TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-07

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Almost all Boolean functions require exponential circuit size

Statement

As $n \to \infty$ , the proportion of Boolean functions on $n$ variables having circuit size at most $2^{n} / (10 n)$ tends to $0$ . In particular, almost all Boolean functions require circuit size $Ω (2^{n} / n)$ .

Facts & Assumptions

Given: the fixed circuit basis of the page.

[L1]

Size-at-most- $s$ circuits represent at most $2^{O (s \log (n + s))}$ functions, with the explicit gate-record count in the proof, by Counting bounded-size Boolean circuits.

[L2]

There are $2^{2^{n}}$ Boolean functions on $n$ inputs, by The set $A^{B}$ of functions $B \to A$ between finite sets is finite, with $∣ A^{B} ∣ = ∣ A ∣^{∣ B ∣}$ .

Proof

technique · direct

1.1

Put $s = ⌊ 2^{n} / (10 n) ⌋$ and let $b$ be the fixed number of available gate types. The explicit encoding in [L1] gives at most $(s + 1) (n + s) (b (n + s)^{2})^{s}$ circuits. Since $b$ is constant, for all sufficiently large $n$ the base-two logarithm is at most $s (2 n + O (1)) + O (n) < 2^{n} / 3$ .

L1givenalgebra

2.1

Dividing the resulting bound $2^{2^{n} / 3}$ by the total $2^{2^{n}}$ from [L2] gives at most $2^{- 2 \cdot 2^{n} / 3}$ , which tends to $0$ . Hence a uniformly random truth table has size greater than $2^{n} / (10 n)$ with probability tending to $1$ .

L2step 1.1algebra

3.1

This is an existence-and-density argument over truth tables; it does not select or compute an explicit hard family.

step 2.1∎

Depends on

Used by

Shannon counting gives an efficiently explicit circuit-hard function False statement

Dependency tree · two levels

14 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Arora and Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)
Lance Fortnow, Counting Complexity (standard reference, not scraped)