Unification of concept descriptions was introduced by Baader and Narendran as a tool for detecting redundancies in knowledge bases. It was shown that unification in the small description logic FL 0 , which allows for conjunction, value restriction, and the top concept only, is already ExpTime complete. The present paper shows that the complexity does not increase if one additionally allows for composition, union, and transitive closure of roles. It also shows that matching (which is polynomial in FL 0 ) is PSpace-complete in the extended description logic.

We construct a finite language L such that the largest language commuting with L is not recursively enumerable. This gives a negative answer to the question raised by Conway in 1971 and also strongly disproves Conway’s conjecture on context-freeness of maximal solutions of systems of semi-linear inequalities. 1

We survey the known results on two old open problems on commutation of languages. The first problem, raised by Conway in 1971, is asking if the centralizer of a rational language must be rational as well – the centralizer of a language is the largest set of words commuting with that language. The second problem, proposed by Ratoandromanana in 1989, is asking for a characterization of those languages commuting with a given code – the conjecture is that the commutation with codes may be characterized as in free monoids. We present here simple proofs for the known results on these two problems. 1

We study in this thesis several problems related to commutation on sets of words and on formal power series. We investigate the notion of semilinearity for formal power series in commuting variables, introducing two families of series - the semilinear and the bounded series - both natural generalizations of the semilinear languages, and we study their behaviour under rational operations, morphisms, Hadamard product, and difference. Turning to commutation on sets of words, we then study the notions of centralizer of a language - the largest set commuting with a language -, of root and of primitive root of a set of words. We answer a question raised by Conway more than thirty years ago - asking whether or not the centralizer of any rational language is rational - in the case of periodic, binary, and ternary sets of words, as well as for rational c-codes, the most general results on this problem. We also prove that any code has a unique primitive root and that two codes commute if and only if they have the same primitive root, thus solving two conjectures of Ratoandromanana, 1989. Moreover, we prove that the commutation with an c-code X can be characterized similarly as in free monoids: a language commutes with X if and only if it is a union of powers of the primitive root of X.

We consider the decidability of existence of solutions to language equations involving the operations of shue and deletion along trajectories. These operations generalize the operations of concatenation, insertion, shue, quotient, sequential and scattered deletion, as well as many others. Our results are constructive in the sense that if a solution exists, it can be eectively represented. We show both positive and negative decidability results.

We prove two results on commutation of languages. First, we show that the maximal language commuting with a three element language, i.e. its centralizer,

Abstract. We survey results, both positive and negative, on regularity of maximal solutions of systems of implicit language equations and inequalities. These results concern inequalities with constant right-hand sides, one-sided linear inequalities, inequalities with restrictions on constants, and commutation equations and inequalities. In addition, we present some of these results in a generalized form in order to underline common principles. 1.

Language equations are equations in which the constants are languages, ranging over some specified class, and the operations are drawn from the standard canon of language operations (union, complementation, concatenation, and star). In previous work, various types of language equations have been studied, with the primary objective of determining (existence and uniqueness of) solutions. A solution of a system of language equations in the n variables X1,..., Xn is a vector of n languages (S1,..., Sn) such that simultaneously substituting Si for every occurrence of the variable Xi yields a system of m language identities. Here, we are interested in derivative equations. These are language equations where not only the variables may occur, but also their derivatives X /v, for v an arbitrary word. The interpretation is that any occurrence of X /v in an equation will be replaced by the left quotient with respect to v of the language S that is substituted for the variable X. We will study two types of systems of derivative equations in n variables, namely implicit equations which are of the form and explicit equations which are of the form αi(X1,..., Xn) = Li, i = 1,..., m, αi(X1,..., Xn) = Xji, i = 1,..., m,

Unification of concept descriptions has been introduced by Baader and Narendran [5] as a new inference service for detecting and avoiding redundancies in Description Logic knowledge bases. The technical results in [5] are concerned with unification in the small Description Logic FL 0 , which allows for conjunction of concepts, value restriction, and the top concept. It is shown that uni cation of FL 0 -concept descriptions is equivalent to unification modulo the equational theory ACUIh, which is concerned with a binary associative-commutative idempotent function symbol with a unit and finitely many unary function symbols that behave like endomorphisms for the binary symbol and the unit. Since ACUIh is a commutative/monoidal theory [6], unification modulo this theory can in turn be reduced to solving linear equations in a certain semiring. For the theory ACUIh, this semiring consists of finite sets of words over a finite alphabet...