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Probabilistic automatic complexity of finite strings

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arXiv:2402.13376v2 Announce Type: replace-cross
Abstract: We introduce a new complexity measure for finite strings using probabilistic finite-state automata (PFAs), in the same spirit as existing notions employing DFAs and NFAs, and explore its properties. The PFA complexity $A_P(x)$ is the least number of states of a PFA for which $x$ is the most likely string of its length to be accepted. The variant $A_{P,\delta}(x)$ adds a real-valued parameter $\delta$ specifying a lower bound on the gap in acceptance probabilities between $x$ and other strings. We relate $A_P$ to the DFA and NFA complexities, obtain a complete classification of binary strings with $A_P=2$, and prove $A_{P,\delta}(x)$ is computable for every $x$ and for cofinitely many $\delta$ (depending on $x$). Finally, we discuss several other variations on $A_P$ with a view to obtaining additional desirable properties.

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