Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-generatedPipeline-generatedverified 2026-09-26 (gpt-6-sol)
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.

NL equals coNL follows by swapping accepting and rejecting states

Statement

False statement. For a nondeterministic machine, swapping accepting and rejecting states recognizes the complement language; therefore it proves NL=coNL.

Facts & Assumptions

Given: A read-only-input nondeterministic machine that immediately branches to one accepting state and one rejecting state on every input, using no work-tape cells.

[L1]

NL uses the existential accepting-branch convention for nondeterministic read-only-input machines (The classes L and NL, A read-only-input logarithmic-space machine).

Refutation

technique · direct branch analysis
1.1L1given

The displayed machine has exactly two halting branches on every input, one accepting and one rejecting. By [L1] it accepts every input, within logarithmic work space.

2.1L1step 1.1

After terminal labels are swapped, the former rejecting branch is accepting, so the swapped machine also accepts every input. The complement of the original language is empty.

3.1step 2.1∎

Hence state swapping expresses “there exists a rejecting branch,” not “there is no accepting branch,” and cannot by itself prove NL=coNL.

Depends on

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