Intersection Type Assignment System Research Articles

From semantics to types: The case of the imperative λ-calculus

We study the logical semantics of an untyped λ-calculus equipped with operators representing read and write operations from and to a global store. Such a logic consists of an intersection type assignment system, which we derive from the denotational semantics of the calculus, based on the monadic approach to model computational λ-calculi.The system is obtained by constructing a filter model in the category of ω-algebraic lattices, such that the typing rules can be recovered out of the term interpretation. By construction, the so-obtained type system satisfies the “type-semantics” property and completeness.

On sets of terms having a given intersection type

Working in a variant of the intersection type assignment system of Coppo, Dezani-Ciancaglini and Venneri [1981], we prove several facts about sets of terms having a given intersection type. Our main result is that every strongly normalizing term M admits a *uniqueness typing*, which is a pair $(\Gamma,A)$ such that 1) $\Gamma \vdash M : A$ 2) $\Gamma \vdash N : A \Longrightarrow M =_{\beta\eta} N$ We also discuss several presentations of intersection type algebras, and the corresponding choices of type assignment rules. Moreover, we show that the set of closed terms with a given type is uniformly separable, and, if infinite, forms an adequate numeral system. The proof of this fact uses an internal version of the B\"ohm-out technique, adapted to terms of a given intersection type.

Kripke Semantics for Intersection Formulas

We propose a notion of the Kripke-style model for intersection logic. Using a game interpretation, we prove soundness and completeness of the proposed semantics. In other words, a formula is provable (a type is inhabited) if and only if it is forced in every model. As a by-product, we obtain another proof of normalization for the Barendregt–Coppo–Dezani intersection type assignment system.

Characterizing of Strong Normalization for Λμ-Calculus

λμ-calculus is introduced by Parigot as an extension isomorphic to an alternative presentation of classical natural deduction. Since then, many properties of it have been studied and, in particular, it does not enjoy the separation property shown by David and Py. Λμ-calculus is proposed by de Groote and developed by Saurin as an extension of λμ-calculus Saurin demonstrates that the separation property holds for the Λμ -calculus. Bakel gives a characterization of strong normalization of λμ-calculus in the view of intersection type. In this paper, we will extend the intersection type assignment system to the Λμ -calculus, and show that it characterizes those terms that are strongly normalizing. The system satisfies the subject reduction and subject expansion properties.

The approximation theorem for the Λμ-calculus

We consider a notion of approximation for terms of de Groote–Saurin Λμ-calculus. Then, we introduce an intersection type assignment system for that calculus which is invariant under subject conversion. The type assignment system also induces a filter model, which is an extensional Λμ-model in the sense of Nakazawa and Katsumata. We then establish the approximation theorem, stating that a type can be assigned to a term in the system if and only if it can be assigned to same of its approximations.

Essential and relational models

Intersection type assignment systems can be used as a general framework for building logical models of λ-calculus that allow to reason about the denotation of terms in a finitary way. We defineessentialmodels (a new class of logical models) through a parametric type assignment system using non-idempotent intersection types. Under an interpretation of terms based on typings instead than the usual one based on types, every suitable instance of the parameters induces a λ-model, whose theory is sensible. We prove that this type assignment system provides a logical description of a family of λ-models arising from a category of sets and relations.

Probabilistic coherence spaces are fully abstract for probabilistic PCF

Probabilistic coherence spaces (PCoh) yield a semantics of higher-order probabilistic computation, interpreting types as convex sets and programs as power series. We prove that the equality of interpretations in Pcoh characterizes the operational indistinguishability of programs in PCF with a random primitive. This is the first result of full abstraction for a semantics of probabilistic PCF. The key ingredient relies on the regularity of power series. Along the way to the theorem, we design a weighted intersection type assignment system giving a logical presentation of PCoh.

