DL-Safe Rules

While SROIQ(D), the description logic of OWL 2 DL, is very expressive, it can only express axioms of a certain tree structure, because OWL 2 DL corresponds to a decidable subset of first-order predicate logic. There are decidable rule-based formalisms, such as function-free Horn rules, which are not restricted in this regard.

Definition 1 (Rule). A rule R is given as HB1, …, Bn (n ≥ 0) , where H, B1, …, Bn are atoms, H is called the head (conclusion or consequent) and B1, …, Bn the body (premise or antecedent).

While some OWL 2 axioms correspond to rules, such as class inclusion and property inclusion, some classes can be decomposed as rules, and property chain axioms provide rule-like axioms, there are rules that cannot be expressed in OWL 2 rules. For example, a rule head with two variables cannot be represented as a subclass axiom, or a rule body that contains a class expression cannot be described by a subproperty axiom. To add the additional expressivity of rules to OWL 2 DL, ontologies can be extended with SWRL rules which, however, make ontologies undecidable. The solution is to apply DL-safe rules, wherein each variable must occur in a non-DL-atom in the rule body, i.e., DL-safe rules are SWRL rules restricted to known individuals. DL-safe rules are very expressive and decidable at the same time.

Definition 2 (DL-Safe Rule). Let KB be a SROIQ(D) knowledge base, and let NP be a set of predicate symbols such that NCNRaNRcNP. A DL-atom is an atom of the form A(s), where ANC, or of the form r(s, t), where RNRaNRc. A rule R is called DL-safe if each variable in r occurs in a non-DL-atom in the rule body.