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19
Separating AC 0 from depth2 majority circuits
 In Proc. of the 39th Symposium on Theory of Computing (STOC
, 2007
"... Abstract. We construct a function in AC 0 that cannot be computed by a depth2 majority circuit of size less than exp(Θ(n 1/5)). This solves an open problem due to Krause and Pudlák (1994) and matches Allender’s classic result (1989) that AC 0 can be efficiently simulated by depth3 majority circuit ..."
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Abstract. We construct a function in AC 0 that cannot be computed by a depth2 majority circuit of size less than exp(Θ(n 1/5)). This solves an open problem due to Krause and Pudlák (1994) and matches Allender’s classic result (1989) that AC 0 can be efficiently simulated by depth3 majority circuits. To obtain our result, we develop a novel technique for proving lower bounds on communication complexity. This technique, the Degree/Discrepancy Theorem, is of independent interest. It translates lower bounds on the threshold degree of a Boolean function into upper bounds on the discrepancy of a related function. Upper bounds on the discrepancy, in turn, immediately imply lower bounds on communication and circuit size. In particular, our work yields the first known function in AC 0 with exponentially small discrepancy, exp(−Ω(n 1/5)). Key words. Majority circuits, constantdepth AND/OR/NOT circuits, communication complexity, discrepancy, threshold degree of Boolean functions. AMS subject classifications. 03D15, 68Q15, 68Q17
Communication lower bounds using dual polynomials
 Bulletin of the EATCS
"... Representations of Boolean functions by real polynomials play an important role in complexity theory. Typically, one is interested in the least degree of a polynomial p(x1,..., xn) that approximates or signrepresents a given Boolean function f (x1,..., xn). This article surveys a new and growing bo ..."
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Cited by 30 (9 self)
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Representations of Boolean functions by real polynomials play an important role in complexity theory. Typically, one is interested in the least degree of a polynomial p(x1,..., xn) that approximates or signrepresents a given Boolean function f (x1,..., xn). This article surveys a new and growing body of work in communication complexity that centers around the dual objects, i.e., polynomials that certify the difficulty of approximating or signrepresenting a given function. We provide a unified guide to the following results, complete with all the key proofs: • Sherstov’s Degree/Discrepancy Theorem, which translates lower bounds on the threshold degree of a Boolean function into upper bounds on the discrepancy of a related function; • Two different methods for proving lower bounds on boundederror communication based on the approximate degree: Sherstov’s pattern matrix method and Shi and Zhu’s block composition method; • Extension of the pattern matrix method to the multiparty model, obtained by Lee and Shraibman and by Chattopadhyay and Ada, and the resulting improved lower bounds for disjointness; • David and Pitassi’s separation of NP and BPP in multiparty communication complexity for k � (1 − ɛ) log n players.
THE PATTERN MATRIX METHOD
, 2009
"... We develop a novel and powerful technique for communication lower bounds, the pattern matrix method. Specifically, fix an arbitrary function f: {0, 1} n → {0, 1} and let Af be the matrix whose columns are each an application of f to some subset of the variables x1, x2,..., x4n. We prove that Af ha ..."
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Cited by 29 (8 self)
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We develop a novel and powerful technique for communication lower bounds, the pattern matrix method. Specifically, fix an arbitrary function f: {0, 1} n → {0, 1} and let Af be the matrix whose columns are each an application of f to some subset of the variables x1, x2,..., x4n. We prove that Af has boundederror communication complexity Ω(d), where d is the approximate degree of f. This result remains valid in the quantum model, regardless of prior entanglement. In particular, it gives a new and simple proof of Razborov’s breakthrough quantum lower bounds for disjointness and other symmetric predicates. We further characterize the discrepancy, approximate rank, and approximate trace norm of Af in terms of wellstudied analytic properties of f, broadly generalizing several recent results on smallbias communication and agnostic learning. The method of this paper has recently enabled important progress in multiparty communication complexity.
A Separation of NP and coNP in Multiparty Communication Complexity
 THEORY OF COMPUTING
, 2010
"... We prove that coNP � MA and in particular NP ̸ = coNP in the numberonforehead model of multiparty communication complexity for up to k = (1−ε)logn players, where ε> 0 is any constant. Specifically, we construct an explicit function F: ({0,1} n) k → {0,1} with conondeterministic complexity O(l ..."
