OpenAI research collection · 2 manuscripts · Source checked 2026-10-19
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.
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We prove that two-way nondeterministic finite automata cannot be complemented with a polynomial number of states independent of the alphabet. For each n ≥ 4 we construct an n-state automaton over a finite alphabet whose complement requires at least \tfrac12 2^{\lfloor(n-4)/127\rfloor}-1 states.
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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 h+3 states and no left moves, whereas every equivalent s-state two-way deterministic automaton satisfies 4(s+2)^2\ge2^{\lfloor(h-2)/31\rfloor}. Partial transition…
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 h+3 states and no left moves, whereas every equivalent s-state two-way deterministic automaton satisfies 4(s+2)^2\ge2^{\lfloor(h-2)/31\rfloor}. 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.
This PDF is listed in OpenAI's formalization catalogue. See the scope and dependencies before interpreting coverage.
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