Extending Description Logic EL++ with Linear Constraints on the Probability of Axioms

TL;DR

This paper introduces a probabilistic extension to EL++ description logic by assigning probabilities to axioms, providing an NP-complete consistency detection method and algorithms for inferring probability bounds, thus balancing expressivity and computational complexity.

Contribution

It presents a novel probabilistic extension of EL++ with linear algebraic algorithms for consistency checking and probability inference, maintaining polynomial-time properties.

Findings

Consistency detection is NP-complete.

A linear algebraic deterministic algorithm is proposed.

Algorithms for inferring probability bounds are developed.

Abstract

One of the main reasons to employ a description logic such as EL or EL++ is the fact that it has efficient, polynomial-time algorithmic properties such as deciding consistency and inferring subsumption. However, simply by adding negation of concepts to it, we obtain the expressivity of description logics whose decision procedure is {ExpTime}-complete. Similar complexity explosion occurs if we add probability assignments on concepts. To lower the resulting complexity, we instead concentrate on assigning probabilities to Axioms (GCIs). We show that the consistency detection problem for such a probabilistic description logic is NP-complete, and present a linear algebraic deterministic algorithm to solve it, using the column generation technique. We also examine and provide algorithms for the probabilistic extension problem, which consists of inferring the minimum and maximum probabilities for a new axiom, given a consistent probabilistic knowledge base.