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# The Big-O Problem for Labelled Markov Chains and Weighted Automata

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LIPIcs.CONCUR.2020.41.pdf
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## Acknowledgements

The authors would like to thank to Engel Lefaucheux, Joël Ouaknine, and James Worrell for discussions during the development of this work.

## Cite As

Dmitry Chistikov, Stefan Kiefer, Andrzej S. Murawski, and David Purser. The Big-O Problem for Labelled Markov Chains and Weighted Automata. In 31st International Conference on Concurrency Theory (CONCUR 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 171, pp. 41:1-41:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)
https://doi.org/10.4230/LIPIcs.CONCUR.2020.41

## Abstract

Given two weighted automata, we consider the problem of whether one is big-O of the other, i.e., if the weight of every finite word in the first is not greater than some constant multiple of the weight in the second. We show that the problem is undecidable, even for the instantiation of weighted automata as labelled Markov chains. Moreover, even when it is known that one weighted automaton is big-O of another, the problem of finding or approximating the associated constant is also undecidable. Our positive results show that the big-O problem is polynomial-time solvable for unambiguous automata, coNP-complete for unlabelled weighted automata (i.e., when the alphabet is a single character) and decidable, subject to Schanuel’s conjecture, when the language is bounded (i.e., a subset of w_1^* … w_m^* for some finite words w_1,… ,w_m). On labelled Markov chains, the problem can be restated as a ratio total variation distance, which, instead of finding the maximum difference between the probabilities of any two events, finds the maximum ratio between the probabilities of any two events. The problem is related to ε-differential privacy, for which the optimal constant of the big-O notation is exactly exp(ε).

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Probabilistic computation
##### Keywords
• weighted automata
• labelled Markov chains
• probabilistic systems

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