Open Problems in Analysis of Boolean Functions

A list of open problems in the field of analysis of boolean functions, compiled February 2012 for the Simons Symposium.

💡 Research Summary

The paper “Open Problems in Analysis of Boolean Functions” is a curated list of the most compelling unanswered questions in the field of Boolean function analysis as of February 2012, prepared for the Simons Symposium. It is organized into thematic sections that reflect the major sub‑areas of the discipline: Fourier analysis, influence and sensitivity, noise stability, sharp thresholds, computational complexity, and learning theory.

The first section focuses on the Fourier spectrum of Boolean functions. Central to this part is the Kahn‑Kalai‑Linial (KKL) theorem and its conjectured refinements. While KKL guarantees that every variable’s influence is bounded below by a universal constant times the total influence divided by the logarithm of the number of variables, the exact constants, the behavior for non‑monotone functions, and extensions to higher‑order influences remain open. Related open problems include the Bourgain sharp‑threshold conjecture, which seeks quantitative bounds linking the size of the Fourier tail to the presence of a sharp threshold, and Friedgut’s junta theorem, whose optimal parameters are still unknown.

The second section deals with noise sensitivity and stability. The Benjamini‑Kalai‑Schramm (BKS) conjecture predicts that any Boolean function with low total influence must be noise sensitive, a statement proved only for specific families such as percolation crossing events. The “Majority is Stablest” theorem, proved in Gaussian space, raises the question of whether analogous optimal stability results hold for broader classes of discrete functions, especially under biased measures. The paper also lists conjectures concerning the exact noise‑stability constants for half‑spaces, the relationship between noise sensitivity and critical exponents in statistical physics models, and the possibility of a universal “noise‑sensitivity exponent” governing all monotone functions.

The third section examines the interplay between Fourier degree, sparsity, and computational models. Mansour’s conjecture asserts that any AC⁰ circuit of size s has a Fourier spectrum that can be approximated by a polynomial with at most s^{O(1)} non‑zero coefficients. While upper bounds on the degree are known, the conjectured sparsity bound remains elusive. Another prominent problem is the “Fourier degree vs. decision‑tree depth” question, which asks whether a Boolean function with low Fourier degree must also admit a shallow decision tree, a relationship that would have profound implications for learning algorithms.

The fourth section is devoted to sharp thresholds and phase transitions. Friedgut’s theorem on sharp thresholds for monotone properties provides a qualitative statement, but determining the precise quantitative thresholds (the “critical window”) for various graph properties is still an open challenge. The Keller‑Mossel conjecture proposes a structural characterization of functions that exhibit a sharp threshold, suggesting that such functions must be close to a junta or a low‑degree polynomial. The paper also highlights open problems concerning the universality of threshold phenomena across different product spaces and the role of hypercontractivity in sharpening existing bounds.

The final section connects the theoretical questions to algorithmic applications, especially learning theory. The “learning juntas” problem asks whether there exists a polynomial‑time algorithm that, given random examples of an unknown k‑junta, can identify the relevant variables and reconstruct the function with high probability. While sub‑exponential algorithms are known, a truly polynomial solution is missing. Related to this are conjectures about noise‑sensitivity based learning: can one exploit a function’s noise stability to design more efficient learning algorithms under adversarial or random noise? The paper also mentions open problems in property testing (e.g., testing whether a function is a low‑degree polynomial) and compressed sensing for Boolean functions, where one seeks to recover a sparse Fourier representation from a limited number of queries.

For each problem, the authors provide a brief status report, the most promising techniques attempted so far (e.g., hypercontractivity, invariance principles, Gaussian isoperimetry, combinatorial regularity lemmas), and a rough assessment of difficulty. They rank the problems according to perceived tractability within the next decade, distinguishing those that may yield to refined analytic tools from those that likely require new conceptual breakthroughs. The paper concludes by emphasizing the deep interconnections among the listed problems: progress on any one of them often translates into advances in several others, underscoring the field’s rich, interdisciplinary nature.