CLARKSON UNIVERSITY Multiplicity Automata, Polynomials and the Complexity of Small-Depth Boolean Circuits A Thesis by Erion Plaku Department of Mathematics and Computer Science Submitted in partial fulfillment of the requirements for the degree of Master of Science, Computer Science April 23, 2002 Accepted by the Graduate School Date Dean CLARKSON UNIVERSITY DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE The undersigned have examined the thesis entitled “Multiplicity Automata, Polynomials and the Complexity of Small-Depth Boolean Circuits” presented by Erion Plaku a candidate for the degree of Master of Science, and hereby certify that it is worthy of acceptance. Date: April 23, 2002 Advisor: Alexis Maciel Examining Committee: Christino Tamon Chris Lynch ii Abstract We are interested in the computational power of Boolean circuits. Our objective is to provide lower bounds for different circuit classes while emphasizing connections to learning and pseudorandomness. The main contribution of this thesis is the development of a simple method based on multiplicity automata which we use to prove lower bounds on the size of some classes of Ω(n) 0 0 k {0} circuits. We show 2 lower bounds for AC1 ◦ CC [p ] ◦ SYM and AND ◦ MODm ◦ SYM Ω(n/ log n) {0} circuits and 2 lower bounds for AND ◦{MODm , EXACT}◦ SYM circuits. We also Ω(n) 0 k 0 k show 2 lower bounds for OR ◦ CCd[p ] ◦ SYMǫn and AND ◦ CCd[p ] ◦ SYMǫn circuits −4kd computing the ANDn and ORn functions, respectively, for any ǫ ≤ p . All our lower 0 k 0 ki bounds remain the same even if CC [p ] circuits are replaced by {CC [pi ]: pi,ki ∈ O(1)} circuits. We show how to compute a pseudorandom function generator secure against polynomial- time adversaries by q(SUM ◦ EXACT)+ and q(OR ◦ EXACT)+ circuits. Consequently, these circuit classes are not weakly learnable in polynomial time even when membership queries are allowed Other contributions include the use of polynomials to obtain lower bounds on circuits computing the square-free function. We show that the square-free function cannot be com- {0} puted by depth 2 circuits with AND⌊0.14 ln n⌋ gates at the input and a modular MODpk or a weighted threshold gate at the output. We remark that p, k and the weights are not bounded above by any function and can in fact be exponential. These results provide non- trivial lower bounds on the complexity of a number theoretic problem which is closely related to the integer factorization problem. iii Acknowledgements I would like to thank Alexis Maciel, my advisor, for introducing me to the area of Boolean circuits. Without his encouragements and constant support this thesis would have not been possible. I enjoyed working with him. I am thankful to Christino Tamon for the many helpful discussions on multiplicity au- tomata and learning and his prompt answers to a myriad of questions. I would also like to thank Chris Lynch for a careful review of this thesis and his valuable comments. I would like to acknowledge the financial support of NSF through grant CCR-9877150. I thank everyone who has helped me in the research process and especially my friends at the churches in Berat and Koinonia for their prayers and encouragement. Potsdam, New York Erion Plaku April 6, 2002 iv Table of Contents 1 Introduction 1 1.1 ContentandResultsoftheThesis. 3 1.2 OutlineoftheThesis ............................... 5 1.3 Notation and Basic Definitions . 5 2 Boolean Circuits 9 2.1 ModelofComputation .............................. 9 2.2 ModularGates .................................. 13 2.2.1 Properties of Modular Gates with Prime Power Moduli . 13 2.2.2 Generalvs. RegularModularGates. 15 2.3 ThresholdGates ................................. 16 2.4 ExactGates.................................... 18 2.5 SymmetricGates ................................. 19 3 Polynomial Method in Circuit Complexity 20 3.1 Representing Polynomials . 21 3.1.1 Polynomials over Zm ........................... 21 3.1.2 ThresholdRepresentation . 22 3.2 RandomRestrictions ............................... 23 3.3 TheWeakDegreeandtheThreshold . 24 3.3.1 TheThresholdofaPolynomial . 25 {r} 3.3.2 The MODq Function .......................... 26 3.3.3 The Quadratic Residuacity Function . 27 3.4 Complexity of the Square-Free Function . .. 29 3.4.1 PreviousWork .............................. 29 3.4.2 NumberofPrimesandBinomialBound. 30 3.4.3 An Important Property of the Square-Free Function . .... 30 3.4.4 StrongRepresentation . .. .. 32 3.4.5 One-SidedRepresentation . 34 v vi 3.4.6 ThresholdRepresentation . 35 3.4.7 SomeRemarksandOtherProperties . 36 4 Multiplicity Automata and Boolean Circuits 38 4.1 Multiplicity Automata . 39 4.1.1 ModelofComputation .......................... 40 4.1.2 HankelMatrix............................... 