eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2020-08-26
41:1
41:19
10.4230/LIPIcs.CONCUR.2020.41
article
The Big-O Problem for Labelled Markov Chains and Weighted Automata
Chistikov, Dmitry
1
https://orcid.org/0000-0001-9055-918X
Kiefer, Stefan
2
Murawski, Andrzej S.
2
Purser, David
3
4
https://orcid.org/0000-0003-0394-1634
Centre for Discrete Mathematics and its Applications (DIMAP) and Department of Computer Science, University of Warwick, Coventry, UK
Department of Computer Science, University of Oxford, UK
Centre for Discrete Mathematics and its Applications (DIMAP) and, Department of Computer Science, University of Warwick, Coventry, UK
Max Planck Institute for Software Systems, Saarland Informatics Campus, Saarbrücken, Germany
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).
https://drops.dagstuhl.de/storage/00lipics/lipics-vol171-concur2020/LIPIcs.CONCUR.2020.41/LIPIcs.CONCUR.2020.41.pdf
weighted automata
labelled Markov chains
probabilistic systems