this paper is to make the similarity between Knuth-Bendix completion and the Buchberger algorithm explicit, by describing a general algorithm called S-normalized completion where S is a parameter, such that both algorithms are Normalized Rewriting: an alternative to Rewriting modulo a Set of Equations 3 instances of this general algorithm for a particular choice of S. This has been achieved in two steps.

We introduce normalised rewriting, a new rewrite relation. It generalises former notions of rewriting modulo E, dropping some conditions on E. For example, E can now be the theory of identity, idempotency, the theory of Abelian groups, the theory of commutative rings. We give a new completion algorithm for normalised rewriting. It contains as an instance the usual AC completion algorithm, but also the wellknown Buchberger's algorithm for computing standard bases of polynomial ideals. We investigate the particular case of completion of ground equations, In this case we prove by a uniform method that completion modulo E terminates, for some interesting E. As a consequence, we obtain the decidability of the word problem for some classes of equational theories. We give implementation results which shows the efficiency of normalised completion with respect to completion modulo AC. 1 Introduction Equational axioms are very common in most sciences, including computer science. Equations can ...

The concept of algebraic simplification is of great importance for the field of symbolic computation in computer algebra. In this paper we review some fundamental concepts concerning reduction rings in the spirit of Buchberger. The most important properties of reduction rings are presented. The techniques for presenting monoids or groups by string rewriting systems are used to define several types of reduction in monoid and group rings. Grobner bases in this setting arise naturally as generalizations of the corresponding known notions in the commutative and some noncommutative cases. Several results on the connection of the word problem and the congruence problem are proven. The concepts of saturation and completion are introduced for monoid rings having a finite convergent presentation by a semi-Thue system. For certain presentations, including free groups and context-free groups, the existence of finite Grobner bases for finitely generated right ideals is shown and a procedure to com...

The problem of choosing efficient algorithms and good admissible orders for computing Gröbner bases in noncommutative algebras is considered. Gröbner bases are an important tool that make many problems in polynomial algebra computationally tractable. However, the computation of Grobner bases is expensive, and in noncommutative algebras is not guaranteed to terminate. The algorithm, together with the order used to determine the leading term of each polynomial, are known to affect the cost of the computation, and are the focus of this thesis. A Gröbner basis is a set of polynomials computed, using Buchberger's algorithm, from another set of polynomials. The noncommutative form of Buchberger's algorithm repeatedly constructs a new polynomial from a triple, which is a pair of polynomials whose leading terms overlap and form a nontrivial common multiple. The algorithm leaves a number of details underspecified, and can be altered to improve its behavior. A significant improvement is the devel...

We propose a new inference system for automated deduction with equality and associative commutative operators. This system is an extension of the ordered paramodulation strategy. However, rather than using associativity and commutativity as the other axioms, they are handled by the AC-unification algorithm and the inference rules. Moreover, we prove the refutational completeness of this system without needing the functional reflexive axioms or ACaxioms. Such a result is obtained by semantic tree techniques. We also show that the inference system is compatible with simplification rules.

: We present extended term rewriting systems as a means to describe a simplification relation for an equational specification with a built-in domain of external objects. Even if the extended term rewriting system is canonical, the combined relation including built-in computations of `ground terms' needs neither be terminating nor confluent. We investigate restrictions on the extended term rewriting systems and the built-in domains under which these properties hold. A very important property of extended term rewriting systems is decomposition freedom. Among others decomposition free extended term rewriting systems allow for efficient simplifications. Some interesting algebraic applications of canonical simplification relations are presented. 1 Introduction There has always been mutual interest in the areas of computer algebra and term rewriting systems as can be seen for example from the calls of papers of the main conferences in the two areas (ISSAC and RTA resp.) which each...

der Technischen Fakult"at der Universit"at des Saarlandes Saarbr"ucken

We prove that there exists an ACD-ground theory --- an equational theory defined by a set of ground equations plus the associativity and commutativity of two binary symbols and +, and the distributivity of over + --- for which the word problem is undecidable. 1 Introduction Equations are ubiquitous in mathematics and the sciences. The word problem of a given a set of equations (that is the problem of deciding if an identity is a consequence of the equations), or equivalently of its equational theory, is undecidable in general. But there are known classes of equational theories which have a decidable word problem, in particular, ground equational theories. The most famous examples of theories with undecidable word problem are given by sets of ground equations over word algebras. Such theories can be considered as associative-ground theories over a certain term algebra, whose signature contains only constants besides the binary (associative) symbol. Their word problem is known to be...