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Cited by 13 (3 self)
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We prove that coNP � MA and in particular NP ̸ = coNP in the numberonforehead model of multiparty communication complexity for up to k = (1−ε)logn players, where ε> 0 is any constant. Specifically, we construct an explicit function F: ({0,1} n) k → {0,1} with conondeterministic complexity O(logn) and MerlinArthur complexity nΩ(1). The problem was open for k >= 3.
Multiparty Communication Complexity and Threshold Circuit Size of AC⁰
, 2008
"... We prove an nΩ(1) /2O(k) lower bound on the randomized kparty communication complexity of readonce depth 4 AC0 functions in the numberonforehead (NOF) model for up to Θ(log n) players. These are the first nontrivial lower bounds for general NOF multiparty communication complexity for any AC0 fu ..."
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Cited by 11 (2 self)
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We prove an nΩ(1) /2O(k) lower bound on the randomized kparty communication complexity of readonce depth 4 AC0 functions in the numberonforehead (NOF) model for up to Θ(log n) players. These are the first nontrivial lower bounds for general NOF multiparty communication complexity for any AC0 function for ω(log log n) players. For nonconstant k the bounds are larger than all previous lower bounds for any AC0 function even for simultaneous communication complexity. Our lower bounds imply the first superpolynomial lower bounds for the simulation of AC0 by general MAJ ◦ SYMM ◦ AND circuits, showing that the wellknown quasipolynomial simulations of AC0 by such circuits are qualitatively optimal, even for readonce formulas of small constant depth. We also exhibit a readonce depth 5 formula in NP cc k − BPPcc k
On the Virtue of Succinct Proofs: Amplifying Communication Complexity Hardness to TimeSpace Tradeoffs in Proof Complexity (Extended Abstract)
 STOC’12, MAY 19–22
, 2012
"... An active line of research in proof complexity over the last decade has been the study of proof space and tradeoffs between size and space. Such questions were originally motivated by practical SAT solving, but have also led to the development of new theoretical concepts in proof complexity of intr ..."
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Cited by 11 (5 self)
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An active line of research in proof complexity over the last decade has been the study of proof space and tradeoffs between size and space. Such questions were originally motivated by practical SAT solving, but have also led to the development of new theoretical concepts in proof complexity of intrinsic interest and to results establishing nontrivial relations between space and other proof complexity measures. By now, the resolution proof system is fairly well understood in this regard, as witnessed by a sequence of papers leading up to [BenSasson and Nordström 2008, 2011] and [Beame, Beck, and Impagliazzo 2012]. However, for other relevant proof systems in the context of SAT solving, such as polynomial calculus (PC) and cutting planes (CP), very little has been known. Inspired by [BN08, BN11], we consider CNF encodings of socalled pebble games played on graphs and the approach of making such pebbling formulas harder by simple syntactic modifications. We use this paradigm of hardness amplification to make progress on the relatively longstanding open question of proving timespace tradeoffs for PC and CP. Namely, we exhibit a family of modified pebbling formulas {Fn} ∞ n=1 such that: • The formulas Fn have size Θ(n) and width O(1). • They have proofs in length O(n) in resolution, which generalize to both PC and CP. • Any refutation in CP or PCR (a generalization of PC) in length L and space s must satisfy s log L � 4 √ n. A crucial technical ingredient in these results is a new twoplayer communication complexity lower bound for composed search problems in terms of block sensitivity, a contribution that we believe to be of independent interest.
Lower bounds on quantum multiparty communication complexity
"... Abstract—A major open question in communication complexity is if randomized and quantum communication are polynomially related for all total functions. So far, no gap larger than a power of two is known, despite significant efforts. We examine this question in the numberontheforehead model of mul ..."
