Negation Without Negation in Probabilistic Logic Programming

Probabilistic logic programs without negation can have cycles (with a preference for false), but cannot represent all conditional distributions. Probabilistic logic programs with negation can represent arbitrary conditional probabilities, but with cycles they create logical inconsistencies. We show how allowing negative noise probabilities allows us to represent arbitrary conditional probabilities without negations. Noise probabilities for non-exclusive rules are difficult to interpret and unintuitive to manipulate; to alleviate this we define “probability-strengths” which provide an intuitive additive algebra for combining rules. For acyclic programs we prove what constraints on the strengths allow for proper distributions on the non-noise variables and allow for all non-extreme distributions to be represented. We show how arbitrary CPDs can be converted into this form in a canonical way. Furthermore, if a joint distribution can be compactly represented by a cyclic program with negations, we show how it can also be compactly represented with negative noise probabilities and no negations. This allows algorithms for exact inference that do not support negations to be applicable to probabilistic logic programs with negations.