Alphabeta MathThe IP = PSPACE Theorem: Examples and Counterexamples
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1 · Prerequisites
Abstract Algebra
Combinatorics
Computability Theory
- Arithmetization and the Sum-Check Protocol
- Formal Languages, Encodings, and Decision Problems
- Interactive Proof Systems and Public Coins
- P, NP, coNP, and Polynomial Reductions
- Resource Bounds and Machine Invariance
- Space Complexity, Savitch's Theorem, and TQBF
- The IP = PSPACE Theorem
- Turing Machines, Configurations, and Computation
Linear Algebra
Number Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Example
A two-quantifier field transcript
Example
A concrete multilinearization calculation
Example
Perfect completeness through a TQBF reduction
Counterexample
Exponential degree without multilinearization
Sources
- A. Shen, IP = PSPACE: Simplified Proof, JACM 39(4) 1992, pp. 878–880
- Arora and Barak, Computational Complexity: A Modern Approach, §8.5.3, author-hosted draft
- Arora and Barak, Computational Complexity: A Modern Approach, Remark 8.19, author-hosted draft
- Arora and Barak, Computational Complexity: A Modern Approach, §8.5.3 (degree growth warning) and Remark 8.19, author-hosted draft