) Maria C. F. Ferreira 1 and Delia Kesner 2 and Laurence Puel 2 1 Dep. de Inform'atica, Fac. de Ciencias e Tecnologia, Univ. Nova de Lisboa, Quinta da Torre, 2825 Monte de Caparica, Portugal, cf@fct.unl.pt. 2 CNRS & Lab. de Rech. en Informatique, Bat 490, Univ. de Paris-Sud, 91405 Orsay Cedex, France, fkesner,puelg@lri.fr. Abstract. We study preservation of fi-strong normalization by d and dn , two confluent -calculi with explicit substitutions defined in [10]; the particularity of these calculi is that both have a composition operator for substitutions. We develop an abstract simulation technique allowing to reduce preservation of fi-strong normalization of one calculus to that of another one, and apply said technique to reduce preservation of fi-strong normalization of d and dn to that of f , another calculus having no composition operator. Then, preservation of fi-strong normalization of f is shown using the same technique as in [2]. As a consequence, d and dn become the fir...

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 ...

We define a simplification ordering on terms which is AC-compatible and total on nonAC -equivalent ground terms, without any restrictions on the signature like the number of AC-symbols or free symbols. Unlike previous work by Narendran and Rusinowitch [NR91], our AC-RPO ordering is not based on polynomial interpretations, but on a simple extension of the well-known RPO ordering (with a total (arbitrary) precedence on the function symbols). This solves an open question posed e.g. by Bachmair [Bac92]. A second difference is that this ordering is also defined on terms with variables, which makes it applicable in practice for complete theorem proving strategies with built-in AC-unification and for orienting non-ground rewrite systems. The ordering is defined in a simple way by means of rewrite rules, and can be easily implemented, since its main component is RPO. 1 Introduction Automated termination proofs are well-known to be crucial for using rewriting-like methods in theorem proving an...

. We introduce rewriting with two sets of rules, the first interpreted equationally and the second not. A semantic view considers equational rules as defining an equational theory and reduction rules as defining a rewrite relation modulo this theory. An operational view considers both sets of rules as similar. We introduce sufficient properties for these two views to be equivalent (up to different notions of equivalence). The paper ends with a collection of example showing the effectiveness of this approach. Rewriting can be viewed simultaneously as the most basic symbol-manipulating method, and as a very expressive specification framework, given the expressive power of rewriting modulo equations. It is a primary candidate to the role of a general logical framework [Mes92, MOM93]. Historically, rewriting has been given an equational semantics, saying that a rewrite rule u \Gamma! v is interpreted as u is equal to v. This is the case for instance when defining functions or solving the w...

. Rewriting is a general paradigm for expressing computations in various logics, and we focus here on rewriting techniques in equational logic. When used at the proof level, rewriting provides with a very powerful methodology for proving completeness results, a technique that is illustrated here. We also consider whether important properties of rewrite systems such as confluence and termination can be proved in a modular way. Finally, we stress the links between rewriting and tree automata. Previous surveys include [21; 18; 37; 12; 45; 46]. The present one owes much to [21]. Keywords. completion, confluence, critical pair, ground reducibility, inductive completion, local confluence, modularity, narrowing, order-sorted algebras, rewrite rule, rewriting, term algebra, termination, tree automata. 1 Introduction The use of equations is traditional in mathematics. Its use in computer science has culminated with the success of algebraic specifications, a method of specifying software by enc...

We introduce a generic definition of reduction orderings on term algebras containing associative-commutative (hereafter denoted AC) operators. These orderings are compatible with the AC theory hence makes them suitable for use in deduction systems where AC operators are built-in. Furthermore, they have the nice property of being total on AC classes of ground terms, a required property for example to avoid failure in ACcompletion, or to insure completeness of ordered strategies in first-order theorem proving with built-in AC operators. We show that the two definitions already known of such total and AC-compatible orderings [24, 25] are actually instances of our definition. Finally, we find new such orderings which have more properties, first an ordering based on an integer polynomial interpretation, answering positively to a question left open by Narendran and Rusinowitch, and second an ordering which allow to orient the distributivity axiom in the usual way, answering positively to a ...

. Theories with associative and commutative (AC) operators, such as arithmetic, process algebras, boolean algebras, sets, : : : are ubiquitous in software and hardware verification. These AC operators are difficult to handle by automatic deduction since they generate complex proofs. In this paper, we present new techniques for combining induction and AC reasoning, in a rewrite-based theorem prover. The resulting system has proved to be quite successful for verification tasks. Thanks to its careful rewriting strategy, it needs less interaction on typical verification problems than well known tools like NQTHM, LP or PVS . We also believe that our approach can easily be integrated as an efficient tactic in other proof systems. 1 Introduction Powerful tools based on model checking have been developed for the verification of finite-state systems [6]. Their extensions to some classes of infinite-state systems has only produced moderate success. Therefore deductive methods offer a promising ...