Analysis of Boolean Functions

Abstract: Scribe notes from the 2012 Barbados Workshop on Computational Complexity. A series of lectures on Analysis of Boolean Functions by Ryan O'Donnell, with a guest lecture by Per Austrin.

• The paper presents a comprehensive framework for analyzing Boolean functions via Fourier expansion and linearity testing, including insights from Arrow's theorem.
• The paper demonstrates that noise stability and the small set expansion theorem reveal the behavior of Boolean functions under random perturbations.
• The paper applies advanced methods like the KKL theorem and Poincaré inequality to quantify variable influences, impacting computational complexity and approximation challenges.

Analysis of Boolean Functions

The document presented provides lecture notes from a workshop on Computational Complexity held in Barbados, with a particular emphasis on the analysis of Boolean functions. The lectures were delivered by Ryan O'Donnell, with contributions from Per Austrin and scribe notes by Li-Yang Tan. The material covers various aspects of Boolean functions, such as their Fourier expansion, the implications of Arrow's theorem, and applications to property testing, voting, pseudorandomness, Gaussian geometry, and the hardness of approximation.

Summary of Key Topics

1. Linearity Testing and Arrow's Theorem

The initial lectures explore the principles of linearity testing. A Boolean function $f: \{-1,1\}^n \to \{-1,1\}$ is linear if there exists a set $S \subseteq [n]$ such that $f(x) = \sum_{i \in S} x_i$. The Blum-Luby-Rubinfeld (BLR) linearity test is highlighted, which verifies whether a function is close to being linear.

The notes also discuss Arrow's Impossibility Theorem in the context of Boolean functions, illustrating how social choice theory interrelates with computational complexity. Specifically, the theorem proves that no voting system can convert individual preferences into a community-wide ranking while simultaneously meeting certain reasonable criteria.

2. Fourier Expansion

O'Donnell's lectures provide a thorough exploration of the Fourier expansion of Boolean functions. Every Boolean function $f$ can be uniquely expressed as a multilinear polynomial over real numbers. The Fourier coefficients and their significance are discussed, including the concepts of orthonormality and the implications of Parseval's identity and Plancherel's theorem.

3. Noise Stability and Small Set Expansion

A significant portion of the notes addresses noise stability, an important metric in the study of Boolean functions. The concept of $\rho$-correlated pairs provides insights into the behavior of functions under random perturbations. The lectures also cover Sheppard's formula, which relates to the stability of the majority function $MAJ$ under noise.

The small set expansion theorem is stated, demonstrating that small subsets of the Boolean hypercube have a high probability of crossing the boundary under random walks. This has been proven to be an effective tool in understanding the combinatorial structure of Boolean functions.

4. Advanced Fourier Analysis and Poincaré Inequality

The lectures advance into sophisticated areas of Fourier analysis, presenting results like the Kahn-Kalai-Linial (KKL) theorem. The KKL theorem effectively establishes lower bounds on the influence of variables on Boolean functions. The Poincaré inequality for the Boolean hypercube is examined, illustrating how it can confine the variance of functions.

Implications and Future Directions

The material covered in these lectures has broad implications for theoretical computer science, particularly in fields related to property testing, hardness of approximation, and pseudorandomness. The algebro-combinatorial techniques used to analyze Boolean functions are foundational in many modern algorithms and complexity theory results.

Particularly noteworthy is the application of these concepts to the Unique Games Conjecture (UGC) and the Majority Is Stablest theorem (MIST). These results continue to influence ongoing research in approximation algorithms and computational complexity.

Future developments in this area might include:

• Refinement of Testing Algorithms: Improved algorithms for testing Boolean functions and their properties could enhance practical applications in machine learning and cryptography.
• Broader Applications of Fourier Analysis: Extending these techniques to other domains such as quantum computing and data privacy.
• Enhanced Understanding of Noise Sensitivity: Deeper insights into the noise stability of complex functions might yield robust solutions for fault-tolerant computing systems.

Conclusion

The notes from the Barbados Workshop provide an extensive and detailed exploration of the analysis of Boolean functions, encapsulating both foundational theories and advanced concepts. They offer valuable insights for researchers exploring computational complexity, providing a springboard for further exploration and innovation in this pivotal area of computer science.