In this article we will make a study around the complexity of reasoning for the satisfiability and subsumption problems in some of the most important languages of the DL family. We will interpret those limits as coming from various sources of complexities that we will isolate. Will be analyzed reasoning both with simple concept descriptions and with respect to a TBox, and will be discussed the complexity of instance checking of the ABox. We will show how a tableaux algorithm actually works in order to decide the satisfiability problem in a concrete DL language using the transformation rules to expand the concept descriptions, then we will expand it for other more complex tasks stated above. Finally we will present a list of complexity results in some of the most known DLs from the literature together with their proof, as it has been found by the scholars that studied them. For those who want introductory material into the Description Logics (DL) domain are invited to see our previous 2 articles where we first introduced the domain as a basis knowledge representation mechanism for ontology languages on the Semantic Web [24], and in the second the most important tasks that these formalisms must be able to do, those of inference and reasoning over the knowledge bases [25].

Logical representation formalisms, Description Logics, Tableaux algorithms, Reasoning