Alphabeta Math
CounterexampleConstruction: AI-generatedVerification: 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.

A looping DPDA shows naive final-state complementation can fail

Statement refuted

The statement "swapping the accepting states of a DPDA automatically complements its language" is false.

Facts & Assumptions

Given: The one-state ε-loop DPDA from FALSE: swapping the accepting states of a DPDA automatically complements its language.

[A1]

The statement refuted is: swapping the accepting states of a DPDA automatically complements its language.

[L1]

The cited false statement already shows that the original looping DPDA and its state-swapped variant both recognize the empty language, by FALSE: swapping the accepting states of a DPDA automatically complements its language.

Counterexample

technique · direct
1.1

The original machine recognizes ∅, because it loops forever on every marked input. After swapping the accepting status of its only state, the machine still has exactly the same infinite computation on every marked input and still recognizes ∅.

L1given
2.1

But the complement of ∅ is the full language Σ∗, so the swapped machine does not recognize the complement of the original language.

step 1.1
3.1

Therefore this looping DPDA is a counterexample to [A1].

A1step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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