Alphabeta MathAlphabet Reduction and the PCP Theorem: Examples and Counterexamples
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1 · Prerequisites
Abstract Algebra
Combinatorics
Computability Theory
- Alphabet Reduction and the PCP Theorem
- Arithmetization and the Sum-Check Protocol
- Boolean Circuits and Nonuniform Complexity
- Expander Graphs and Constraint Graphs
- Formal Languages, Encodings, and Decision Problems
- Gap Amplification and Assignment Testing
- P, NP, coNP, and Polynomial Reductions
- Resource Bounds and Machine Invariance
- Turing Machines, Configurations, and Computation
Real Analysis
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- limsup, liminf, and Subsequential Limits
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Suprema and Infima
2 · Logical flowchart
3 · Items
Example
A single equality edge through robust composition
Example
Three repetitions of a three-quarters-sound PCP
Counterexample
A powered graph whose alphabet grows
Example
Four coordinates of a Walsh–Hadamard codeword
Sources
- Irit Dinur, The PCP Theorem by Gap Amplification, §5.1, Definition 5.1 and Lemma 1.8 (completeness direction), printed pp. 17–18
- Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach, §18.5.2, proof of Lemma 18.30 (completeness direction), printed pp. 378–379
- Arora and Barak, Computational Complexity: A Modern Approach, §18.1 Note 3 to Theorem 18.2, printed p. 354
- Arora and Barak, Computational Complexity: A Modern Approach, §18.5.1, Lemma 18.31, printed pp. 371–372
- Arora and Barak, Computational Complexity: A Modern Approach, §18.4.1, printed pp. 363–364