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On the Stability by Union of Reducibility Candidates
 In Proceedings of FoSSaCS’07, volume 4423 of LNCS
"... Abstract. We investigate some aspects of proof methods for the termination of (extensions of) the secondorder λcalculus in presence of union and existential types. We prove that Girard’s reducibility candidates are stable by union iff they are exactly the nonempty sets of terminating terms which ..."
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Abstract. We investigate some aspects of proof methods for the termination of (extensions of) the secondorder λcalculus in presence of union and existential types. We prove that Girard’s reducibility candidates are stable by union iff they are exactly the nonempty sets of terminating terms which are downwardclosed w.r.t. a weak observational preorder. We show that this is the case for the Currystyle secondorder λcalculus. As a corollary, we obtain that reducibility candidates are exactly the Tait’s saturated sets that are stable by reduction. We then extend the proof to a system with product, coproduct and positive isorecursive types. 1
Callbyvalue nondeterminism in a linear logic type discipline
 in "LFCS  Logical Foundations of Computer Science  2013
, 2013
"... Abstract. We consider the callbyvalue λcalculus extended with a mayconvergent nondeterministic choice and a mustconvergent parallel composition. Inspired by recent works on the relational semantics of linear logic and nonidempotent intersection types, we endow this calculus with a type syste ..."
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Abstract. We consider the callbyvalue λcalculus extended with a mayconvergent nondeterministic choice and a mustconvergent parallel composition. Inspired by recent works on the relational semantics of linear logic and nonidempotent intersection types, we endow this calculus with a type system based on the socalled Girard’s second translation of intuitionistic logic into linear logic. We prove that a term is typable if and only if it is converging, and that its typing tree carries enough information to give a bound on the length of its lazy callbyvalue reduction. Moreover, when the typing tree is minimal, such a bound becomes the exact length of the reduction.
On the preciseness of subtyping in session types
 In Proceedings of the 16th International Symposium on Principles and Practice of Declarative Programming (PPDP
, 2014
"... Subtyping in concurrency has been extensively studied since early 1990s as one of the most interesting issues in type theory. The correctness of subtyping relations has been usually provided as the soundness for type safety. The converse direction, the completeness, has been largely ignored in spit ..."
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Subtyping in concurrency has been extensively studied since early 1990s as one of the most interesting issues in type theory. The correctness of subtyping relations has been usually provided as the soundness for type safety. The converse direction, the completeness, has been largely ignored in spite of its usefulness to define the greatest subtyping relation ensuring type safety. This paper formalises preciseness (i.e. both soundness and completeness) of subtyping for mobile processes and studies it for the synchronous and the asynchronous session calculi. We first prove that the wellknown session subtyping, the branchingselection subtyping, is sound and complete for the synchronous calculus. Next we show that in the asynchronous calculus, this subtyping is incomplete for typesafety: that is, there exist session types T and S such that T can safely be considered as a subtype of S, but T 6 S is not derivable by the subtyping. We then propose an asynchronous subtyping system which is sound and complete for the asynchronous calculus. The method gives a general guidance to design rigorous channelbased subtypings respecting desired safety properties.
Preciseness of Subtyping on Intersection and Union Types?
"... 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 dif ..."
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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. 1
Intersection Types, λmodels, and Böhm Trees
"... This paper is an introduction to intersection type disciplines, with the aim of illustrating their theoretical relevance in the foundations of λcalculus. We start by describing the wellknown results showing the deep connection between intersection type systems and normalization properties, i.e. ..."
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This paper is an introduction to intersection type disciplines, with the aim of illustrating their theoretical relevance in the foundations of λcalculus. We start by describing the wellknown results showing the deep connection between intersection type systems and normalization properties, i.e., their power of naturally characterizing solvable, normalizing, and strongly normalizing pure λterms. We then explain the importance of intersection types for the semantics of λcalculus, through the construction of filter models and the representation of algebraic lattices. We end with an original result that shows how intersection types also allow to naturally characterize tree representations of unfoldings of λterms (Böhm trees).
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
On the characterization of models of H∗
"... We give a characterization, with respect to a large class of models of untyped λcalculus, of those models that are fully abstract for headnormalization, i.e., whose equational theory is H∗. An extensional Kmodel D is fully abstract if and only if it is hyperimmune, i.e., nonwell founded chains o ..."
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We give a characterization, with respect to a large class of models of untyped λcalculus, of those models that are fully abstract for headnormalization, i.e., whose equational theory is H∗. An extensional Kmodel D is fully abstract if and only if it is hyperimmune, i.e., nonwell founded chains of elements of D cannot be captured by any recursive function.
Semantics of the reFLect Language
, 2004
"... is a new functional language, developed at Intel for use in hardware design and veri cation. It contains features intended to facilitate the construction, analysis, and manipulation of the language's own programs. It is also intended to be the executable subset of the term language of a theore ..."
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is a new functional language, developed at Intel for use in hardware design and veri cation. It contains features intended to facilitate the construction, analysis, and manipulation of the language's own programs. It is also intended to be the executable subset of the term language of a theorem prover based on higher order logic.