC0 (Extended Abstract)
Yuval Filmus,Or Meir,Avishay Tal +2 more
TL;DR: This work extends the shrinkage result of H̊astad to hold under a far wider family of random restrictions and their generalization — random projections, and proves that the KRW conjecture holds for inner functions for which the unweighted quantum adversary bound is tight.
Posted Content
Shallow Quantum Circuits with Uninitialized Ancillary Qubits.
TL;DR: Near-logarithmic-depth quantum circuits with only O(\log n)$ initialized ancillary qubits such that they include unbounded fan-out gates on a small number of qubits and unbounded Toffoli gates are considered, but it is shown that they cannot compute the parity function on $n$ bits, even when they are augmented by $n^{O(1)}$ uninitialized qubits.
References
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Book
The Theory of Error-Correcting Codes
TL;DR: This book presents an introduction to BCH Codes and Finite Fields, and methods for Combining Codes, and discusses self-dual Codes and Invariant Theory, as well as nonlinear Codes, Hadamard Matrices, Designs and the Golay Code.
Journal ArticleDOI
A Computing Procedure for Quantification Theory
Martin Davis,Hilary Putnam +1 more
TL;DR: In the present paper, a uniform proof procedure for quantification theory is given which is feasible for use with some rather complicated formulas and which does not ordinarily lead to exponentiation.
Journal ArticleDOI
The ellipsoid method and its consequences in combinatorial optimization
TL;DR: The method yields polynomial algorithms for vertex packing in perfect graphs, for the matching and matroid intersection problems, for optimum covering of directed cuts of a digraph, and for the minimum value of a submodular set function.
Book
Communication Complexity
Eyal Kushilevitz,Noam Nisan +1 more
TL;DR: This chapter surveys the theory of two-party communication complexity and presents results regarding the following models of computation: • Finite automata • Turing machines • Decision trees • Ordered binary decision diagrams • VLSI chips • Networks of threshold gates.
Journal ArticleDOI
On the Shannon capacity of a graph
TL;DR: It is proved that the Shannon zero-error capacity of the pentagon is \sqrt{5} and a well-characterized, and in a sense easily computable, function is introduced which bounds the capacity from above and equals the capacity in a large number of cases.