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.

Equational unification can be an effective tool for the analysis of cryptographic protocols. This, for example

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 ...

. Although the use of expansionary j-rewrite has become increasingly common in recent years, one area where j-contractions have until now remained the only possibility is in the more powerful type theories of the -cube. This paper rectifies this situation by applying j-expansions to the Calculus of Constructions --- we discuss some of the difficulties posed by the presence of dependent types, prove that every term rewrites to a unique long fij-normal form and deduce the decidability of fij-equality, typeability and type inhabitation as corollaries. 1 Introduction Extensional equality for the simply typed -calculus requires j-conversion, whose interpretation as a rewrite rule has traditionally been as a contraction x : T:fx ) f where x 6 2 FV(t). When combined with the usual fi-reduction, the resulting rewrite relation is strongly normalising and confluent, and thus reduction to normal form provides a decision procedure for the associated equational theory. However j-contractions beh...

Abstract. Many security protocols fundamentally depend on the algebraic properties of cryptographic operators. It is however difficult to handle these properties when formally analyzing protocols, since basic problems like the equality of terms that represent cryptographic messages are undecidable, even for relatively simple algebraic theories. We present a framework for security protocol analysis that can handle algebraic properties of cryptographic operators in a uniform and modular way. Our framework is based on two ideas: the use of modular rewriting to formalize a generalized equational deduction problem for the Dolev-Yao intruder, and the introduction of two parameters that control the complexity of the equational unification problems that arise during protocol analysis by bounding the depth of message terms and the operations that the intruder can perform when analyzing messages. We motivate the different restrictions made in our model by highlighting different ways in which undecidability arises when incorporating algebraic properties of cryptographic operators into formal protocol analysis. 1

this paper we overcome these drawbacks by working with clauses with symbolic constraints (Kirchner et al., 1990; Nieuwenhuis and Rubio, 1992; Rubio, 1994; Nieuwenhuis and Rubio, 1995) . A constrained clause C [[ T ]] is a shorthand for the set of ground instances of the clause part C satisfying the constraint T . In a constrained equation

An abstract framework of canonical inference is used to explore how different proof orderings induce different variants of saturation and completeness. Notions like completion, paramodulation, saturation, redundancy elimination, and rewrite-system reduction are connected to proof orderings. Fairness of deductive mechanisms is defined in terms of proof orderings, distinguishing between (ordinary) “fairness, ” which yields completeness, and “uniform fairness, ” which yields saturation.

Solving goals—like deciding word problems or resolving constraints—is much easier in some theory presentations than in others. What have been called “completion processes”, in particular in the study of equational logic, involve finding appropriate presentations of a given theory to solve easily a given class of problems. We provide a general proof-theoretic setting within which completion-like processes can be modelled and studied. This framework centers around well-founded orderings of proofs. It allows for abstract definitions and very general characterizations of saturation processes and redundancy criteria.

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.

Rewriting with associativity, commutativity and identity has been an open problem for a long time. In 1989, Baird, Peterson and Wilkerson introduced the notion of constrained rewriting, to avoid the problem of non-termination inherent to the use of identities. We build up on this idea in two ways: by giving a complete set of rules for completion modulo these axioms; by showing how to build appropriate orderings for proving termination of constrained rewriting modulo associativity, commutativity and identity. 1 Introduction Equations are ubiquitous in mathematics and the sciences. Among the most common equations are associativity, commutativity and identity (existence of a neutral element). Rewriting is an efficient way of reasoning with equations, introduced by Knuth and Bendix [12]. When rewriting, equations are used in one direction chosen once and for all. Unfortunately, orientation alone is not a complete inference rule: given a set of equational axioms E, there may be equal terms...