Result 134, Theoretical computer science

Generalized star height at most three

Every regular language over a finite alphabet has a generalized regular expression with at most three nested Kleene stars, allowing union, concatenation and complement over the same alphabet. This establishes an absolute bound independent of automaton size, resolving the uniform-boundedness version of the generalized star-height problem.

Lean formalization Proof

The bigger picture

Why it matters

An unreviewed manuscript reports that every language recognizable by a finite-state machine can be described with at most three nested repetition operators, if complement is allowed. The bound does not depend on the machine's size.

What changes?

A regular language is a set of finite words recognized by a finite-state machine. Generalized regular expressions describe such sets using union, concatenation, repetition by Kleene star, and complement. Star height counts the deepest nesting of stars, not their total number. The latest manuscript reports a bound of three for every regular language over any finite alphabet, improving the earlier reported bounds of thirteen and four. The alphabet stays unchanged, and complement means all finite words over that same alphabet.

What does that help mathematicians do?

If established, this rules out regular languages that require arbitrarily deep nested repetition when complement is available, settling the uniform-boundedness question. Researchers could therefore separate a machine's potentially large number of states from the repetition depth needed to describe its language. The claim does not establish that three is necessary, nor does it bound the total length of the resulting expression.

Are there practical applications?

The immediate value is foundational: the reported construction connects finite-monoid computations, algebraic summaries of how word pieces combine, with expressions of bounded repetition depth. It uses a prefix code of word pieces. The abstracts give no expression-size or running-time guarantees, so the result alone does not demonstrate faster pattern matching or a practical conversion method.

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.

3 manuscripts

Finite Monoid Computations and a Uniform Generalized Star-Height Bound

September 25, 2026 44 pages

Every regular language over a finite alphabet has a generalized regular expression of star height at most thirteen over that same alphabet. We prove this uniform bound by representing finite monoid computations as affine updates and recovering them through twelve successive split constructions.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026,
  author = {{OpenAI}},
  title = {{Finite Monoid Computations and a Uniform Generalized Star-Height Bound}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026/Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026.pdf}{OAI:Finite-Monoid-Computations-and-a-Uniform-Generalized-Star-Height-Bound-September-25-2026}},
  year = {2026}
}

Generalized Star Height at Most Four

September 25, 2026 34 pages

Every regular language over a finite alphabet has generalized star height at most four over that same alphabet. We give a complete construction using an affine correction that hides one interval product, a finite clock, and several scales for moving boundaries through periodic words.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Generalized-Star-Height-at-Most-Four-September-25-2026,
  author = {{OpenAI}},
  title = {{Generalized Star Height at Most Four}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Generalized-Star-Height-at-Most-Four-September-25-2026/Generalized-Star-Height-at-Most-Four-September-25-2026.pdf}{OAI:Generalized-Star-Height-at-Most-Four-September-25-2026}},
  year = {2026}
}

Generalized Star Height at Most Three

September 25, 2026 38 pages

Every regular language over a finite alphabet has generalized star height at most three, with complement taken in the same free monoid. We express finite-monoid computations using a prefix code of word pieces.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:Generalized-Star-Height-at-Most-Three-September-25-2026,
  author = {{OpenAI}},
  title = {{Generalized Star Height at Most Three}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Generalized-Star-Height-at-Most-Three-September-25-2026/article.pdf}{OAI:Generalized-Star-Height-at-Most-Three-September-25-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/134.md.

Generalized star height at most three

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

Scope

Generalized star height measures the nesting of Kleene stars in regular expressions that also allow Boolean operations. The formalization proves that every regular language over a finite alphabet has a generalized expression over the same alphabet of star height at most three. This is stronger than the bound of thirteen in the accompanying finite-monoid paper. The selected statement asserts the uniform expression bound, without separately encoding every construction step of that paper.

The formalization proves that every regular language over a finite alphabet has a generalized regular expression over that same alphabet with star height at most three. Generalized expressions allow Boolean operations as well as concatenation and Kleene star. The bound of three implies the accompanying paper's bound of four; the selected statement concerns the expression bound itself.

Generalized star height measures the nesting of Kleene stars in regular expressions that also allow Boolean operations. The formalization proves that every regular language over a finite alphabet has a generalized regular expression over that same alphabet of star height at most three. Complements are taken in the same free monoid. This is the paper's uniform bound.

Comparator links

Result Comparator statement
Uniform generalized star-height bound of three GeneralizedStarHeight.lean
Generalized star height at most three GeneralizedStarHeight.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 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.