Alphabeta MathHow 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 machine with two branches shows that one satisfying tableau need not encode every branch
Statement refuted
The Cook-Levin reduction works by explicitly enumerating every nondeterministic branch of the machine.
Facts & Assumptions
Given: A nondeterministic machine that, on the empty input, branches immediately into a left branch that accepts in one step and a right branch that rejects in one step.
A bounded computation tableau records one branch rather than the whole nondeterministic tree, by A bounded computation tableau for a nondeterministic Turing-machine run.
The false slogan being refuted is the one recorded in FALSE: the Cook-Levin reduction enumerates all nondeterministic branches.
The Cook-Levin formula is assembled from one fixed family of cell constraints and from start, accept, and local-window constraints, by The exactly-one-symbol constraints have polynomial size and The start, accept, and transition constraints have polynomial size.
Counterexample
The formula for is produced from the fixed constraint families in [L3] by iterating over tableau cells and local windows. This construction never generates either computation branch.
The left branch has an accepting tableau of constant size, so assigning the tableau variables according to that branch satisfies the formula. By [L1], this assignment encodes only the accepting branch; the rejecting branch is not encoded by it. Together with the branch-free construction in step 1.1, this refutes the enumeration slogan [L2].
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
11 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)