Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passjudge 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.

FALSE: acceptance by final state and by empty stack are literally identical

Statement

False claim: for a fixed PDA, acceptance by final state and acceptance by empty stack are the same condition with no machine conversion.

Facts & Assumptions

Given: A PDA whose start state q0 is accepting, whose initial stack symbol is Z0, and which has no transitions at all.

[A1]

The statement refuted is: for one fixed PDA, final-state acceptance and empty-stack acceptance are literally identical.

[L1]

By Acceptance by final state for a PDA, the empty word is accepted by final state whenever the start state is already accepting.

[L2]

By Acceptance by empty stack for a PDA, empty-stack acceptance requires a computation that consumes the input and empties the stack.

[L3]

By Acceptance by final state and by empty stack are equivalent for PDAs, every language accepted in either mode is accepted by some PDA in the other mode; this language-class equivalence does not assert that the two modes agree for one fixed PDA.

Refutation

technique · direct
1.1

In the displayed PDA, the empty word is accepted by final state in zero moves because the start state is accepting.

L1given
2.1

The same machine does not accept the empty word by empty stack, because the stack begins with Z0 and there is no transition that removes it. So the required empty-stack configuration never occurs.

L2step 1.1
3.1

Thus the two acceptance notions differ on this fixed machine, contradicting [A1]. The correct general statement is the language-class equivalence from [L3].

A1L3step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

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