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Polynomial complementation of two-way nondeterministic finite automata

Theorem 1.1: for every n≥4n\ge4 there is an nn-state 2NFA AnA_n for relation-product liveness on n−2n-2 points such that every 2NFA for its complement has at least 122⌊(n−4)/127⌋−1\tfrac12 2^{\lfloor(n-4)/127\rfloor}-1 states. Corollary: every 2DFA for the same language needs 122⌊(n−4)/127⌋\tfrac12 2^{\lfloor(n-4)/127\rfloor} states when n≥131n\ge131. The alphabet has 2(n−2)22^{(n-2)^2} letters, so the result says nothing about complementation over a fixed alphabet, about transition-table size, or about uniform logarithmic-space classes.

Exact FrontierDelta

Prior state unknown→disproved

Scope and record

Occurred: Sep 25, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: lean-checked. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Polynomial complementation of two-way nondeterministic finite automata · 2NFA complementation

Confidence: Not scored

Registry verification: lean checked · announcement · candidate

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Attribution

VibeMathed
registry · event recorded by

Unreleased internal OpenAI model
model · ai model contributor · OpenAI

Artifacts and verifiers

Lean proof (OAI.TwoWayComplementation.complementation_lower_bound_finite)

lean artifact · pending

Artifact ↗
Comparator statement: TwoWayComplementation.lean

lean artifact · pending

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Compute record

No linked compute attempts recorded.

Lineage and corrections

Polynomial complementation of two-way nondeterministic finite automata parent of this event

This event attributed to Unreleased internal OpenAI model

Lean proof (OAI.TwoWayComplementation.complementation_lower_bound_finite) evidence for this event

Companion: An exponential two-way deterministic state lower bound for one-way liveness evidence for this event

Theorem 1.1: for every n≥4n\ge4 there is an nn-state 2NFA AnA_n for relation-product liveness on n−2n-2 points such that every 2NFA for its complement has at least 122⌊(n−4)/127⌋−1\tfrac12 2^{\lfloor(n-4)/127\rfloor}-1 states. Corollary: every 2DFA for the same language needs 122⌊(n−4)/127⌋\tfrac12 2^{\lfloor(n-4)/127\rfloor} states when n≥131n\ge131. The alphabet has 2(n−2)22^{(n-2)^2} letters, so the result says nothing about complementation over a fixed alphabet, about transition-table size, or about uniform logarithmic-space classes. parent of this event

Comparator statement: TwoWayComplementation.lean evidence for this event

Polynomial complementation of two-way nondeterministic finite automata evidence for this event

VibeMathed record: Polynomial complementation of two-way nondeterministic finite automata evidence for this event

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Polynomial complementation of two-way nondeterministic finite automata — Mathematical Frontier Network