Using Curry-Howard isomorphism, we extend the typed lambda-calculus with intersection and union types, and its corresponding proof-functional logic, previously defined by the authors, with subtyping and explicit coercions. We show the extension of the lambda-calculus to be isomorphic to the Barbanera-Dezani-de'Liguoro type assignment system and we provide a sound interpretation of the proof-functional logic with the NJ(β) logic, using Mints' realizers. We finally present a sound and complete algorithm for subtyping in presence of intersection and union types. The algorithm is conceived to work for the (sub)type theory Ξ.
We present an ongoing implementation of a dependent-type theory (∆-framework) based on the Edinburgh Logical Framework LF, extended with Proof-functional logical connectives such as intersection, union, and strong (or minimal relevant) implication. Their combination opens up new possibilities of formal reasoning on proof-theoretic semantics. We provide some examples in the extended type theory and we outline a type checker. The theory of the system is under investigation. Once validated in vitro, the proof-functional type theory could be successfully plugged into existing truth-functional proof-systems.
Session types are a well-established framework for the specification of interactions between components of a distributed systems. An important issue is how to determine the type for an open system, i.e., obtained by assembling subcomponents, some of which could be missing. To this end, we introduce partial sessions and partial (multiparty) session types. Partial sessions can be composed, and the type of the resulting system is derived from those of its components without knowing any suitable global type nor the types of missing parts. To deal with this incomplete information, partial session types represent the subjective views of the interactions from participants’ perspectives; when sessions are composed, different partial views can be merged if compatible, yielding a unified view of the session. Incompatible types, due to, e.g., miscommunications or deadlocks, are detected at the merging phase. In fact, in this theory the distinction between global and local types vanishes. We apply these types to a process calculus for which we prove subject reduction and progress, so that well-typed systems never violate the prescribed constraints. In particular, we introduce a generalization of the progress property, in order to accommodate the case when a partial session cannot progress not due to a deadlock, but because some participants are still missing. Therefore, partial session types support the development of systems by incremental assembling of components.
Proof-functional logical connectives allow reasoning about the structure of logical proofs, in this way giving to the latter the status of first-class objects. This is in contrast to classical truth-functional connectives where the meaning of a compound formula is dependent only on the truth value of its subformulas. In this paper we present a typed lambda calculus, enriched with strong products, strong sums, and a related proof-functional logic. This calculus, directly derived from a typed calculus previously defined by two of the current authors, has been proved isomorphic to the well-known Barbanera-Dezani-Ciancaglini-de'Liguoro type assignment system. We present a logic L ∩∪ featuring two proof-functional connectives, namely strong conjunction and strong disjunction. We prove the typed calculus to be isomorphic to the logic L ∩∪ and we give a realizability semantics using Mints' realizers [Min89] and a completeness theorem. A prototype implementation is also described.
We present the ∆-calculus, an explicitly typed λ-calculus with strong pairs, projections and explicit type coercions. The calculus can be parametrized with different intersection type theories T , e.g. the Coppo-Dezani, the Coppo-Dezani-Sallé, the Coppo-Dezani-Venneri and the Barendregt-Coppo-Dezani ones, producing a family of ∆-calculi with related intersection typed systems. We prove the main properties like Church-Rosser, unicity of type, subject reduction, strong normalization, decidability of type checking and type reconstruction. We state the relationship between the intersection type assignment systems à la Curry and the corresponding intersection typed systems à la Church by means of an essence function translating an explicitly typed ∆-term into a pure λ-term one. We finally translate a ∆-term with type coercions into an equivalent one without them; the translation is proved to be coherent because its essence is the identity. The generic ∆-calculus can be parametrized to take into account other intersection type theories as the ones in the Barendregt et al. book. Keywords Intersection types, Lambda calculus à la Church and à la Curry, Proof-functional logics 1 Work supported by the COST Action CA15123 EUTYPES "The European research network on types for programming and verification".
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