LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-12

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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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For a fixed machine, each tableau cell ranges over a constant-size extended alphabet

Statement

Fix a nondeterministic Turing machine $N$ with state set $Q$ and tape alphabet $Γ$ . Then every cell of a bounded computation tableau for $N$ ranges over the constant-size set $Δ_{N} : = Γ \cup (Q \times Γ) .$

Facts & Assumptions

Given: A fixed machine $N$ with state set $Q$ and tape alphabet $Γ$ .

[L1]

In a bounded computation tableau, each row is a configuration and each row has exactly one state-tagged tape symbol marking the head position, by A bounded computation tableau for a nondeterministic Turing-machine run.

Proof

technique · direct

1.1

A tableau cell either lies away from the head, in which case it stores only a tape symbol from $Γ$ , or it is the unique scanned cell of that row, in which case it must record both the scanned tape symbol and the current state, hence an element of $Q \times Γ$ . By [L1], these are the only possibilities.

L1given

2.1

Therefore every cell value lies in $Δ_{N} = Γ \cup (Q \times Γ)$ . Because the machine $N$ is fixed before any input is considered, both $Q$ and $Γ$ are fixed finite sets, so $∣ Δ_{N} ∣ = ∣ Γ ∣ + ∣ Q ∣ ∣ Γ ∣$ is a constant independent of the input length.

step 1.1∎

Depends on

A bounded computation tableau for a nondeterministic Turing-machine run

Used by

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

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