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Abstract—A major open question in communication complexity is if randomized and quantum communication are polynomially related for all total functions. So far, no gap larger than a power of two is known, despite significant efforts. We examine this question in the numberontheforehead model of multiparty communication complexity. We show that essentially all lower bounds known on randomized complexity in this model also hold for quantum communication. This includes bounds of size Ω(n/2 k) for the kparty complexity of explicit functions, bounds for the generalized inner product function, and recent work on the multiparty complexity of disjointness. To the best of our knowledge, these are the first lower bounds of any kind on quantum communication in the general numberontheforehead model. We show this result in the following way. In the twoparty case, there is a lower bound on quantum communication complexity in terms of a norm γ2, which is known to subsume nearly all other techniques in the literature. For randomized complexity there is another natural bound in terms of a different norm µ which is also one of the strongest techniques available. A deep theorem in functional analysis, Grothendieck’s inequality, implies that γ2 and µ are equivalent up to a constant factor. This connection is one of the major obstacles to showing a larger gap between randomized and quantum communication complexity in the twoparty case. The lower bound technique in terms of the norm µ was recently extended to the multiparty numberontheforehead model. Here we show how the γ2 norm can be also extended to lower bound quantum multiparty complexity. Surprisingly, even in this general setting the two lower bounds, on quantum and classical communication, are still very closely related. This implies that separating quantum and classical communication in this setting will require the development of new techniques. The relation between these extensions of µ and γ2 is proved by a multidimensional version of Grothendieck’s inequality. KeywordsCommunication complexity, quantum computing, numberontheforehead model
Hardness Amplification in Proof Complexity
, 2009
"... We present a generic method for converting any family of unsatisfiable CNF formulas that require large resolution rank into CNF formulas whose refutation requires large rank for proof systems that manipulate polynomials or polynomial threshold functions of degree at most k. (The latter are known as ..."
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We present a generic method for converting any family of unsatisfiable CNF formulas that require large resolution rank into CNF formulas whose refutation requires large rank for proof systems that manipulate polynomials or polynomial threshold functions of degree at most k. (The latter are known as Th(k) proofs.) As special cases, such systems include: LovászSchrijver systems (LS, LS+), high degree analogues of LovászSchrijver (LS(k), LS+(k)), Cutting Planes and high degree versions of Cutting Planes (CP(k)), as well as SheraliAdams and Lasserre proofs. We introduce two very general families of proof systems, denoted by T cc (k) and R cc (k). The proof lines of T cc (k) are arbitrary Boolean functions, each of which can be evaluated by an efficient kparty randomized communication protocol. T cc (k) proofs are very powerful and include Th(k − 1) proofs as a special case. R cc (k) proofs generalize T cc (k) proofs and require only that each inference be checkable (in a certain weak sense) by an efficient kparty randomized communication protocol. Our main results are the following:
On the Communication Complexity of ReadOnce AC 0 Formulae
"... Abstract—We study the 2party randomized communication complexity of readonce AC 0 formulae. For balanced ANDOR trees T with n inputs and depth d, we show that the communication complexity of the function f T (x, y) = T (x◦y) is Ω(n/4 d) where (x◦y)i is defined so that the resulting tree also has ..."
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Abstract—We study the 2party randomized communication complexity of readonce AC 0 formulae. For balanced ANDOR trees T with n inputs and depth d, we show that the communication complexity of the function f T (x, y) = T (x◦y) is Ω(n/4 d) where (x◦y)i is defined so that the resulting tree also has alternating levels of AND and OR gates. For each bit of x, y, the operation ◦ is either AND or OR depending on the gate in T to which it is an input. Using this, we show that for general ANDOR trees T with n inputs and depth d, the communication complexity of f T (x, y) is n/2 Ω(d log d). These results generalize the classical results on the communication complexity of setdisjointness [1], [2] (where T is an ORgate) and recent results on the communication complexity of the TRIBES functions [3] (where T is a depth2 readonce formula). Our techniques build on and extend the information complexity methodology [4], [5], [3] for proving lower bounds on randomized communication complexity. Our analysis for trees of depth d proceeds in two steps: (1) reduction to measuring the information complexity of binary depthd trees, and (2) proving lower bounds on the information complexity of binary trees. In order to execute this program, we carefully construct input distributions under which both these steps can be carried out simultaneously. We believe the tools we develop will prove useful in further studies of information complexity in particular, and communication complexity in general. KeywordsCommunication complexity, Information complexity, ANDOR trees, Lower bounds I.