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Probabilistic Automatic Complexity Is At Most Three

Gill introduced the probabilistic automatic complexity $A_P(w)$ of a string: the least number of states of a probabilistic finite automaton for which $w$ is the unique most probably accepted string of its length. He asked whether $A_P$ is unbounded, no string with $A_P>3$ being known. The paper proves $A_P(w)\le 3$ for every string over every finite alphabet, with an explicit three-state witness.

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theoretical-computer-scienceJul 28, 2026Significance 7/100Registry: unreviewed

Probabilistic Automatic Complexity Is At Most Three

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Gill introduced the probabilistic automatic complexity $A_P(w)$ of a string: the least number of states of a probabilistic finite automaton for which $w$ is the unique most probably accepted string of its length. He asked whether $A_P$ is unbounded, no string with $A_P>3$ being known. The paper proves $A_P(w)\le 3$ for every string over every finite alphabet, with an explicit three-state witness.

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Gill introduced the probabilistic automatic complexity $A_P(w)$ of a string: the least number of states of a probabilistic finite automaton for which $w$ is the unique most probably accepted string of its length. He asked whether $A_P$ is unbounded, no string with $A_P>3$ being known. The paper proves $A_P(w)\le 3$ for every string over every finite alphabet, with an explicit three-state witness.

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Probabilistic Automatic Complexity Is At Most Three — Mathematical Frontier Network