Why Rules Are Complex: Real-Valued Probabilistic Logic Programs Are Not Fully Expressive

Abstract

This paper explores what can and cannot be represented by probabilistic logic programs (PLPs). Propositional PLPs can represent any distribution because they can be acyclic. For relational domains with fixed populations, the probabilistic parameters can be derived as the solutions to polynomial equations. Unfortunately, sometimes they only have complex-valued solutions. Thus PLPs, even with arbitrarily real-valued parameters, cannot represent all distributions. Moreover, they cannot approximate all distributions. Allowing the parameters to be complex numbers, we present a natural truly-cyclic canonical representation that with probability 1 can represent all distributions for propositional or relational domains with fixed populations, and, unlike standard representations, has no redundant parameters.