LemmaStatement: AI-adaptedProof: AI-generatedprecheck passaudited 2026-09-12

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.

The start, accept, and transition constraints have polynomial size

Statement

For a fixed machine and a polynomially bounded tableau, the Boolean formulas forcing the correct start row, forcing an accepting state somewhere, and forcing legal row-to-row transitions all have polynomial total size.

Facts & Assumptions

Given: A tableau of side length $T + 1$ for a fixed machine $N$ on input $w$ .

[L1]

A local $2 \times 3$ window is legal exactly when it matches one allowed local update of the machine, by Local 2x3 tableau windows and legality.

[L2]

Each tableau cell ranges over a constant-size alphabet for fixed $N$ , by For a fixed machine, each tableau cell ranges over a constant-size extended alphabet.

Proof

technique · direct

1.1givenconstruct

The start constraint fixes each cell of the first row to its predetermined initial symbol, so it contributes one constant-size literal condition per column and therefore $O (T)$ total size. The accept constraint is the disjunction over all cells that one of them carries an accepting-state tag, so it has $O (T^{2})$ size.

1.2L1L2givenconstruct

For each adjacent-row position and each illegal $2 \times 3$ pattern, add a clause forbidding that pattern. By [L1] and [L2], there are $O (T^{2})$ window positions and only constantly many patterns to inspect at each position, each producing a constant-size clause.

2.1step 1.1step 1.2∎

Combining steps 1.1 and 1.2 shows that the start, accept, and transition families together have polynomial total size.

Depends on

Used by

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

MIT 18.404J / 6.840J, Lecture 16: Cook-Levin Theorem (standard reference, not scraped)