Towards an Intersection Typed System à la Church

Topics in Theoretical Computer Science

Stolze

2017

Self Cite

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.

Section: Introductionmentioning

confidence: 99%

A Decidable Subtyping Logic for Intersection and Union Types

Topics in Theoretical Computer Science

Stolze

2017

Self Cite

“…The reader will find a good number of examples showing some typing in the intersection type system in [LR07]. As an example of the present system using intersection and union types in an essential way, we treat the example (due to Pierce) that shows the failure of subject reduction for simple, non parallel, reduction in [BDCD95].…”

Section: Example Of Typing For λ ∧∨ Tmentioning

confidence: 99%

“…The languages proposed in these papers have been designed with various purposes, and they do not satisfy one or more of our desiderata above. A fuller discussion of this related work can be found in [LR07].…”

Section: Introductionmentioning

confidence: 99%

“…In this paper we extend the system of [LR07] to a calculus with union types. This is non-trivial, essentially because the standard typing rule for ∨-elimination is, as has been noted by many authors, so awkward.…”

Section: Introductionmentioning

confidence: 99%

“…A solution to this problem was introduced in [LR07] where a calculus is designed whose terms comprise two parts, carrying computational and logical information respectively. The first component (the marked-term) is a simply typed λ-term, but types are variable-marks.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Logic and Computation in a Lambda Calculus with Intersection and Union Types

Dougherty

Logic for Programming, Artificial Intelligence, and Reasoning

2010

Self Cite

We present an explicitly typed lambda calculus "à la Church" based on the union and intersection types discipline; this system is the counterpart of the standard type assignment calculus "à la Curry." Our typed calculus enjoys Subject Reduction and confluence, and typed terms are strongly normalizing when the universal type is omitted. Moreover both type checking and type reconstruction are decidable. In contrast to other typed calculi, a system with union types will fail to be "coherent" in the sense of Tannen, Coquand, Gunter, and Scedrov: different proofs of the same typing judgment will not necessarily have the same meaning. In response, we introduce a decidable notion of equality on type-assignment derivations inspired by the equational theory of bicartesian-closed categories.

A Realizability Interpretation for Intersection and Union Types

Dougherty

de’Liguoro

Programming Languages and Systems

et al. 2016

Self Cite

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.