A New Tractable Description Logic under Categorical Semantics

Biomedical ontologies contain numerous concept or role names involving negative knowledge such as lacks_part, absence_of. Such a representation with labels rather than logical constructors would not allow a reasoner to interpret lacks_part as a kind of negation of has_part. It is known that adding negation to the tractable Description Logic (DL) EL allowing for conjunction, existential restriction and concept inclusion makes it intractable since the obtained logic includes implicitly disjunction and universal restriction which interact with other constructors. In this paper, we propose a new extension of EL with a weakened negation allowing to represent negative knowledge while retaining tractability. To this end, we introduce categorical semantics of all logical constructors of the DL SH including EL with disjunction, negation, universal restriction, role inclusion and transitive roles. The categorical semantics of a logical constructor is usually described as a set of categorical properties referring to several objects without using set membership. To restore tractability, we have to weaken semantics of disjunction and universal restriction by identifying \emph{independent} categorical properties that are responsible for intractability, and dropping them from the set of categorical properties. We show that the logic resulting from weakening semantics is more expressive than EL with the bottom concept, transitive roles and role inclusion.

Reasoning in the Description Logic ALC under Category Semantics

We present in this paper a reformulation of the usual set-theoretical semantics of the description logic $\mathcal{ALC}$ with general TBoxes by using categorical language. In this setting, $\mathcal{ALC}$ concepts are represented as objects, concept subsumptions as arrows, and memberships as logical quantifiers over objects and arrows of categories. Such a category-based semantics provides a more modular representation of the semantics of $\mathcal{ALC}$. This feature allows us to define a sublogic of $\mathcal{ALC}$ by dropping the interaction between existential and universal restrictions, which would be responsible for an exponential complexity in space. Such a sublogic is undefinable in the usual set-theoretical semantics, We show that this sublogic is {\sc{PSPACE}} by proposing a deterministic algorithm for checking concept satisfiability which runs in polynomial space.