Semantic Types and Approximation for Featherweight Java

We consider semantics for the class-based object-oriented calculus Featherweight Java (without casts) based upon approximation. We also define an intersection type assignment system for this calculus and show that it satisfies subject reduction and expansion, i.e. types are preserved under reduction and its converse. We establish a link between type assignment and the approximation semantics by showing an approximation result, which leads to a sufficient condition for the characterisation of head-normalisation and termination. We show the expressivity of both our calculus and our type system by defining an encoding of Combinatory Logic into our calculus and showing that this encoding preserves typeability. We also show that our system characterises the normalising and strongly normalising terms for this encoding. We thus demonstrate that the great analytic capabilities of intersection types can be applied to the context of class-based object orientation.

Intersection Types from a Proof-theoretic Perspective

In this work we present a proof-theoretical justification for the intersection type assignment system IT by means of the logical system Intersection Synchronous Logic ISL. ISL builds classes of equivalent deductions of the implicative and conjunctive fragment of the intuitionistic logic NJ. ISL results from decomposing intuitionistic conjunction into two connectives: a synchronous conjunction, that can be used only among equivalent deductions of NJ, and an asynchronous one, that can be applied among any sets of deductions of NJ. A term decoration of ISL exists so that it matches both: the IT assignment system, when only the synchronous conjunction is used, and the simple types assignment with pairs and projections, when the asynchronous conjunction is used. Moreover, the proof of strong normalization property for ISL is a simple consequence of the same property in NJ and hence strong normalization for IT comes for free.

Intersection Logic in sequent calculus style

The intersection type assignment system has been designed directly as deductive system for assigning formulae of the implicative and conjunctive fragment of the intuitionistic logic to terms of lambda-calculus. But its relation with the logic is not standard. Between all the logics that have been proposed as its foundation, we consider ISL, which gives a logical interpretation of the intersection by splitting the intuitionistic conjunction into two connectives, with a local and global behaviour respectively, being the intersection the local one. We think ISL is a logic interesting by itself, and in order to support this claim we give a sequent calculus formulation of it, and we prove that it enjoys the cut elimination property.

Completeness and partial soundness results for intersection and union typing for [formula omitted

This paper studies intersection and union type assignment for the calculus λ ¯ μ μ ̃ (Curien and Herbelin, 2000 [16]), a proof-term syntax for Gentzen’s classical sequent calculus, with the aim of defining a type-based semantics, via setting up a system that is closed under conversion. We will start by investigating what the minimal requirements are for a system, for λ ¯ μ μ ̃ to be complete (closed under redex expansion); this coincides with System M ∩ ∪ , the notion defined in Dougherty et al. (2004) [18]; however, we show that this system is not sound (closed under subject reduction), so our goal cannot be achieved. We will then show that System M ∩ ∪ is also not complete, but can recover from this by presenting System M c as an extension of M ∩ ∪ (by adding typing rules) and showing that it satisfies completeness; it still lacks soundness. We show how to restrict M ∩ ∪ so that it satisfies soundness as well by limiting the applicability of certain type assignment rules, but only when limiting reduction to (confluent) call-by-name or call-by-value reduction; in restricting the system this way, we sacrifice completeness. These results when combined show that, with respect to full reduction, it is not possible to define a sound and complete intersection and union type assignment system for λ ¯ μ μ ̃ .

Logical Semantics for Stability

Type assignment systems for λ -calculus based on intersection types are a general framework for building models of λ -calculus (known as filter-models) which are useful tools for reasoning in a finitary way about the denotational interpretation of terms. Indeed the denotation of a term is the set of types derivable for it and a type is a “finite piece” of information on such a denotation. This approach to the λ -calculus semantics is called in the literature logical semantics , and it has been intensively studied in relation with λ -models in the Scott's domain setting. In this paper we define two intersection type assignment systems for λ -calculus, parametric with respect to a coherence relation between types. We prove that, when the instantiation of the parameter satisfies a given condition, our two type systems induce models of λ -calculus, that we call clique-models . Lastly we show that such systems give a logical characterization of two classes of models built on the category of Girard's coherence spaces and stable functions

Intersection-types à la Church

In this paper, we present Λ ∧ t , a fully typed λ-calculus based on the intersection-type system discipline, which is a counterpart à la Church of the type assignment system as invented by Coppo and Dezani. The relationship between Λ ∧ t and the intersection type assignment system is the standard isomorphism between typed and type assignment system, and so the typed language inherits from the untyped system all the good properties, like subject reduction and strong normalization. Moreover, both type checking and type reconstruction are decidable.

