The Big-O Problem

TL;DR

This paper investigates the computational complexity of the big-O problem for weighted automata, revealing undecidability in general but identifying specific cases where it is polynomial-time solvable or decidable under certain conjectures.

Contribution

It establishes the undecidability of the big-O problem for weighted automata and provides complexity results for special automaton classes, including unambiguous and bounded languages.

Findings

Undecidable in general for weighted automata.

Polynomial-time solvable for unambiguous automata.

Decidable under Schanuel's conjecture for bounded languages.

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 $w_1^*\dots w_m^*$ for some finite words $w_1,\dots,w_m$) or when the automaton has finite ambiguity. 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 $\varepsilon$-differential privacy, for which the optimal constant of the big-O notation is exactly $\exp(\varepsilon)$.