CounterexampleConstruction: Literature-sourcedVerification: 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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Unary circuits for an undecidable language are not uniformly generated

Statement

Let $H = {n : M_{n} (n) halts}$ , let $U = {1^{n} : n \in H}$ , and define $L = {x : 1^{∣ x ∣} \in U}$ . The family whose $n$ th circuit is the constant bit $1_{H} (n)$ has constant size but has no polynomial-time uniform generator.

Facts & Assumptions

Given: the displayed diagonal halting tally language.

[L1]

The length-language construction has constant circuits but is undecidable. by Some undecidable languages have polynomial-size circuits.

Counterexample

technique · direct

1.1

By [L1], $L$ is undecidable and the chosen $C_{n}$ computes $L$ on every $n$ -bit input. Each $C_{n}$ is one constant-output gate.

L1given

2.1

If an algorithm $G$ generated $C_{n}$ from $1^{n}$ in polynomial time, then on input $1^{n}$ we could run $G$ , evaluate its output circuit on $1^{n}$ , and answer whether $n \in H$ . This would decide the diagonal halting set, contrary to [L1].

L1step 1.1contradiction

3.1

Thus the displayed constant-size family is the promised concrete counterexample to uniform generation.

step 1.1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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)