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CS-160 · computer science · mathematics · Computational complexity

Quadratic sensitivity versus block sensitivity

Status OpenLLM status Open unverified

Statement

Definitions and mathematical statement

Statement recorded in the supplied source.
Is there an absolute C such that every nonconstant total Boolean function satisfies $bs(f)\leq C$ $s(f)^{2}$, where s is sensitivity and bs block sensitivity?

Short English statement

Is there an absolute C such that every nonconstant total Boolean function satisfies $bs(f)\leq C$ $s(f)^{2}$, where s is sensitivity and bs block sensitivity?

Sources

Statement licence: CC-BY-4.0 · Source file

Contributions 1

CS-160-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for CS-160

The displayed question has a negative answer under the manuscript's stated construction. Its counterexample meets the conditions in this problem and defeats the universal assertion, so this is a full disproof claim. The supporting result is manuscript 1 (family 132; 2026-09-25).

Quadratic sensitivity versus block sensitivity

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 132; 2026-09-25), the asserted result is that total Boolean functions can have block sensitivity exceeding sensitivity to a fixed exponent above two, disproving a universal quadratic relation.

Mathematical statement of the claimed result

Manuscript 1.
For a total Boolean function, let $s(f)$ be its maximum number of sensitive individual coordinates and $\operatorname{bs}(f)$ its maximum number of pairwise disjoint sensitive blocks. The claim states \[ \forall C>0\ \exists n\ge1\ \exists f:\{0,1\}^n\to\{0,1\}:\quad f\text{ nonconstant},\quad\operatorname{bs}(f)>C s(f)^2. \] Its explicit family has, for every integer $d\ge1$, \[ \operatorname{bs}(f)/s(f)^2\ge 2^d/[4(d+2)^2]. \]

Outline of the argument

For manuscript 1 (family 132; 2026-09-25), nested predicates are composed using labeled tournament rows. Strong child acceptance and weaker outgoing rejection requirements keep one-bit changes scarce, while disjoint zero-input blocks multiply through the recursion; a fixed composition converts the growing ratio into an exponent gap.

Formal verification

For manuscript 1 (family 132; 2026-09-25), OpenAI lists Lean statements for this assertion: total Boolean functions can have block sensitivity exceeding sensitivity to a fixed exponent above two, disproving a universal quadratic relation.

The selected statement includes the clause used for this problem.

The linked family Comparator checks are SensitivitySeparation.lean.

Their named declarations include OAI.Paper320.quantitative_separation.

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

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