Circuits, communication and polynomials

• Authors: Arkadev Chattopadhyay
• Venue: Circuits, communication and polynomials
• Year: 2009

In this thesis, we prove unconditional lower bounds on resources needed to compute explicit functions in the following three models of computation: constant-depth boolean circuits, multivariate polynomials over commutative rings and the 'Number on the Forehead' model of multiparty communication. Apart from using tools from diverse areas, we exploit the rich interplay between these models to make progress on questions arising in the study of each of them. Boolean circuits are natural computing devices and are ubiquitous in the modern electronic age. We study the limitation of this model when the depth of circuits is fixed, independent of the length of the input. The power of such constant-depth circuits using gates computing modular counting functions remains undetermined, despite intensive efforts for nearly twenty years. We make progress on two fronts: let m be a number having r distinct prime factors none of which divides ℓ. We first show that constant depth circuits employing AND/OR/MODm gates cannot compute efficiently the MAJORITY and MODℓ function on n bits if 'few' MODm gates are allowed, i.e. they need size nW&parl0;1s&parl0;log n&parr0;1/&parl0;r-1&parr0;&parr0; if s MODm gates are allowed in the circuit. Second, we analyze circuits that comprise only MOD m gates, We show that in sub-linear size (and arbitrary depth), they cannot compute AND of n bits. Further, we establish that in that size they can only very poorly approximate MODℓ. Our first result on circuits is derived by introducing a novel notion of computation of boolean functions by polynomials. The study of degree as a resource in polynomial representation of boolean functions is of much independent interest. Our notion, called the weak generalized representation, generalizes all previously studied notions of computation by polynomials over finite commutative rings. We prove that over the ring Zm , polynomials need Wlogn 1/r-1 degree to represent, in our sense, simple functions like MAJORITY and MODℓ. Using ideas from arguments in communication complexity, we simplify and strengthen the breakthrough work of Bourgain showing that functions computed by o(log n)-degree polynomials over Zm do not even correlate well with MODℓ. Finally, we study the 'Number on the Forehead' model of multiparty communication that was introduced by Chandra, Furst and Lipton [CFL83]. We obtain fresh insight into this model by studying the class CCk of languages that have constant k-party deterministic communication complexity under every possible partition of input bits among parties. This study is motivated by Szegedy's [Sze93] surprising result that languages in CC2 can all be extremely efficiently recognized by very shallow boolean circuits. In contrast, we show that even CC 3 contains languages of arbitrarily large circuit complexity. On the other hand, we show that the advantage of multiple players over two players is significantly curtailed for computing two simple classes of languages: languages that have a neutral letter and those that are symmetric. Extending the recent breakthrough works of Sherstov [She07, She08b] for two-party communication, we prove strong lower bounds on multiparty communication complexity of functions. First, we obtaina bound of n Ω(1) on the k-party randomized communication complexity of a function that is computable by constant-depth circuits using AND/OR gates, when k is a constant. The bound holds as long as protocols are required to have better than inverse exponential (i.e. 2-no1 ) advantage over random guessing. This is strong enough to yield lower bounds on the size of an important class of depth-three circuits: circuits having a MAJORITY gate at its output, a middle layer of gates computing arbitrary symmetric functions and a base layer of arbitrary gates of restricted fan-in. Second, we obtain nΩ(1) lower bounds on the k-party randomized (bounded error) communication complexity of the Disjointness function. This resolves a major open question in multiparty communication complexity with applications to proof complexity. Our techniques in obtaining the last two bounds, exploit connections between representation by polynomials over teals of a boolean function and communication complexity of a closely related function.