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

How statement and proof provenance work

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Counting bounded-size Boolean circuits

Statement

For $n, s \geq 1$ , the number of Boolean functions on $n$ inputs computed by circuits of size at most $s$ over the fixed page basis is at most $2^{O (s \log (n + s))} .$

Facts & Assumptions

Given: integers $n, s \geq 1$ .

[L1]

The circuit basis is fixed and has arity at most two, by Boolean circuits: basis, fan-in, size, and depth.

[L2]

Independent finite choices multiply, by The product rule: $∣ A \times B ∣ = ∣ A ∣ ∣ B ∣$ , and $∣ \prod_{i < m} A_{i} ∣ = \prod_{i < m} ∣ A_{i} ∣$ .

Proof

technique · direct

1.1L1given

A topologically numbered gate has at most a fixed number $b$ of type choices and at most two input-wire indices among at most $n + s$ available wires. Thus each gate record has at most $b (n + s)^{2}$ possibilities; choosing an output wire adds at most $n + s$ possibilities.

2.1L2step 1.1algebra

By [L2], circuits with exactly $r \leq s$ gates therefore number at most $(n + s) [b (n + s)^{2}]^{r}$ . Summing over $0 \leq r \leq s$ gives at most $(s + 1) (n + s) [b (n + s)^{2}]^{s} = 2^{O (s \log (n + s))}$ because $s \geq 1$ .

3.1step 2.1∎

Distinct encodings may compute the same function, so this circuit count is also an upper bound on the number of represented Boolean functions.

Depends on

Used by

Shannon counting gives an efficiently explicit circuit-hard function False statement
Almost all Boolean functions require exponential circuit size Theorem

Dependency tree · two levels

13 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)