Result 129, Theoretical computer science

Exponential state costs for two-way automata

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.

Lean formalization Proof

The bigger picture

Why it matters

Finite automata have finitely many internal states. These manuscripts claim that moving backward through an input cannot always cheaply replace nondeterminism, the ability to choose among possible moves, and that reversing acceptance can also require exponentially more states.

What changes?

The manuscripts report two barriers over growing finite alphabets. For every n >= 4, some n-state two-way nondeterministic automaton needs at least 2^floor((n-4)/127)/2 - 1 states to recognize its complement, the inputs it rejects. For every h >= 2, one-way liveness on h points has an (h+3)-state nondeterministic automaton, but every equivalent s-state two-way deterministic automaton satisfies 4(s+2)^2 >= 2^floor((h-2)/31). This simulation bound allows partial transition rules, stay moves and nonaccepting infinite computations.

What does that help mathematicians do?

One-way liveness asks whether a sequence of relations between h points permits any complete path through the sequence. The claimed bound makes this a concrete example where choosing moves nondeterministically is exponentially more state-efficient than deterministic computation, even with backward movement. Together, the results rule out polynomial state guarantees independent of alphabet size for both simulation and complementation. They do not settle the corresponding fixed-alphabet questions.

Are there practical applications?

The immediate value is foundational for automata theory and the study of memory requirements. States count a machine's internal memory configurations. These bounds identify unavoidable costs for general automaton transformations, rather than shortcomings of particular construction methods. They do not establish comparable costs for specific practical instances or measure implementation performance.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

An exponential state lower bound for two-way nondeterministic complementation

September 25, 2026 18 pages Main result formalized in Lean

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 122⌊(n−4)/127⌋−1\tfrac12 2^{\lfloor(n-4)/127\rfloor}-1 states.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:An-exponential-state-lower-bound-for-two-way-nondeterministic-complementation-September-25-2026,
  author = {{OpenAI}},
  title = {{An exponential state lower bound for two-way nondeterministic complementation}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-exponential-state-lower-bound-for-two-way-nondeterministic-complementation-September-25-2026/paper.pdf}{OAI:An-exponential-state-lower-bound-for-two-way-nondeterministic-complementation-September-25-2026}},
  year = {2026}
}

An exponential two-way deterministic state lower bound for one-way liveness

September 25, 2026 20 pages Main result formalized in Lean

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+3h+3 states and no left moves, whereas every equivalent s-state two-way deterministic automaton satisfies 4(s+2)2≥2⌊(h−2)/31⌋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.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:An-exponential-two-way-deterministic-state-lower-bound-for-one-way-liveness-September-25-2026,
  author = {{OpenAI}},
  title = {{An exponential two-way deterministic state lower bound for one-way liveness}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/An-exponential-two-way-deterministic-state-lower-bound-for-one-way-liveness-September-25-2026/main.pdf}{OAI:An-exponential-two-way-deterministic-state-lower-bound-for-one-way-liveness-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/129.md.

Exponential state costs for two-way automata

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The paper asks how many states are needed to complement or determinize two-way nondeterministic finite automata. The formalization gives, for every n≥4n\ge4, an explicit nn-state automaton whose complement requires at least 122⌊(n−4)/127⌋−1\tfrac12 2^{\lfloor(n-4)/127\rfloor}-1 states. For the same family of relational languages and every n≥131n\ge131, every equivalent deterministic two-way automaton requires at least 122⌊(n−4)/127⌋\tfrac12 2^{\lfloor(n-4)/127\rfloor} states.

The alphabet grows with nn, so neither complementation nor determinization has a polynomial state bound uniform over alphabets. The model has two endmarkers, left, right, and stay moves, and acceptance by a finite run. The separate results about one-way liveness and the Sakoda–Sipser problem are outside this scope.

The formalization gives a family of one-way nondeterministic automata with h+3h+3 states whose languages require exponentially many states for two-way deterministic recognition. For every h≥2h\ge2, a deterministic recognizer with ss states satisfies 2⌊(h−2)/31⌋≤4(s+2)22^{\lfloor(h-2)/31\rfloor}\le4(s+2)^2 under both acceptance conventions considered in the paper; one convention has the sharper factor 4(s+1)24(s+1)^2. The alphabet may grow with hh, so no alphabet-independent polynomial simulation bound holds.

Comparator links

Result Comparator statement
Two-way nondeterministic complementation lower bound TwoWayComplementation.lean
Same-family determinization lower bound TwoWayDeterminization.lean
Exponential deterministic state lower bound for liveness OneWayLiveness.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 7 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.