ExampleConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedverified 2026-09-26 (gpt-6-sol)

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.

A two-branch computation and its parsimonious tableau formula

Statement

On empty input, let $N$ make one nondeterministic choice $b \in {0, 1}$ , record $b$ in its state, and accept. Its two accepting paths give two legal padded tableaux, and the exact Cook--Levin encoding has exactly two satisfying assignments.

Facts & Assumptions

Given: the displayed one-choice machine $N$ .

[L1]

The Cook--Levin construction can be made parsimonious by The Cook--Levin construction can be made parsimonious.

Verification

technique · direct

1.1givenconstruct

Use two distinct intermediate states $q_{0}^{'}$ and $q_{1}^{'}$ to record the first choice: from the initial state the machine branches to $q_{b}^{'}$ , then deterministically enters its single accepting state. These are its only two computation paths. Their first successor rows have different state tags, and padding repeats the final accepting configuration, so they give two and only two legal accepting tableaux.

2.1L1step 1.1

In the exact encoding of [L1], every declared variable is a tableau cell variable and there are no auxiliary variables. Each of the two legal padded tableaux fixes every declared variable, and [L1] gives the converse from any satisfying assignment to exactly one accepting path. Hence the two paths correspond bijectively to the formula's satisfying assignments.

3.1step 1.1step 2.1∎

Therefore the source accepting-path count and the formula's satisfying- assignment count are both exactly $2$ .

Depends on

The Cook--Levin construction can be made parsimonious

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)