Complexity Theory Retrospective II

1 Time, Hardware, and Uniformity.- 1 Introduction.- 2 Background: Descriptive Complexity.- 3 First Uniformity Theorem.- 4 Variables That Are Longer Than log nBits.- 5 Uniformity: The Third Dimension.- 6 Variables That Are Shorter Than log nBits.- 7 Conclusions.- 2 Quantum Computation.- 1 The Need for Quantum Mechanics.- 2 Basic Principles of Quantum Mechanics.- 2.1 Probability Amplitudes.- 2.2 Qubits and How to Observe Them.- 2.3 Digression on Quantum Cryptography.- 2.4 Evolution of a Quantum System.- 2.5 Quantum Registers.- 3 Computing with Quantum Registers.- 4 Separating Two Classes of Functions.- 5 Shor’s Factoring Algorithm.- 6 Building a Quantum Computer.- 3 Sparse Sets versus Complexity Classes.- 1 Introduction.- 2 Earlier Results for Turing Reductions.- 2.1 Sparse Sets and Polynomial Size Circuits.- 2.2 The Karp-Lipton Theorem.- 2.3 Long’s Extension.- 3 Earlier Results for Many-One Reductions.- 3.1 The Isomorphism Conjecture for NP.- 3.2 Mahaney’s Theorem.- 4 Bounded Truth Table Reduction of NP.- 4.1 Extensions.- 5 The Hartmanis Conjecture for P.- 5.1 Ogihara’s Language and Randomized NC2.- 5.2 Deterministic Construction.- 5.3 The Finale: NC1 Simulation.- 6 Conclusions.- 4 Counting Complexity.- 1 Introduction.- 2 Preliminaries.- 3 Counting Functions.- 3.1 Algebraic Properties of Counting Functions.- 3.2 A Randomized sign Function.- 3.3 Counting Functions and the Polynomial-Time Hierarchy.- 4 Counting Classes.- 4.1 Classifying Counting Classes.- 4.2 Counting Operators.- 4.3 The Polynomial-Time Hierarchy.- 4.4 Closure Properties of PP.- 5 Relativization.- 6 Other Work.- 6.1 Circuits.- 6.2 Lowness.- 6.3 Characterizing Specific Problems.- 6.4 Interactive Proof Systems.- 6.5 Counting in Space Classes.- 6.6 Other Research.- 5 A Taxonomy of Proof Systems.- 1 Introduction.- 2 A Technical Exposition.- 2.1 Interactive Proof Systems.- 2.2 MIP and PCP.- 2.3 Computationally Sound Proof Systems.- 2.4 Other Types of Proof Systems.- 2.5 Comparison.- 3 The Story.- 3.1 The Evolution of Proof Systems.- 3.2 PCP and Approximation.- 3.3 Interactive Proofs and Program Checking.- 3.4 Zero-Knowledge Proofs.- 6 Structural Properties of Complete Problems for Exponential Time.- 1 Introduction.- 2 Strong Reductions to Complete Sets.- 3 Immunity for Complete Problems.- 4 Differences between Complete Sets.- 5 Other Properties and Open Problems.- 5.1 Properties of “Weak” Complete Sets.- 5.2 Polynomial-Time Complete Recursively Enumerable Sets.- 5.3 A Short List of Open Problems.- 7 The Complexity of Obtaining Solutions for Problems in NP and NL.- 1 Introduction.- 2 Computing Optimal Solutions: The Class FPNP.- 3 Bounded Queries to NP.- 4 Computing Solutions Uniquely: The Class NPSV.- 5 Nonadaptive Queries to NP: The Class FPNPtt.- 6 A Look inside Nondeterministic Logspace.- 7 Conclusions.- 8 Biological Computing.- 1 Introduction.- 2 The One-Molecule Processor.- 3 A Brief Introduction to Biochemistry.- 3.1 DNA, RNA, and Proteins.- 3.2 Protein Synthesis.- 4 Computational Molecules.- 4.1 CNA.- 4.2 tCNA.- 4.3 The Synthesis of tCNA.- 5 The Microarchitecture of CNA Computers.- 6 A Brief Discussion of Adleman’s Model Versus Our Model.- 7 Conclusions.- 9 Computing with Sublogarithmic Space.- 1 Are Sublogarithmic Space Classes of Any Interest?.- 2 The Alternating Sublogarithmic Space World.- 3 Adding Randomness.- 4 Special Limitations of Machines with a Sublogarithmic Space Bound.- 4.1 Technical Preliminaries.- 4.2 Inputs with a Periodic Structure.- 4.3 Fooling ATMs.- 5 A Survey of Lower Space Bound Proofs.- 5.1 Languages for Separating the Levels of the Alternation Hierarchy.- 5.2 ATMs with a Constant Number of Alternations.- 5.3 Unbounded Alternation.- 5.4 Closure Properties.- 5.5 Lower Bounds for Context-Free Languages.- 6 Conclusions and Open Problems.- 10 The Quantitative Structure of Exponential Time.- 1 Introduction.- 2 Preliminaries.- 3 Resource-Bounded Measure.- 4 Incompressibility and Bi-Immunity.- 5 Complexity Cores.- 6 Small Span Theorems.- 7 Weakly Hard Problems.- 8 Upper Bounds for Hard Problems.- 9 Nonuniform Complexity, Natural Proofs, and Pseudorandom Generators.- 10 Weak Stochasticity.- 11 Density of Hard Languages.- 12 Strong Hypotheses.- 13 Conclusions and Open Directions.- 11 Polynomials and Combinatorial Definitions of Languages.- 1 Introduction.- 2 Polynomials.- 3 Representation Schemes and Language Classes.- 4 Strong versus Weak Representation.- 5 Known Upper and Lower Bounds on Degree.- 6 Polynomials for Closure Properties.- 7 Probabilistic Polynomials.- 8 Other Combinatorial Structures.- 12 Average-Case Computational Complexity Theory.- 1 Introduction.- 2 Average Polynomial Time.- 3 Average-Case Completeness.- 3.1 Polynomial-Time Reductions.- 3.2 Polynomial-Time Computable Distributions.- 3.3 Uniform Distributions.- 3.4 Distribution Controlling Lemma.- 3.5 Distributional NP-Completeness.- 3.6 Average Polynomial-Time Reductions.- 3.7 Distributional Search Problems.- 4 Randomization.- 4.1 Flat Distributions and Incompleteness.- 4.2 Randomized Average Polynomial Time.- 4.3 Randomizing Reductions and Completeness.- 4.4 Polynomial-Time Sampling.- 4.5 Randomized Turing Reductions.- 5 Hierarchies of Average-Case Complexity.- 5.1 Average-Time Hierarchies.- 5.2 Fast Convergence of Average Time.- 5.3 Averaging on Ranking of Distributions.- 6 A Brief Survey of Other Results.