3 research outputs found
The big-O problem for labelled markov chains and weighted automata
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)
Deciding Differential Privacy of Online Algorithms with Multiple Variables
We consider the problem of checking the differential privacy of online
randomized algorithms that process a stream of inputs and produce outputs
corresponding to each input. This paper generalizes an automaton model called
DiP automata (See arXiv:2104.14519) to describe such algorithms by allowing
multiple real-valued storage variables. A DiP automaton is a parametric
automaton whose behavior depends on the privacy budget . An automaton
will be said to be differentially private if, for some , the
automaton is -differentially private for all values of
. We identify a precise characterization of the class of all
differentially private DiP automata. We show that the problem of determining if
a given DiP automaton belongs to this class is PSPACE-complete. Our PSPACE
algorithm also computes a value for when the given automaton is
differentially private. The algorithm has been implemented, and experiments
demonstrating its effectiveness are presented
The Big-O problem for labelled Markov chains and weighted automata
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)