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Despite the fact that call-by-value λ-calculus was defined by Plotkin in 1977, we believe that its theory of program approximation is still at the beginning. A problem that is often encountered when studying its operational semantics is that, during the reduction of a λ-term, some redexes remain stuck (waiting for a value). Recently, Carraro and Guerrieri proposed to endow this calculus with permutation rules, naturally arising in the context of linear logic proof-nets, that succeed in unblocking a certain number of such redexes. In the present paper we introduce a new class of models of call-by-value λ-calculus, arising from non-idempotent intersection type systems. Beside satisfying the usual properties as soundness and adequacy, these models validate the permutation rules mentioned above as well as some reductions obtained by contracting suitable λI-redexes. Thanks to these (perhaps unexpected) features, we are able to demonstrate that every model living in this class satisfies an Approximation Theorem with respect to a refined notion of syntactic approximant. While this kind of results often require impredicative techniques like reducibility candidates, the quantitative information carried by type derivations in our system allows us to provide a combinatorial proof. [ABSTRACT FROM AUTHOR]

For a long time, intersection types have been admired for their surprising ability to complete the simply typed lambda calculus. Intersection types are an example of an implicit typing feature which can describe program behavior without manifesting itself within the syntax of a program. Dual to intersections, union types are another implicit typing feature which extends the completeness property of intersection types in the lambda calculus to full-fledged programming languages. However, the formalization of union types can easily break other desirable meta-theoretical properties of the type system. But why should unions be troublesome when their dual, intersections, are not? We look at the issues surrounding the design of type systems for both intersection and union types through the lens of duality by formalizing them within the symmetric language of the classical sequent calculus. In order to formulate type systems which have all of our properties of interest—soundness, completeness, and type safety—we also look at the impact of evaluation strategy on typing. As a result, we present two dual type systems—one for call-by-value and one for call-by-name evaluation—which have all three properties. We also consider the possibility of classical non-deterministic evaluation, for which there is a choice between two different systems depending on which properties are desired: a full type system which is complete, and a simplified type system which is sound and type safe. [ABSTRACT FROM AUTHOR]

We present a new type system for verifying the security of reference implementations of cryptographic protocols written in a core functional programming language. The type system combines prior work on refinement types, with union, intersection, and polymorphic types, and with the novel ability to reason statically about the disjointness of types. The increased expressivity enables the analysis of important protocol classes that were previously out of scope for the type-based analyses of reference protocol implementations. In particular, our types can statically characterize: (i) more usages of asymmetric cryptography, such as signatures of private data and encryptions of authenticated data; (ii) authenticity and integrity properties achieved by showing knowledge of secret data; (iii) applications based on zero-knowledge proofs. The type system comes with a mechanized proof of correctness and an efficient type-checker. [ABSTRACT FROM AUTHOR]

With the eye on defining a type-based semantics, this paper defines intersection and union type assignment for the sequent calculus 𝒳, a substitution-free language that enjoys the Curry-Howard correspondence with respect to the implicative fragment of Gentzen's sequent calculus for classical logic. We investigate the minimal requirements for such a system to be complete (i.e. closed under redex expansion), and show that the non-logical nature of both intersection and union types disturbs the soundness (i.e. closed uder reduction) properties. This implies that this notion of intersection-union type assignment needs to be restricted to satisfy soundness as well, making it unsuitable to define a semantics. We will look at two (confluent) notions of reduction, called Call-by-Name and Call-by-Value, and prove soundness results for those. [ABSTRACT FROM AUTHOR]

This paper gives a characterisation, via intersection types, of the strongly normalising proof-terms of an intuitionistic sequent calculus (where LJ easily embeds). The soundness of the typing system is reduced to that of a well known typing system with intersection types for the ordinary λ-calculus. The completeness of the typing system is obtained from subject expansion at root position. Next we use our result to analyze the characterisation of strong normalisability for three classes of intuitionistic terms: ordinary λ-terms, ΛJ-terms (λ-terms with generalised application), and λx-terms (λ-terms with explicit substitution). We explain via our system why the type systems in the natural deduction format for ΛJ and λx known from the literature contain extra, exceptional rules for typing generalised application or substitution; and we show a new characterisation of the β-strongly normalising λ-terms, as a corollary to a PSN-result, relating the λ-calculus and the intuitionistic sequent calculus. Finally, we obtain variants of our characterisation by restricting the set of assignable types to sub-classes of intersection types, notably strict types. In addition, the known characterisation of the β-strongly normalising λ-terms in terms of assignment of strict types follows as an easy corollary of our results. [ABSTRACT FROM AUTHOR]

Expansion is a crucial operation for calculating principal typings in intersection type systems. Because the early definitions of expansion were complicated, E-variables were introduced in order to make the calculations easier to mechanise and reason about. Recently, E-variables have been further simplified and generalised to also allow calculating other type operators than just intersection. There has been much work on semantics for type systems with intersection types, but none whatsoever before our work, on type systems with E-variables. In this paper we expose the challenges of building a semantics for E-variables and we provide a novel solution. Because it is unclear how to devise a space of meanings for E-variables, we develop instead a space of meanings for types that is hierarchical. First, we index each type with a natural number and show that although this intuitively captures the use of E-variables, it is difficult to index the universal type ω with this hierarchy and it is not possible to obtain completeness of the semantics if more than one E-variable is used. We then move to a more complex semantics where each type is associated with a list of natural numbers and establish that both ω and an arbitrary number of E-variables can be represented without losing any of the desirable properties of a realisability semantics. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]