42 4.1.3 Functions f : Σn →K .......................... 44 4.1.4 Some Simple Functions with Exponential Rank . 44 4.1.5 ClosureProperties ............................ 47 4.2 Simulating Circuits by Multiplicity Automata . 49 4.2.1 Simulating Symmetric Gates . 49 A 4.2.2 Simulating MODpk Gates......................... 49 4.2.3 Simulating CC0[pk] ◦ SYMCircuits ................... 50 4.3 LowerBounds................................... 51 4.3.1 CC0[pk] ◦ SYMCircuits.......................... 52 0 0 k 4.3.2 AC1 ◦ CC [p ] ◦ SYMCircuits ...................... 52 {0} 4.3.3 AND ◦{MODm , EXACT}◦ SYMCircuits............... 53 4.3.4 LowerBoundsforAND,ORFunctions . 54 5 Computing Pseudorandom Function Generators 57 5.1 APRFGbasedontheGDH-Assumption . 57 5.2 Computing the PRFG in q(SUM ◦ EXACT)+ and q(OR ◦ EXACT)+ .... 62 5.3 HardnessofLearning............................... 65 5.4 RemarksonNaturalProofs . .. .. 67 6 Conclusions and Further Research 70 Bibliography 71 Chapter 1 Introduction The goal of complexity theory is to characterize the amount of resources needed for the computation of specific functions. Common resources include sequential time, sequential space, number of gates in Boolean circuits, parallel time in a multiprocessor machine, etc. The exact complexity of a function is determined by the amount of resources that is both sufficient and necessary for its computation. Sufficiency implies an upper bound on the amount of resources needed to compute the function for every instance of the input. Necessity implies a lower bound, that is, for some instance of the input, at least a certain amount of resources is required to compute the function. The amount of resources that is needed to compute a function allows for an elegant classification of functions according to their computational complexity. Researchers have developed the notion of complexity classes, where a complexity class is defined by specifying (a) the type of computation model M (b) the resource R which we wish to measure in this model, and (c) an upper bound U on this resource. A complexity class, then, consists of all functions requiring at most an amount U of resource R for their computation in the model M. Thus, to find the complexity of a function we determine to which complexity classes it belongs (by providing upper bounds on the resource) and to which complexity classes it does not belong (by providing lower bounds). Naturally, we say that a function is easy if the upper bound on the amount of resources needed to compute it is small, and we say that the function is hard if the lower bound is large. For instance, functions belonging to P are said to be easy and NP-complete functions are said to be hard, although not provably so. The most fundamental question in complexity theory is whether P is different from NP. After many years of extensive research the question remains unanswered, although most researchers believe that P = NP. No function that can be computed in nondeterministic polynomial time is known to require more than deterministic polynomial time. In fact, no nonlinear lower bounds have been proven on general models of computation for any functions in NP. 1 Chapter 1. Introduction 2 This lack of progress has led researchers to consider restricted models of computation with the hope that these restricted models would enable them to constrain the problem and develop methods to derive strong lower bounds. In turn, these methods would provide a bet- ter understanding of the model of computation, and, by gradually removing the restrictions, nonlinear lower bounds would be proven for the general model of computation. In this thesis we will consider the Boolean circuit model of computation. A Boolean circuit is a directed acyclic graph whose nodes compute certain Boolean functions from some basis of computation Ω. A Boolean function f : {0, 1}∗ →{0, 1} is computed by a sequence N of circuits {Cn}n∈N, where for every n ∈ , circuit Cn computes the function f restricted to inputs of length n. Boolean circuits are a general model of computation, since non-uniform Boolean circuits can compute any functions. Furthermore, a Turing machine running in time T (n) can be simulated by circuits of size O (T (n) log T (n)). Consequently, strong lower bounds on the size of Boolean circuits computing a function f imply strong lower bounds on the computational time of f on a Turing machine. Unlike Turing machines, Boolean circuits are easy to define and their fixed structure is more amendable to mathematical analysis. It remains an open problem, however, to derive nonlinear lower bounds on the size of general circuits computing some explicit Boolean function. Another attractive feature of the Boolean circuit model of computation is that it can be easily restricted, and it is in these restricted