Silvia Ghilezan

Strong Normalization and Typability with Intersection Types

A simple proof is given of the property that the set of strongly normalizing lambda terms coincides with the set of lambda terms typable in certain intersection type assignment systems.

Lambda terms for natural deduction, sequent calculus and cut elimination

It is well known that there is an isomorphism between natural deduction derivations and typed lambda terms. Moreover, normalising these terms corresponds to eliminating cuts in the equivalent sequent calculus derivations. Several papers have been written on this topic. The correspondence between sequent calculus derivations and natural deduction derivations is, however, not a one-one map, which causes some syntactic technicalities. The correspondence is best explained by two extensionally equivalent type assignment systems for untyped lambda terms, one corresponding to natural deduction (λN) and the other to sequent calculus (λL). These two systems constitute different grammars for generating the same (type assignment relation for untyped) lambda terms. The second grammar is ambiguous, but the first one is not. This fact explains the many-one correspondence mentioned above. Moreover, the second type assignment system has a ‘cut-free’ fragment (λLcf). This fragment generates exactly the typeable lambda terms in normal form. The cut elimination theorem becomes a simple consequence of the fact that typed lambda terms possess a normal form.

The "Relevance" of Intersection and Union Types

The aim of this paper is to investigate a Curry-Howard interpre- tation of the intersection and union type inference system for Combinatory Logic. Types are interpreted as formulas of a Hilbert-style logic L, which turns out to be an extension of the intuitionistic logic with respect to provable dis- junctive formulas (because of new equivalence relations on formulas), while the implicational-conjunctive fragment of L is still a fragment of intuitionistic logic. Moreover, typable terms are translated in a typed version, so that ∨-∧- typed combinatory logic terms are proved to completely codify the associated logical proofs.

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.

Strong normalization of the dual classical sequent calculus

We investigate some syntactic properties of Wadler’s dual calculus, a term calculus which corresponds to classical sequent logic in the same way that Parigot’s λμ calculus corresponds to classical natural deduction. Our main result is strong normalization theorem for reduction in the dual calculus; we also prove some confluence results for the typed and untyped versions of the system.

Characterizing strong normalization in a language with control operators

We investigate some fundamental properties of the reduction relation in the untyped term calculus derived from Curien and Herbelins λμμ. The original λμμ has a system of simple types, based on sequent calculus, embodying a Curry-Howard correspondence with classical logic; the significance of the untyped calculus of raw terms is that it is a Turing-complete language for computation with explicit representation of control as well as code. We define a type assignment system for the raw terms satisfying: a term is typable if and only if it is strongly normalizing. The intrinsic symmetry in the λμμ calculus leads to an essential use of both intersection and union types; in contrast to other union-types systems in the literature, our system enjoys the Subject Reduction property.

Intersection types for the resource control lambda calculi

We propose intersection type assignment systems for two resource control term calculi: the lambda calculus and the sequent lambda calculus with explicit operators for weakening and contraction. These resource control calculi, λ® and λ®Gtz, respectively, capture the computational content of intuitionistic natural deduction and intuitionistic sequent logic with explicit structural rules. Our main contribution is the characterisation of strong normalisation of reductions in both calculi. We first prove that typability implies strong normalisation in λ® by adapting the reducibility method. Then we prove that typability implies strong normalisation in λ®Gtz by using a combination of well-orders and a suitable embedding of λ®Gtz-terms into λ®-terms which preserves types and enables the simulation of all its reductions by the operational semantics of the λ®-calculus. Finally, we prove that strong normalisation implies typability in both systems using head subject expansion.

Types for role-based access control of dynamic web data

We introduce a role-based access control calculus for modelling dynamic web data and a corresponding type system. It is an extension of the Xdp calculus proposed by Gardner and Maffeis. In our framework, a network is a parallel composition of locations, where each location contains processes with roles and a data tree whose edges are associated with roles. Processes can communicate, migrate from a location to another, use the data, change the data and the roles in the local tree. In this way, we obtain a model that controls process access to data. We propose a type system which ensures that a specified network policy is respected during computations. Finally, we show that our calculus obeys the following security properties: (1) all data trees and processes with roles in a location agree with the location policy; (2) a process can migrate only to a location with whose policy it agrees; (3) a process with roles can read and modify only data which are accessible to it; (4) a process with roles can enable and disable roles in agreement with the location policy.

Two behavioural lambda models

We build two inverse limit lambda models which characterize completely sets of terms having similar computational behaviour. More precisely for each one 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. This is proved by using the finitary logical description of the models obtained by defining suitable intersection type assignment systems.

Combining behavioural types with security analysis

Todays software systems are highly distributed and interconnected, and they increasingly rely on communication to achieve their goals; due to their societal importance, security and trustworthiness are crucial aspects for the correctness of these systems. Behavioural types, which extend data types by describing also the structured behaviour of programs, are a widely studied approach to the enforcement of correctness properties in communicating systems. This paper offers a unified overview of proposals based on behavioural types which are aimed at the analysis of security properties.