Solving an equation in an algebra of terms is known as unification. Solving more complex formulas combining equations and involving in particular negation is called disunification. With such a broad definition, many works fall into the scope of disunification. The goal of this paper is to survey these works and bring them together in a same framework. R'esum'e On appelle habituellement (algorithme d') unification un algorithme de r'esolution d'une 'equation dans une alg`ebre de termes. La r'esolution de formules plus complexes, comportant en particulier des n'egations, est appel'ee ici disunification. Avec une d'efinition aussi 'etendue, de nombreux travaux peuvent etre consid'er'es comme portant sur la disunification. L'objet de cet article de synth`ese est de rassembler tous ces travaux dans un meme formalisme. Laboratoire de Recherche en Informatique, Bat. 490, Universit'e de Paris-Sud, 91405 ORSAY cedex, France. E-mail: comon@lri.lri.fr i Contents 1 Syntax 5 1.1 Basic Defini...

www.loria.fr/{~cirstea,~ckirchne,~lliquori} Abstract. In this paper we give a simple and uniform presentation of the rewriting calculus, also called Rho Calculus. In addition to its simplicity, this formulation explicitly allows us to encode complex structures such as lists, sets, and objects. We provide extensive examples of the calculus, and we focus on its ability to represent some object oriented calculi, namely the Lambda Calculus of Objects of Fisher, Honsell, and Mitchell, and the Object Calculus of Abadi and Cardelli. Furthermore, the calculus allows us to get object oriented constructions unreachable in other calculi. In summa, we intend to show that because of its matching ability, the Rho Calculus represents a lingua franca to naturally encode many paradigms of computations. This enlightens the capabilities of the rewriting calculus based language ELAN to be used as a logical as well as powerful semantical framework. 1

The functional strategy has been widely used implicitly (Haskell, Miranda, Lazy ML) and explicitly (Clean) as an efficient, intuitively easy to understand reduction strategy for term (or graph) rewriting systems. However, little is known of its formal properties since the strategy deals with priority rewriting which significantly complicates the semantics. Nevertheless, this paper shows that some formal results about the functional strategy can be produced by studying the functional strategy entirely within the standard framework of orthogonal term rewriting systems. A concept is introduced that is one of the key aspects of the efficiency of the functional strategy: transitive indexes . The corresponding class of transitive term rewriting systems is characterized. An efficient normalizing strategy is given for these rewriting systems. It is shown that the functional strategy is normalizing for the class of left-incompatible term rewriting systems. 1. Introduction An interesting commo...

We give an informal review of the problem of eliminating negation in term algebras and its applications. The initial results appear to be very specialized with complex combinatorial proofs. Nevertheless they have applications and relevance to a number of important areas: unification, learning, abstract data types and rewriting systems, constraints and constructive negation in logic languages. 1 Initial Motivation: Learning Plotkin [36] proposed a formal model for inductive inference which was based upon Popplestone 's suggestion that Since unification is useful in automatic deduction, its dual might prove helpful for induction. A similar formalism was independently introduced by Reynolds [38], who was more concerned with its algebraic properties than with its applications. The algebraic properties were further investigated by Huet [9, 10], who also studied the case of the infinitary Herbrand universe. The key result in this theory is that, for any set of terms, there exists a...

Abstract. Kamperman and Walters proposed the notion of a simulation of one rewrite system by another one, whereby each term of the simulating rewrite system is related to a term in the original rewrite system. In this paper it is shown that if such a simulation is sound and complete and preserves termination, then the transformation of the original into the simulating rewrite system constitutes a correct step in the compilation of the original rewrite system. That is, the normal forms of a term in the original rewrite system can then be obtained by computing the normal forms of a related term in the simulating rewrite system. 1

Interpretation versus Demand Propagation Wadler [1987] uses abstract interpretation over a four-point domain for reasoning about strictness on lists. The four points correspond to undefined list (represented by value 0), infinite lists and lists with some tail undefined (value 1), lists with at least one head undefined (value 2), and all lists (value 3). Burn's work [Burn 1987] on evaluation transformers also uses abstract interpretation on the above-mentioned domain for strictness analysis. He defines four evaluators that correspond to the four points mentioned above in the sense that the ith evaluator will fail to produce an answer when given a list with the abstract value i \Gamma 1.