DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 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 quantified formula game for a quantified Boolean formula

Definition

Let

$Φ = Q_{1} x_{1} Q_{2} x_{2} \dots Q_{m} x_{m} ψ (x_{1}, \dots, x_{m})$

be a quantified Boolean formula. Its quantified formula game is the two-player perfect-information game in which:

whenever $Q_{i} = \exists$ , Player E chooses the value of $x_{i}$ ;
whenever $Q_{i} = \forall$ , Player A chooses the value of $x_{i}$ .

After all variables are assigned, the quantifier-free matrix $ψ$ is evaluated. Player E wins if $ψ$ is true under the chosen assignment, and Player A wins otherwise.

Remarks

The existential player tries to force truth of the formula.
The universal player tries to force falsity.

Depends on

Quantified Boolean formulas and the language TQBF

Used by

Dependency tree · one level

1 result within one dependency step 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

Sanjeev Arora and Boaz Barak, Computational Complexity: A Modern Approach (standard reference, not scraped)