Description Logic is a formalism that is widely used in the framework of Knowledge Representation and Reasoning in Artificial Intelligence. They are based on Classical Logic in order to guarantee the correctness of the inferences on the required reasoning tasks. It is indeed a fragment of First Order Predicate Logic whose language is strictly related to the one of Modal Logic. Fuzzy Description Logic is the generalization of the classical Description Logic framework thought for reasoning with vague concepts that often arise in practical applications.

Fuzzy Description Logic has been investigated since the last decade of the $20^{th}$ century. During the first fifteen years of investigation their semantics has been based on Fuzzy Set Theory. A semantics based on Fuzzy Set Theory, however, has been shown to have some counter-intuitive behavior, due to the fact that the truth function for the implication used is not the residuum of the truth function for the conjunction. In the meanwhile, Fuzzy Logic has been given a formal framework based on Many-valued Logic. This framework, called Mathematical Fuzzy Logic, has been proposed has the kernel of a well mathematically founded Fuzzy Logic.

In this dissertation we propose a Fuzzy Description Logic whose semantics is based on Mathematical Fuzzy Logic as its mathematically well settled kernel. To this end we provide a novel notation that is strictly related to the notation that is used in Mathematical Fuzzy Logic. After having settled the notation, we investigate the hierarchies of description languages over different $t$-norm based semantics and the reductions that can be performed between reasoning tasks. The new framework that we establish gives us the possibility to systematically investigate the relation of Fuzzy Description Logic to Fuzzy First Order Logic and Fuzzy Modal Logic. Next we provide some (un)decidability results for the case of infinite $t$-norm based semantics with or without knowledge bases. Finally we investigate the complexity bounds of reasoning tasks without knowledge bases for basic Fuzzy Description Logics over finite $t$-norms.