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 *H* ← *B*_{1}, …, *B*_{n} (n ≥ 0) , where *H*, *B*_{1}, …, *B*_{n} are atoms, *H* is called the head (conclusion or consequent) and *B*_{1}, …, *B*_{n} 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 *N _{P}* be a set of predicate symbols such that

*NC*∪

*N*∪

_{Ra}*N*⊆

_{Rc}*N*. A DL-atom is an atom of the form

_{P}*A*(

*s*), where

*A*∈

*N*, or of the form

_{C}*r*(

*s*,

*t*), where

*R*∈

*N*∪

_{Ra}*N*. A rule

_{Rc}*R*is called DL-safe if each variable in

*r*occurs in a non-DL-atom in the rule body.