论文标题
Big-O问题
The Big-O Problem
论文作者
论文摘要
给定两个加权自动机，我们考虑一个问题是一个问题是另一个是另一个的大o，即，如果第一个中每个有限词的权重不大于第二个重量的常数倍数。 我们表明，即使是将加权自动机的实例化为标记的马尔可夫链，问题是不可决定的。此外，即使知道一个加权自动机是另一个加权自动机，查找或近似相关常数的问题也是不可确定的。 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 $）或自动机具有有限的歧义。 在标记的马尔可夫连锁店上，可以将问题作为比率总变化距离来重述，该距离没有找到任何两个事件的概率之间的最大差异，而是发现任何两个事件的概率之间的最大比率。该问题与$ \ varepsilon $ -Differential隐私有关，为此，Big-O符号的最佳常数正好是$ \ exp（\ varepsilon）$。
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 $\exp(\varepsilon)$.
