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[#P2834] Logarithmic DFA separation of binary words

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A flat mathematical diagram showing two binary words following different paths through a DFA.
A schematic view of two binary words following different paths through a DFA.

Problem. For distinct words \(x,y\in\{0,1\}^n\), let \(s(x,y)\) be the minimum number of states in a deterministic finite automaton that accepts exactly one of \(x,y\), and let \(S(n)=\max_{x\ne y}s(x,y)\). Is \(S(n)=O(\log n)\)?

1Context

The problem asks how little finite memory suffices to distinguish two known inputs. Modular position counts and short pattern statistics can be stored as independent separators, while hard pairs record which entire classes of small automata fail.

2Problem setup

Definition 1 (A deterministic finite automaton has a finite state set, one start state, a set of accepting states, and one outgoing transition for each state and input symbol). A deterministic finite automaton has a finite state set, one start state, a set of accepting states, and one outgoing transition for each state and input symbol.

Definition 2 (An automaton separates \(x\) and \(y\) when it accepts one word and rejects the other). An automaton separates \(x\) and \(y\) when it accepts one word and rejects the other.

Definition 3 (The assertion \(S(n)=O(\log n)\). The assertion \(S(n)=O(\log n)\) means that an absolute constant \(C\) exists with \(S(n)\le C\log n\) for every sufficiently large \(n\).

Remark 1. The problem asks how little finite memory suffices to distinguish two known inputs. Modular position counts and short pattern statistics can be stored as independent separators, while hard pairs record which entire classes of small automata fail.

3What counts as a solution

  • Prove that \(s(x,y)\le C\log n\) for an absolute \(C\), every sufficiently large \(n\), and all distinct binary \(n\)-letter words, or give infinitely many pairs with \(s(x,y)/\log n\) unbounded.

1Status

Current status (Current status and unresolved remainder). UNKNOWN as of 2026-07-31. A March 2025 primary preprint proves \(S(n)=O(\log^2 n)\). The checked primary literature and subsequent searches do not prove the logarithmic bound or a superlogarithmic lower bound. Prove that \(s(x,y)\le C\log n\) for an absolute \(C\), every sufficiently large \(n\), and all distinct binary \(n\)-letter words, or give infinitely many pairs with \(s(x,y)/\log n\) unbounded.[1]

1Packet records

2 records

RecordKindAssessment
ResultSupported
TraceReported
Notes and companion materialContext, examples, and computations

Original intake status. UNKNOWN as of 2026-07-31. A March 2025 primary preprint proves \(S(n)=O(\log^2 n)\). The checked primary literature and subsequent searches do not prove the logarithmic bound or a superlogarithmic lower bound.

  • 2026-07-27 status search checked the 2011 separating-words survey, the 2020 bound, the March 2025 polylogarithmic improvement, and 2026 work on minimal DFA witnesses. No source checked settles \(S(n)=O(\log n)\).
  • The strongest neighboring result is the 2025 \(O(\log^2 n)\) construction, improving an earlier \(O(n^{1/3}\log^7 n)\) bound.
  • A reusable positive artifact should specify the distinguishing statistic and compile it into a DFA with an explicit state count. A lower-bound artifact should give hard word pairs and an equivalence relation respected by every smaller DFA.

Recorded example 1. For \(x=0\) and \(y=1\), a two-state DFA separates the words. For arbitrary \(x\ne y\) of length \(n\), a DFA recognizing the singleton language \(\{x\}\) gives the elementary bound \(s(x,y)\le n+2\).

How the 2 records connectTyped relations and evidence flow
How the records connect to the problem

ProblemLogarithmic DFA separation of binary words

  1. Claim 1Current status and unresolved remainderin this packetSupported
  2. Route 1Resolve the stated acceptance conditionaddressesReported

2See also

How to cite

TheoremDB contributors, “Logarithmic DFA separation of binary words,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/binary-word-logarithmic-dfa-separation

This page as plain text: binary-word-logarithmic-dfa-separation.md

This problem includes 2 records joined by 1 typed links, sourced from arxiv.org[1], current as of July 31, 2026.

1References

  1. Packet source. A Sharper Upper Bound for the Separating Words Problem, source checked for the TheoremDB status review (2026-07-31). The preprint proves the current polylogarithmic upper bound; the logarithmic target is the standard remaining separating-words question. This CC0 self-contained record was prepared on 2026-07-27. ↗preprint · primary source · arXiv:2503.23184, checked 2026-07-31 · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. A March 2025 primary preprint proves \(S(n)=O(\log^2 n)\). The checked primary literature and subsequent searches do not prove the logarithmic bound or a superlogarithmic lower bound.Also cited at See dataset.references[0] for the exact external source and locator.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Source named by the research packet.
  2. Zachary Chase, “A new upper bound for separating words”. arXiv:2007.12097 (2020). Status evidence identified in the source record and checked at the linked publication. ↗preprint · primary source · arXiv:2007.12097, checked 2026-07-31 · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. A March 2025 primary preprint proves \(S(n)=O(\log^2 n)\). The checked primary literature and subsequent searches do not prove the logarithmic bound or a superlogarithmic lower bound.Also cited at Full preprint relevant to Logarithmic DFA separation of binary words.Source used to assess the problem's recorded status.For Logarithmic DFA separation of binary words: UNKNOWN as of 2026-07-27. A March 2025 primary preprint proves \(S(n)=O(\log^2 n)\). The checked primary literature and subsequent searches do not prove the logarithmic bound or a superlogarithmic lower bound.
  3. Erik D. Demaine, Sarah Eisenstat, Jeffrey Shallit, and David A. Wilson, “Remarks on separating words”. arXiv:1103.4513 (2011). Status evidence identified in the source record and checked at the linked publication. ↗preprint · primary source · arXiv:1103.4513, checked 2026-07-31 · checked 2026-07-31Source use: original summary.UNKNOWN as of 2026-07-27. A March 2025 primary preprint proves \(S(n)=O(\log^2 n)\). The checked primary literature and subsequent searches do not prove the logarithmic bound or a superlogarithmic lower bound.Also cited at Full preprint relevant to Logarithmic DFA separation of binary words.Source used to assess the problem's recorded status.For Logarithmic DFA separation of binary words: UNKNOWN as of 2026-07-27. A March 2025 primary preprint proves \(S(n)=O(\log^2 n)\). The checked primary literature and subsequent searches do not prove the logarithmic bound or a superlogarithmic lower bound.

Original CC0 record prose for the logarithmic separating-words problem.

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