On Prototypical Indifference and Lifted Inference in Relational Probabilistic Conditional Logic

Semantics for formal models of probabilistic reasoning rely on probability functions that are defined on the interpretations of the underlying classical logic. When this underlying logic is of relational nature, i. e. a fragment of first-order logic, then the space needed for representing these probability functions explicitly is exponential in both the number of predicates and the number of domain elements. Consequently, probabilistic reasoning becomes a demanding task. Here, we investigate lifted inference in the context of explicit model representation with respect to an inference operator that satisfies prototypical indifference, i. e. an inference operator that is indifferent about individuals for which the same information is represented. As reasoning based on the principle of maximum entropy satisfies this property we exemplify our ideas by compactly characterizing the maximum entropy model of a probabilistic knowledge base in a relational probabilistic conditional logic. Our results show that lifted inference is no longer exponential in the number of domain elements when we restrict the language to unary predicates but is still infeasible for the general case.