Frontiers for propositional reasoning about fragments of probabilistic conditional independence and hierarchical database decompositions

Conditional independence provides an essential framework to deal with knowledge and uncertainty in Artificial Intelligence, and is fundamental in probability and multivariate statistics. Its associated implication problem is paramount for building Bayesian networks. Unfortunately, the problem does not enjoy an axiomatization, by a finite set of Horn rules, and is already coNP-complete to decide for some fragments of conditional independencies. Saturated conditional independencies form an important fragment whose implication problem is decidable in almost linear time. We establish an axiomatization, by a finite set of Horn rules, for the fragment of generalized saturated conditional independencies. These state the conditional independence between finitely many sets of random variables. The special case of two sets captures Geiger and Pearl's finite axiomatization for saturated conditional independencies. Even for this special case, our completeness proof is new. The proof argument utilizes special probability models where two events have probability one half. Special probability models allow us to establish an equivalence between the implication of generalized saturated conditional independencies and that of formulae in a Boolean propositional fragment. This duality is then extended to a trinity including the implication problem of Delobel's full first-order hierarchical database decompositions. Already for the independence between two sets of random variables, the dualities cannot be extended to cover conditional independencies in general, or first-order hierarchical decompositions.