Every string has probabilistic automatic complexity at most three

2026-07-28

AI Summary

This study resolves the open question of whether probabilistic automatic complexity is unbounded. By explicitly constructing a three-state probabilistic finite automaton, the authors demonstrate that the probabilistic automatic complexity of any finite string—including binary strings—is at most three, thereby establishing for the first time a universal upper bound for this complexity measure. The work integrates techniques from string encoding, dynamical system modeling, and acceptance functional design to not only fully classify the probabilistic automatic complexity of binary strings but also provide a general method for automaton construction. This result fundamentally settles the boundedness problem for probabilistic automatic complexity.

Abstract

Gill (arXiv:2402.13376) introduced the probabilistic automatic complexity $A_P(w)$ of a finite string $w$: the least number of states of a probabilistic finite automaton (PFA) for which $w$ is the unique most probably accepted string of its length. He asked whether $A_P$ is unbounded, noting that no string with $A_P > 3$ was known (Question 4.14 of that paper). We answer the question by proving that $A_P(w)\le 3$ for every string $w$ over every finite alphabet. The witnessing three-state automaton is explicit: its reduced dynamics tracks the pair $(u,u^2)$, where $u$ is the reversed base-$b$ value of the input, and its acceptance functional is a downward parabola peaked at the value of the target string. Combined with Gill's classification of the binary strings with $A_P=2$, this completely determines $A_P$ on binary strings.

Problem

• probabilistic automatic complexity
• finite automaton
• string complexity
• unboundedness
• most probable string

Innovation

• probabilistic automatic complexity
• probabilistic finite automaton
• string complexity
• automata theory
• computational complexity