Year: 2026
Type: preprint
Abstract: We consider the problem of separating words with deterministic finite automata (DFA) (Goral{č}{í}k and Koubek, 1986). This problem asks: given two distinct words $u,v$ of length at most $n$, what is the size of the smallest DFA that accepts one and rejects the other? The best upper bound on the worst-case over all pairs of words of length at most $n$ is $\tilde{O}(n^{1/3})$ states (Chase, 2021), while the best lower bound is $Ω(\log n)$. In this work, we consider the half-random, half-adversarial case: we show that if $u$ is a uniformly random binary word of length $n$, then with high probability, for any word $v$ not equal to $u$, there is a DFA with $O(\log^{7/3} n \mathrm{poly}\log\log n)$ states that separates $u$ and $v$. Our results are based on a novel analysis that exploits the structural sparsity of random words: we show how to apply block-wise compaction with small deterministic transducers to reduce the separation problem to the case of words with short run-length encodings.
Source: arXiv (Cornell University)
Author: Bathie, Gabriel
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204800399
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