Intersection types for λGtz-calculus

We introduce an intersection type assignment system for Espirito-Santo?s ?Gtz-calculus, a term calculus embodying the Curry-Howard correspondence for the intuitionistic sequent calculus. We investigate basic properties of this intersection type system. Our main result is Subject reduction property.

Compositional characterisations of [formula omitted]-terms using intersection types

We show how to characterise compositionally a number of evaluation properties of λ -terms using Intersection Type assignment systems. In particular, we focus on termination properties, such as strong normalisation, normalisation, head normalisation, and weak head normalisation. We consider also the persistent versions of such notions. By way of example, we consider also another evaluation property, unrelated to termination, namely reducibility to a closed term. Many of these characterisation results are new, to our knowledge, or else they streamline, strengthen, or generalise earlier results in the literature. The completeness parts of the characterisations are proved uniformly for all the properties, using a set-theoretical semantics of intersection types over suitable kinds of stable sets. This technique generalises Krivine's and Mitchell's methods for strong normalisation to other evaluation properties.

Type Assignement for Mobile Objects

We address the problem of formal reasoning about mobile code. We consider an Ambient Calculus, where process syntax includes constructs for object-oriented sequential programming. For the sake of concreteness, and because of practical relevance, we consider objects using message exchange to implement method invocation and overriding. The main contribution of the paper is an intersection type assignment system, obtained by a combination of systems introduced for the Calculus of Mobile Ambients and for the &-calculus. We exploit in the mobility framework a typical feature of the intersection type discipline for object calculi, namely late typing of self. The proposed system is then checked against standard properties of related systems, establishing type invariance and a completeness theorem.

Behavioural inverse limit λ-models

We construct two inverse limit λ-models which completely characterise sets of terms with similar computational behaviours: the sets of normalising, head normalising, weak head normalising λ-terms, those corresponding to the persistent versions of these notions, and the sets of closable, closable normalising, and closable head normalising λ-terms. More precisely, for each of these sets of terms there is a corresponding element in at least one of the two models such that a term belongs to the set if and only if its interpretation (in a suitable environment) is greater than or equal to that element. We use the finitary logical description of the models, obtained by defining suitable intersection type assignment systems, to prove this.

Cut-Elimination in the Strict Intersection Type Assignment System is Strongly Normalizing

This paper defines reduction on derivations (cut-elimination) in the Strict Intersection Type Assignment System of an earlier paper and shows a strong normalization result for this reduction. Using this result, new proofs are given for the approximation theorem and the characterization of normalizability of terms using intersection types.

Infinite intersection types

A type theory with infinitary intersection and union types for an extension of the λ -calculus is introduced. Types are viewed as upper closed subsets of a Scott domain and intersection and union type constructors are interpreted as the set-theoretic intersection and union, respectively, even when they are not finite. The assignment of types to λ -terms extends naturally the basic type assignment system. We prove soundness and completeness using a generalization of Abramsky’s finitary logic of domains. Finally, we apply the framework to applicative transition systems, obtaining a sound a complete infinitary intersection type assignment system for the lazy λ -calculus.

Strongly Normalising Cut-Elimination with Strict Intersection Types

This paper defines reduction on derivations in the strict intersection type assignment system of [2], by generalising cutelimination, and shows a strong normalisation result for this reduction. Using this result, new proofs are given for the approximation theorem and the characterisation of normalisability using intersection types.