We study the strict type assignment system, a restriction of the intersection type discipline [6], and prove that it has the principal type property. We define, for a term, the principal pair (of basis and type). We specify three operations on pairs, and prove that all pairs deducible for can be obtained from the principal one by these operations, and that these map deducible pairs to deducible pairs.

A notion of type assignment on Curryfied Term Rewriting Systems is introduced that uses Intersection Types of Rank 2, and in which all function symbols are assumed to have a type. Type assignment will consist of specifying derivation rules that describe how types can be assigned to terms, using the types of function symbols. Using a modified unification procedure, for each term the principal pair (of basis and type) will be defined in the following sense: from these all admissible pairs can be generated by chains of operations on pairs, consisting of the operations substitution, copying, and weakening. In general, given an arbitrary typeable CuTRS, the subject reduction property does not hold. Using the principal type for the left-hand side of a rewrite rule, a sufficient and decidable condition will be formulated that typeable rewrite rules should satisfy in order to obtain this property. Introduction In the recent years, several paradigms have been investigated for the implementatio...

In this paper we introduce Curryfied Term Rewriting Systems, and a notion of partial type assignment on terms and rewrite rules that uses intersection types with sorts and !. Three operations on types -- substitution, expansion, and lifting -- are used to define type assignment, and are proved to be sound. With this result the system is proved closed for reduction. Using a more liberal approach to recursion, we define a general scheme for recursive definitions and prove that, for all systems that satisfy this scheme, every term typeable without using the type-constant ! is strongly normalizable. We also show that, under certain restrictions, all typeable terms have a (weak) head-normal form, and that terms whose type does not contain ! are normalizable.

This paper studies termination properties of rewrite systems that are typeable using intersection types. It introduces a notion of partial type assignment on Curryfied Term Rewrite Systems, that consists of assigning intersection types to function symbols, and specifying the way in which types can be assigned to nodes and edges between nodes in the tree representation of terms. Two operations on types are specified that are used to define type assignment on terms and rewrite rules, and are proven to be sound on both terms and rewrite rules. Using a more liberal approach to recursion, a general scheme for recursive definitions is presented, that generalizes primitive recursion, but has full Turing-machine computational power. It will be proved that, for all systems that satisfy this scheme, every typeable term is strongly normalizable. Introduction Most functional programming languages, like Miranda [23] or ML [19] for instance, although implemented through an extended Lambda Calculus ...

We consider an intersection type assignment system for term rewriting systems extended with application, and define a notion of (finite) approximation on terms. We then prove that for typeable rewrite systems satisfying a general scheme for recursive definitions, every typeable term has an approximant of the same type. This approximation result, and the proof technique developed to obtain it, allow us to deduce in a direct way a headnormalization, a normalization, and a strong normalization theorem, for different classes of typeable terms. 1

We introduce an intersection type system which is a restriction of the intersection type discipline. This restriction leads to a principal type property for normal forms in the classical sense, while retaining the expressivity of the classical discipline. We characterize the structure of principal types of normal forms and give an algorithm that reconstructs normal forms from types. Having shown the equivalence between principal types and normal forms, we define an expansion operation on types which allows us to recover all possible types for any normalizable -term. The contribution of this work is a new and simpler presentation of the intersection type discipline through a purely syntactic and completely characterized notion of principal types.

Abstract. In this paper we study normalization properties of rewrite systems that are typeable using intersection types with and with sorts. We prove two normalization properties of typeable systems. On one hand, for all systems that satisfy a variant of the Jouannaud-Okada Recursion Scheme, every term typeable with a type that is not is head normalizable. On the other hand, non-Curryfied terms that are typeable with a type that does not contain, are normalizable.

Interaction nets have proved to be a useful tool for the study of computational aspects of different formalisms (e.g. -calculus, term rewriting systems), but they are also a programming paradigm in themselves, and this is actually how they were introduced by Lafont. In this paper we consider semi-simple interaction nets as a programming language, and present a type assignment system using intersection types. First we show that interactions preserve types (i.e. the system enjoys subject reduction), and we compare this type assignment system with the intersection systems for -calculus and term rewriting systems. Then we define a recursion scheme that ensures termination of all interaction sequences. By relaxing the scheme and using the type assignment system, we derive another sufficient condition for termination of interaction nets. Finally, we show that although the type system based on general intersection types is not decidable, its restriction to rank 2 types is, and we give an algo...

Abstract. We define two type assignment systems for first-order rewriting extended with application,-abstraction, and-reduction (TRS). The types used in these systems are a combination of (-free) intersection and polymorphic types. The first system is the general one, for which we prove a subject reduction theorem and show that all typeable terms are strongly normalisable. The second is a decidable subsystem of the first, by restricting types to Rank 2. For this system we define, using an extended notion of unification, a notion of principal type, and show that type assignment is decidable.

ion and fi-rule: Types, Approximants and Normalization Steffen van Bakel 1 Franco Barbanera 2 Maribel Fernandez 3 1 Department of Computing, Imperial College, 180 Queens Gate, London SW7 2BZ, svb@doc.ic.ac.uk 2 Dipartimento di Informatica, Universita degli Studi di Torino, Corso Svizzera 185, 10149 Torino, Italia, barba@di.unito.it 3 DMI - LIENS (CNRS URA 1327), Ecole Normale Superieure, 45, rue d'Ulm, 75005 Paris, France, maribel@ens.fr Abstract In this paper we define and study intersection type assignment systems for first-order rewriting extended with application, -abstraction, and fi-reduction (TRS fi). One of the main results presented is that, using a suitable notion of approximation of terms, any typeable term of a TRS fi that satisfies a general scheme for recursive definitions has an approximant of the same type. From this result we deduce, for different classes of typeable terms, a head-normalization and a normalization theorem. Introduction Lambda Calculus (LC...