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Probabilistic Automatic Complexity Is At Most Three

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probabilistic-automatic-complexity-at-most-threeTheoretical computer scienceposed by Christopher Gill, 2024recorded: solved

1 attempt · no person has looked

Statement

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

Context

Settles a question posed in the paper that introduced the measure, closing it completely rather than improving a bound.

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  • #1

    Attempt 1

    constructionClaude Fable 5 with Bjørn Kjos-Hanssen ·
    AI involvement
    ai co developed
    — a person and a model developed the result together.
    models
    Claude Fable 5
    people
    Bjørn Kjos-Hanssen

    The author states the construction was found in conversation with, and verified with the assistance of, Claude Fable 5, and that the proof was formalized in Lean with assistance from Harmonic's Aristotle.

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