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 introduces a notion of partial type assignment on applicative term rewriting systems that is based on a combination of an essential intersection type assignment system, and the type assignment system as defined for ML [16], both extensions of Curry's type assignment system [11]. Terms and rewrite rules will be written as trees, and type assignment will consists of assigning intersection types function symbols, and specifying the way in which types can be assigned to nodes and edges between nodes. The only constraints on this system are local: they are imposed by the relation between the type assigned to a node and those assigned to its incoming and out-going edges. In general, given an arbitrary typeable applicative term rewriting system, the subject reduction property does not hold. We will formulate a sufficient but undecidable condition typeable rewrite rules should satisfy in order to obtain this property. Introduction In the recent years several paradigms hav...

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 ...

This paper studies normalization of typeable terms and the relation between approximation semantics and filter models for Combinator Systems. It presents notions of approximants for terms, intersection type assignment, and reduction on type derivations; the last will be proved to be strongly normalizable. With this result, it is shown that, for every typeable term, there exists an approximant with the same type, and a characterization of the normalization behaviour of terms using their assignable types is given. Then the two semantics are defined and compared, and it is shown that the approximants semantics is fully abstract but the filter semantics is not.

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

This paper discusses the relation between the minimal positive relevant logic B+ and intersection and union type theories. There is a marvellous coincidence between these very differently motivated research areas. First, we show a perfect fit between the Intersection Type Discipline ITD and the tweaking B intersect T of B+ , which saves implication and conjunction but drops disjunction. The filter models of the lambda-calculus (and its intimate partner Combinatory Logic CL) of the first author and her co-authors then become theory models of these calculi. (The logician's Theory is the algebraist's Filter.) The coincidence extends to a dual interpretation of key particles -- the subtype translates to provable ->, type intersection to conjunction, function space -> to implication and whole domain omega to the (trivially added but trivial) truth T. This satisfying ointment contains a fly. For it is right, proper and to be expected that type union U should correspond to the logical disjunction \/ of B+ . But the simulation of functional application by a fusion (or modus ponens product) operation o on theories leaves the key Bubbling lemma of work on ITD unprovable for the \/-prime theories now appropriate for the modelling. The focus of the present paper lies in an appeal to Harrop theories which are (a) prime and (b) closed under fusion. A version of the Bubbling lemma is then proved for Harrop theories, which accordingly furnish a model of lambda and CL.

These notes belong to the course Type Systems for Programming Languages, given to fourth year students in Computing and Joint Mathematics and Computing with some experience in reasoning and logic, and students in the Advanced Masters programme at the Department of Computing, Imperial College, London. The course is intended for students interested in theoretical computer science, who possess some knowledge of logic. No prior knowledge on type systems or proof techniques is assumed, other than being familiar with the principle of induction. Aims • To lay out in detail the design of type assignment systems for programming languages. • To focus on the importance of a sound theoretical framework, in order to be able to reason about properties of a typed program. • To understand the concepts of: type checking, type reconstruction, polymorphism, type derivation, typeability, typing of recursive functions, termination in the context of typeability, and undecidable systems. • To study various systems and various languages, and to compare those and to select.