Completeness of intersection and union type assignment systems for call-by-value λ-models

Abstract. The straightforward elimination of union types is known to break subject reduction, and for some extensions of the lambda-calculus, to break strong normalization as well. Similarly, the straightforward elimination of implicit existential types breaks subject reduction. We propose elimination rules for union types and implicit existential quantification which use a form call-by-value issued from Girard's reducibility candidates. We show that these rules remedy the above mentioned difficulties, for strong normalization and, for the existential quantification, for subject reduction as well. Moreover, for extensions of the lambda-calculus based on intuitionistic logic, we show that the obtained existential quantification is equivalent to its usual impredicative encoding w.r.t. provability in realizability models built from reducibility candidates and biorthogonals.

“…In contrast with [8,6,19], this does not force the whole calculus to be call-by-value. We build on [15,16], where a general notion of Girard's candidates is proposed.…”

Section: Introductionmentioning

confidence: 94%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

On the Values of Reducibility Candidates

Riba

2009

“…These technique has been recently applied to the study of lazy -calculus 4], of parallel extension of -calculus 7,16], and of call-by-value -calculus 17,20]. Further studies of reduction properties via intersection types are reported in 15].…”

Section: Related Workmentioning

confidence: 99%

Characterizing Convergent Terms in Object Calculi via Intersection Types

de’Liguoro

2001

“…This well-established powerful technique is applied to the pure λ-calculus with arrow and intersection types by Barendregt et al [3], to a call-by-value λ-calculus with arrow, intersection and union types by van Bakel et al [1] and by Ishihara and Kurata [16], and to a wide class of calculi with arrow, union and pair types by Vouillon [25].…”

Section: Introductionmentioning

confidence: 99%

Preciseness of Subtyping on Intersection and Union Types

Dezani‐Ciancaglini

Ghilezan

2014

Abstract. The notion of subtyping has gained an important role both in theoretical and applicative domains: in lambda and concurrent calculi as well as in programming languages. The soundness and the completeness, together referred to as the preciseness of subtyping, can be considered from two different points of view: denotational and operational. The former preciseness is based on the denotation of a type which is a mathematical object that describes the meaning of the type in accordance with the denotations of other expressions from the language. The latter preciseness has been recently developed with respect to type safety, i.e. the safe replacement of a term of a smaller type when a term of a bigger type is expected. We propose a technique for formalising and proving operational preciseness of the subtyping relation in the setting of a concurrent lambda calculus with intersection and union types. The key feature is the link between typings and the operational semantics. We then prove soundness and completeness getting that the subtyping relation of this calculus enjoys both denotational and operational preciseness.