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Nitin Saurabh the Institute of Mathematical Sciences, Chennai ALGEBRAIC MODELS OF COMPUTATION By Nitin Saurabh The Institute of Mathematical Sciences, Chennai. A thesis submitted to the Board of Studies in Mathematical Sciences In partial fulllment of the requirements For the Degree of Master of Science of HOMI BHABHA NATIONAL INSTITUTE April 2012 CERTIFICATE Certied that the work contained in the thesis entitled Algebraic models of Computation, by Nitin Saurabh, has been carried out under my supervision and that this work has not been submitted elsewhere for a degree. Meena Mahajan Theoretical Computer Science Group The Institute of Mathematical Sciences, Chennai ACKNOWLEDGEMENTS I would like to thank my advisor Prof. Meena Mahajan for her invaluable guidance and continuous support since my undergraduate days. Her expertise and ideas helped me comprehend new techniques. Her guidance during the preparation of this thesis has been invaluable. I also thank her for always being there to discuss and clarify any matter. I am extremely grateful to all the faculty members of theory group at IMSc and CMI for their continuous encouragement and giving me an opportunity to learn from them. I would like to thank all my friends, at IMSc and CMI, for making my stay in Chennai a memorable one. Most of all, I take this opportunity to thank my parents, my uncle and my brother. Abstract Valiant [Val79, Val82] had proposed an analogue of the theory of NP-completeness in an entirely algebraic framework to study the complexity of polynomial families. Artihmetic circuits form the most standard model for studying the complexity of polynomial computations. In a note [Val92], Valiant argued that in order to prove lower bounds for boolean circuits, obtaining lower bounds for arithmetic circuits should be a rst step. One could hope that techniques from well-developed areas of mathematics may help us to solve fundamental problems in algebraic complexity theory. Therefore, It has attracted a large amount of research in last two decades. As a consequence, there has been a lot of progress in the area. The aim of this thesis is to explore Valiant's approach and to understand dif- ferent models of algebraic computation. In the process, we survey known results and new techniques. We discuss simple combinatorial proofs of known results like VNP = VNPe and study combinatorial characterization of classes VP and VQP. Further, we explore the parallelization of the class VP and discuss a complete problem for the same class. We also cover classical proof of completeness for the permanent family. Later, we study how the complexity of computing certain inte- gers is related to separating constant-free algebraic classes. iv Contents 1 Introduction1 2 Branching Programs3 2.1 Algebraic Branching Programs.....................3 2.2 Reduction and Completeness......................8 2.3 Comparison with other models of computation............9 2.4 Depth reduced circuits......................... 15 3 VP 18 3.1 The Class VP.............................. 18 3.2 Parallel Complexity........................... 22 3.3 Complete polynomials......................... 28 3.4 Classes within VP............................ 32 3.5 Class VQP and Determinant...................... 32 3.6 Symmetric determinantal representation............... 35 4 VNP 39 4.1 Class VNP................................ 39 4.2 Universality and Completeness of the Permanent........... 44 4.3 Characterizing uniform VNP...................... 50 5 Cost of computing Integers 57 5.1 Basic denitions and simple bounds.................. 57 5.2 Connection to Valiant's Theory.................... 60 5.3 Relaxing the hypothesis VP0 = VNP0 ................. 64 6 Conclusion 67 i List of Figures 2.1 An algebraic branching program computing 2 2 .3 : 2x1x2x3 + x1x3 + x1 2.2 A circuit and a formula computing 4(x1 + x2) ............ 10 2.3 Construction of an algebraic branching program from a formula.. 11 2.4 Algebraic branching program to skew circuit............. 13 2.5 Weakly skew circuit to algebraic branching program......... 15 4.1 Graph K; i-coupling of edges c = (u; v) and c0 = (u0; v0) ...... 46 4.2 The Rosette R(4) ............................ 