Steffen van Bakel

Complete restrictions of the intersection type discipline

Abstract In this paper the intersection type discipline as defined in Barendregt (1983) is studied. We will present two different and independent complete restrictions of the intersection type discipline. The first restricted system, the strict type assignment system, is presented in Section 2. Its major feature is the absence of the derivation rule (⩽) and it is based on a set of strict types. We will show that these together give rise to a strict filter lambda model that is essentially different from the one presented in Barendregt. We will show that the strict type assignment system is the nucleus of the full system, i.e. for every derivation in the intersection type discipline there is a derivation in which (⩽) is used only at the very end. Finally we will prove that strict type assignment is complete for inference semantics. The second restricted system is presented in Section 3. Its major feature is the absence of the type ω. We will show that this system gives rise to a filter λ I-model and that type assignment without ω is complete for the λ I-calculus. Finally we will prove that a lambda term is typeable in this system if and only if it is strongly normalizable.

Intersection type assignment systems

Abstract This paper gives an overview of intersection type assignment for the Lambda Calculus, as well as compares in detail variants that have been defined in the past. It presents the essential intersection type assignment system, that will prove to be as powerful as the well-known BCD-system. It is essential in the following sense: it is an almost syntax directed system that satisfies all major properties of the BCD-system, and the types used are the representatives of equivalence classes of types in the BCD-system. The set of typeable terms can be characterized in the same way, the system is complete with respect to the simple type semantics, and it has the principal type property.

Normalization Results for Typeable Rewrite Systems

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.

The language X : circuits, computations and classical logic

We present the syntax and reduction rules for χ, an untyped language that is well suited to describe structures which we call “circuits” and which are made of parts that are connected by wires. To demonstrate that χ gives an expressive platform, we will show how, even in an untyped setting, that we can faithfully embed algebraic objects and elaborate calculi, like the naturals, the λ-calculus, Bloe and Rose’s calculus of explicit substitutions λx, and Parigot’s λμ.

Strict intersection types for the Lambda Calculus

This article will show the usefulness and elegance of strict intersection types for the Lambda Calculus, that are strict in the sense that they are the representatives of equivalence classes of types in the BCD-system [Barendregt et al. 1983]. We will focus on the essential intersection type assignment; this system is almost syntax directed, and we will show that all major properties hold that are known to hold for other intersection systems, like the approximation theorem, the characterization of (head/strong) normalization, completeness of type assignment using filter semantics, strong normalization for cut-elimination and the principal pair property. In part, the proofs for these properties are new; we will briefly compare the essential system with other existing systems.

Rank 2 intersection type assignment in term rewriting

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 GTRS, 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.

Comparing cubes of typed and type assignment systems

Abstract We study the cube of type assignment systems, as introduced in Giannini et al. (Fund. Inform. 19 (1993) 87–126), and confront it with Barendregts typed gl-cube (Barendregt, in: Handbook of Logic in Computer Science, Vol. 2, Clarenden Press, Oxford, 1992). The first is obtained from the latter through applying a natural type erasing function E to derivation rules, that erases type information from terms. In particular, we address the question whether a judgement, derivable in a type assignment system, is always an erasure of a derivable judgement in a corresponding typed system; we show that this property holds only for systems without polymorphism. The type assignment systems we consider satisfy the properties ‘subject reduction’ and ‘strong normalization’. Moreover, we define a new type assignment cube that is isomorphic to the typed one.

Boundary Inference for Enforcing Security Policies in Mobile Ambients

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Logical Equivalence for Subtyping Object and Recursive Types

is an untyped continuation-style formal language with a typed subset that provides a Curry–Howard isomorphism for a sequent calculus for implicative classical logic.