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On the strength of proofirrelevant type theories
 of Lecture Notes in Computer Science
, 2006
"... Vol. 4 (3:13) 2008, pp. 1–20 ..."
Complexity of strongly normalising λterms via nonidempotent intersection types
"... We present a typing system for the λcalculus, with nonidempotent intersection types. As it is the case in (some) systems with idempotent intersections, a λterm is typable if and only if it is strongly normalising. Nonidempotency brings some further information into typing trees, such as a bound o ..."
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We present a typing system for the λcalculus, with nonidempotent intersection types. As it is the case in (some) systems with idempotent intersections, a λterm is typable if and only if it is strongly normalising. Nonidempotency brings some further information into typing trees, such as a bound on the longest βreduction sequence reducing a term to its normal form. We actually present these results in Klop’s extension of λcalculus, where the bound that is read in the typing tree of a term is refined into an exact measure of the longest reduction sequence. This complexity result is, for longest reduction sequences, the counterpart of de Carvalho’s result for linear headreduction sequences.
Realisability and adequacy for (co)induction
"... Abstract. We prove the correctness of a formalised realisability interpretation of extensions of firstorder theories by inductive and coinductive definitions in an untyped λcalculus with fixedpoints. We illustrate the use of this interpretation for program extraction by some simple examples in th ..."
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Abstract. We prove the correctness of a formalised realisability interpretation of extensions of firstorder theories by inductive and coinductive definitions in an untyped λcalculus with fixedpoints. We illustrate the use of this interpretation for program extraction by some simple examples in the area of exact real number computation, and hint at further nontrivial applications in computable analysis. 1
A simple typetheoretic language: MiniTT
"... This paper presents a formal description of a small functional language with dependent types. The language contains data types, mutual recursive/inductive definitions and a universe of small types. The syntax, semantics and type system is specified in such a way that the implementation of a parser, ..."
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This paper presents a formal description of a small functional language with dependent types. The language contains data types, mutual recursive/inductive definitions and a universe of small types. The syntax, semantics and type system is specified in such a way that the implementation of a parser, interpreter and type checker is straightforward. The main difficulty is to design the conversion algorithm in such a way that it works for open expressions. The paper ends with a complete implementation in Haskell (around 400 lines of code).
Filter models: nonidempotent intersection types, orthogonality and polymorphism
"... This paper revisits models of typed λcalculus based on filters of intersection types: By using nonidempotent intersections, we simplify a methodology that produces modular proofs of strong normalisation based on filter models. Nonidempotent intersections provide a decreasing measure proving a key ..."
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This paper revisits models of typed λcalculus based on filters of intersection types: By using nonidempotent intersections, we simplify a methodology that produces modular proofs of strong normalisation based on filter models. Nonidempotent intersections provide a decreasing measure proving a key termination property, simpler than the reducibility techniques used with idempotent intersections. Such filter models are shown to be captured by orthogonality techniques: we formalise an abstract notion of orthogonality model inspired by classical realisability, and express a filter model as one of its instances, along with two termmodels (one of which captures a now common technique for strong normalisation). Applying the above range of model constructions to Currystyle System F describes at different levels of detail how the infinite polymorphism of System F can systematically be reduced to the finite polymorphism of intersection types.
On the Values of Reducibility Candidates
, 2013
"... Abstract. The straightforward elimination of union types is known to break subject reduction, and for some extensions of the lambdacalculus, to break strong normalization as well. Similarly, the straightforward elimination of implicit existential types breaks subject reduction. We propose eliminati ..."
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Abstract. The straightforward elimination of union types is known to break subject reduction, and for some extensions of the lambdacalculus, 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 callbyvalue 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 lambdacalculus 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. 1
Computability and analysis: the legacy of Alan Turing
, 2012
"... For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a par ..."
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For most of its history, mathematics was algorithmic in nature. The geometric claims in Euclid’s Elements fall into two distinct categories: “problems, ” which assert that a construction can be carried out to meet a given specification, and “theorems, ” which assert that some property holds of a particular geometric