OpenAI · 129-02 · 2026-09-25 · Theoretical computer science
An exponential two-way deterministic state lower bound for one-way liveness
preprints/An-exponential-two-way-deterministic-state-lower-bound-for-one-way-liveness-September-25-2026
Family 129. Exponential state costs for two-way automata.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
Proves exponential lower bounds both for complementing two-way nondeterministic finite automata and for simulating one-way nondeterministic automata by two-way deterministic ones. The latter resolves the Sakoda–Sipser state-succinctness conjecture over growing finite alphabets; both results rule out polynomial state bounds independent of alphabet size.
One-way liveness on h points accepts a word of binary relations when their ordered product is nonempty. For every h ≥ 2, it has a nondeterministic automaton with states and no left moves, whereas every equivalent s-state two-way deterministic automaton satisfies . Partial transition rules, stay moves, and nonaccepting infinite computations are allowed. The alphabets are finite and grow with h, so the result rules out an alphabet-independent polynomial state bound for deterministic two-way simulation, already for one-way nondeterministic sources.
Preprint source and Lean scope
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This preprint is named in the Lean scope note for 129: Exponential state costs for two-way automata. The authors are OpenAI. This companion does not verify the formalization. It is not a premise. The OAI library is not imported. No compilation is asserted. O.1 stays NOT_CLOSED.