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arXiv · 2402.13376

Probabilistic automatic complexity of finite strings

Kenneth Gill
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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 $A_{P,δ}(x)$ adds a real-valued parameter $δ$ specifying a required lower bound on the gap in acceptance probabilities between $x$ and other strings. We prove $A_{P,δ}$ is $δ$-computable for all $δ$, relate $A_P$ to the DFA and NFA complexities, and obtain a complete classification of binary strings with $A_P=2$. Finally, we discuss several other variations on $A_P$ with a view to obtaining additional desirable properties.

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Kenneth Gill. 2024-06-03. Probabilistic automatic complexity of finite strings. https://arxiv.org/abs/2402.13376

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