Optimization and Implementation of Subsumption Algorithms for the Description Logic El with Cyclic Tboxes and General Concept Inclusion Axioms

The subsumption problem in the description logic (DL) EL has been shown to be polynomial regardless of whether cyclic or acyclic TBoxes are used. Recently, it was shown that the problem remains tractable even when admitting general concept inclusion (GCI) axioms. Motivated by its nice complexity and sufficient expressiveness for some applications, we propose three decision procedures for computing subsumption in the DL EL whose run-time is bounded by a lowdegree polynomial. The three decision procedures are for three terminological settings in EL: TBoxes with greatest fixpoint semantics (EL), TBoxes with descriptive semantics (EL), and terminologies with GCIs (EL). For subsumption w.r.t. TBoxes, i.e., EL and EL, we use a characterization through simulations on so-called EL-description graphs—the syntactically normalized representation of EL-TBoxes. With an efficient algorithm for computing simulations on graphs, we show that EL-subsumption can be decided in time cubic in the size of the input TBox. We decide EL-subsumption by reducing the simulation problem on graphs to the satisfiability problem of Horn formulae. Then, we apply a linear-time Horn-SAT algorithm to our Horn formulae. This approach yields a quartic-time decision procedure for EL-subsumption. Concerning terminologies with GCIs, we employ a different normalization and characterize subsumption through so-called implication sets. We show that ELsubsumption can be decided in time cubic in the size of the input terminology, by translating the implication sets into a Horn formula and exploiting the linear-time Horn-SAT algorithm similarly to EL. Besides, we implement these decision procedures in the Common LISP language and evaluate their efficiency using the Gene Ontology as a benchmark. The implementation can be used as terminological reasoners that classify ontologies represented in EL-TBoxes.