This textbook offers a unified and self-contained introduction to the field of term rewriting. It covers all the basic material (abstract reduction systems, termination, confluence, completion, and combination problems), but also some important and closely connected subjects: universal algebra, unification theory, Gröbner bases and Buchberger's algorithm. The main algorithms are presented both informally and as programs in the functional language Standard ML (an appendix contains a quick and easy introduction to ML). Certain crucial algorithms like unification and congruence closure are covered in more depth and Pascal programs are developed. The book contains many examples and over 170 exercises. This text is also an ideal reference book for professional researchers: results that have been spread over many conference and journal articles are collected together in a unified notation, proofs of almost all theorems are provided, and each chapter closes with a guide to the literature.

In this chapter, we introduce and explain the basic notions of Description Logic, including syntax, semantics and reasoning services, and we explain how the latter are used in applications. The concept language of the DL ALCIn this section, we will describe the central notions of Description Logic first on an intuitive level and then on a more precise level. As a running example, we use the domain of university courses and teaching, and we will use a conceptualisation given informally, in graphical form, in Figure 2.1. Please note that this is one way of viewing university teachingwhich might be very different from the reader's way of viewing it. Also, as it is an informal representation, different readers may interpret arrows in different ways; that is, our representation does not come with a well-defined semantics that would inform us in an unambiguous way how to interpret the different arrows. 1 In the next sections, we will describe our way of viewing university teaching in a DL knowledge base, thereby establishing some constraints on the meaning of terms like "Professor" and "teaches" used in Figure 2.1 and throughout this section.In Description Logic, we assume that we want to describe some abstraction of some domain of interest, and that this abstraction is populated by elements. 2 We use three main building blocks to describe these elements:• Concepts represent sets of elements and can be viewed as unary pred-1 Our graphical representation looks somewhat similar to an extended ER diagram, for which such a well-defined semantics has been specified [Che76, CLN94]. 2 We have chosen the term "elements" rather than "individuals" or "objects" to prevent the reader from making false assumptions.

This chapter provides an introduction to Description Logics as a formal language for representing knowledge and reasoning about it. It first gives a short overview of the ideas underlying Description Logics. Then it introduces syntax and semantics, covering the basic constructors that are used in systems or have been introduced in the literature, and the way these constructors can be used to build knowledge bases. Finally, it defines the typical inference problems, shows how they are interrelated, and describes different approaches for effectively solving these problems. Some of the topics that are only briefly mentioned in this chapter will be treated in more detail in subsequent chapters. 47 48 F. Baader, W. Nutt Proposition 2.5 Let T be a terminology, I be an interpretation, and J be the restriction of I to the base symbols of T . Then I is a model of T if, and only if, I is a fixpoint of T J .According to the preceding proposition, a terminology T is definitorial iff every base interpretation J has a unique extension that is a fixpoint of T J .Example 2.6 To get a feel for why cyclic terminologies are not definitorial, we discuss as an example the terminology T Momo that consists only of Axiom (2.4). Consider the base interpretation J defined by

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