We study functional dependencies together with two different probabilistic dependency notions: unary marginal identity and unary marginal distribution equivalence. A unary marginal identity states that two variables x and y are identically distributed. A unary marginal distribution equivalence states that the multiset consisting of the marginal probabilities of all the values for variable x is the same as the corresponding multiset for y. We present a sound and complete axiomatization for the class of these dependencies and show that it has Armstrong relations. The axiomatization is infinite, but we show that there can be no finite axiomatization. The implication problem for the subclass that contains only functional dependencies and unary marginal identities can be simulated with functional dependencies and unary inclusion atoms, and therefore the problem is in polynomial-time. This complexity bound also holds in the case of the full class, which we show by constructing a polynomial-time algorithm.

翻译：我们研究了函数依赖与两种不同的概率依赖概念：一元边际同一性（unary marginal identity）和一元边际分布等价性（unary marginal distribution equivalence）。一元边际同一性指出两个变量x和y具有相同的分布。一元边际分布等价性指出变量x所有取值的边际概率构成的多重集与y对应的多重集相同。我们为这类依赖关系给出了一个完备且有穷公理系统，并证明其具有Armstrong关系。该公理系统是无穷的，但我们证明不存在有穷公理化方法。仅包含函数依赖和一元边际同一性的子类的蕴含问题可通过函数依赖与一元包含原子进行模拟，因此该问题可在多项式时间内解决。这一复杂度界对完整类同样成立，我们通过构造一个多项式时间算法证明了这一点。