48 ii 1 Introduction Valiant introduced algebraic complexity theory to study the complexity of poly- nomial families. Arithmetic circuits provide a natural model of computation that captures the complexity of computing polynomials using algebraic operations. The most fundamental problems in algebraic complexity are related to the complexity of arithmetic circuits. In general arithmetic circuits are quite powerful, and to this day, we do not know explicit examples of polynomials requiring super-polynomial circuit size. Apart from arithmetic circuits, there is another model of arithmetic computation called algebraic branching programs. The expressive power of these models has been a puzzling question for more than ve decades. Nevertheless, advances have been made on understanding restricted models of arithmetic com- putation. The goal of this thesis is to understand dierent models of algebraic computa- tion. This thesis is structured in the following ways. There are four main chapters. • In Chapter 2, we introduce a simple model of computation called algebraic branching programs. We show the completeness of the determinant fam- ily. Then we introduce arithmetic circuits and compare their computational power with algebraic branching programs. Finally, we give a depth ecient construction of arithmetic circuits from algebraic branching programs. • In Chapter 3, we study the computational power of restricted circuits ob- tained by restricting the size of the circuit. Then we discuss an ecient parallel algorithm for the Valiant's class VP. Again we see completeness of the determinant for a dierent class, but under dierent notion of reduc- 1 Chapter 1. Introduction tion. Finally, we end the chapter with a discussion on symmetrizing the determinant polynomial. • In Chapter 4, we begin our discussion by dening the class VNP and then going on to develop a proof of the completeness of the permanent. At the end we take a slight detour to characterize certain uniform versions of Valiant's algebraic classes. • In Chapter 5, we investigate the cost of computing integers in the arithmetic circuit model. We present some simple lower and upper bounds and dis- cuss some examples. We also describe the relationship between the cost of computing certain integer sequences and Valiant's classes. We conclude the thesis and discuss a few open problems in Chapter 6. 2 2 Branching Programs 2.1 Algebraic Branching Programs In this section we will dene a simple model of algebraic computation, called algebraic branching programs. Denition 2.1. An algebraic branching program (ABP) G = (V; E) is a directed acyclic graph with two distinguished vertices, a source s and a sink t. The source s has in-degree 0 and the sink t has out-degree 0. There is a weight function dened on edges, where is a eld or ring. w : E ! F [ fx1; x2;:::g F The weight of a path from s to t is the product of the weights of the edges appearing in the path. The weight of (s; t) in G is the sum of the weights of all paths from s to t. The polynomial computed by an algebraic branching program G is the weight of (s; t) in G (Figure 2.1). The size of an algebraic branching program is the number of vertices in it. Denition 2.2. A sequence of polynomials (fn) belongs to the class VBP if and x2 x1 s x x 3 3 1 x2 x1 t Figure 2.1: An algebraic branching program computing 2 2 : 2x1x2x3 + x1x3 + x1 3 Chapter 2. Branching Programs only if there exist a sequence of algebraic branching programs (Gn) of polynomially bounded size such that Gn computes fn. It has been known for long that the determinant family DETn belongs to the class VBP. We will present here an elegant combinatorial construction from [MV97]. Let A be an n × n matrix. Let GA denote the weighted directed graph rep- resented by A. Let the vertex set of GA be f1; 2; : : : ; ng. The edges in GA are weighted, that is, an edge (i; j) has weight ai;j. Then we know, X det(A) = sign(C)w(C) C is a cycle cover of GA where, w(C) denotes the weight of a cycle cover C. But it seems dicult to get an algebraic branching program to eciently compute determinant from this denition. The ingenuity of the proof lies in extending the summation to a larger set such that the overall contribution from the new terms is zero. To achieve this, we generalize the notion of cycle covers to clow sequences. A clow is an ordered sequence of vertices C = hv1; v2; : : : ; vli such that (vl; v1) is an edge in the graph and also, any (vi; vi+1) is an edge in the graph. We will require that v1 is the least numbered vertex in the clow and occurs only once in the clow. v1 is called the head of the clow. The weight of a clow C, w(C), is the product of the weights of the edges in with multiplicity, that is, Ql−1 . C w(C) = ( i=1 avi;vi+1 ) · avl;v1 A clow sequence is a sequence of clows W = hC1;C2;:::;Cki such that head(C1) < head(C2) < ··· < head(Ck) and the total number of edges, counted with multi- plicity, adds to exactly n. The sign of a clow sequence is dened to be (−1)n+k, where n is the number of vertices and k is the number of clows in the sequence. The weight of a clow sequence Qk . w(W) = i=1 w(Ci) Theorem 2.3 ([MV97]). Let A be a (n × n)-matrix. Then, X det(A) = sign(W)w(W) W is a clow sequence in GA Proof. As alluded before the proof is by showing that the total contribution of the clow sequences that are not cycle covers is zero.
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