Complexity Theory Retrospective II

Readers of Complexity Theory Retrospective (published by Springer-Verlag in 1990) will remember that the collection consisted primarily of articles that first ap­ peared in preliminary form at one of the meetings of the Annual IEEE Conference on Structure in Complexity Theory. In particular, Complexity Theory Retrospective contained final versions of high-quality technical expository presentations, includ­ ing talks that honored Juris Hartmanis on the occasion of his sixtieth birthday. We began planning for the current collection some months before the scheduled meeting in 1994 of the Tenth Annual IEEE Conference on Structure in Complexity Theory. As with the original volume, several of the papers in this book originated as presentations at one of the meetings of the Structure in Complexity Theory con­ ference. We are pleased to provide this forum for final, polished versions of these papers. As it turns out, 1994 was a watershed year for the Structures conference, for at this meeting the conference attendees voted to change the conference name to its current name, the Annual IEEE Conference on Computational Complex­ ity. In voting to remove the expression "Structure in," the conferees recognized the recent explosion of techniques and results in computational complexity, and expressed concern that the original conference name might not accurately reflect the current status. We approached this volume in the same spirit.

Table of Contents:
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.- 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.- 3 Earlier Results for Many-One Reductions.- 4 Bounded Truth Table Reduction of NP.- 5 The Hartmanis Conjecture for P.- 6 Conclusions.- 4 Counting Complexity.- 1 Introduction.- 2 Preliminaries.- 3 Counting Functions.- 4 Counting Classes.- 5 Relativization.- 6 Other Work.- 5 A Taxonomy of Proof Systems.- 1 Introduction.- 2 A Technical Exposition.- 3 The Story.- 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.- 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.- 4 Computational Molecules.- 5 The Microarchitecture of CNA Computers.- 6 A Brief Discussion of Adleman’s Model Versus Our Model.- 7 Conclusions.- 9 Computing withSublogarithmic 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.- 5 A Survey of Lower Space Bound Proofs.- 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.- 4 Randomization.- 5 Hierarchies of Average-Case Complexity.- 6 A Brief Survey of Other Results.