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Strong normalization and equi(co)inductive types
 Proc. of the 8th Int. Conf. on Typed Lambda Calculi and Applications, TLCA 2007, volume 4583 of Lect. Notes in Comput. Sci. SpringerVerlag (2007), 8–22
"... Abstract. A type system for the lambdacalculus enriched with recursive and corecursive functions over equiinductive andcoinductive types is presented in which all welltyped programs are strongly normalizing. The choice of equiinductive types, instead of the more common isoinductive types, in ue ..."
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Abstract. A type system for the lambdacalculus enriched with recursive and corecursive functions over equiinductive andcoinductive types is presented in which all welltyped programs are strongly normalizing. The choice of equiinductive types, instead of the more common isoinductive types, in uences both reduction rules and the strong normalization proof. By embedding iso into equitypes, the latter ones are recognized as more fundamental. A model based on orthogonality is constructed where a semantical type corresponds to a set of observations, and soundness of the type system is proven. 1
Metatheoretical Results for a Modal λCalculus
, 2000
"... This paper presents the proofs of the strong normalization, subject reduction, and ChurchRosser theorems for a presentation of the intuitionistic modal λcalculus S4. It is adapted from Healfdene Goguen's thesis, where these properties are shown for the simply typed λcalculus and for Luo&a ..."
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This paper presents the proofs of the strong normalization, subject reduction, and ChurchRosser theorems for a presentation of the intuitionistic modal λcalculus S4. It is adapted from Healfdene Goguen's thesis, where these properties are shown for the simply typed λcalculus and for Luo's type theory UTT. Following this method, we introduce the notion of typed operational semantics for our system. We define a notion of typed substitution for our system, which has context stacks instead of the usual contexts. This latter peculiarity leads to the main diculties and consequently to the main original features in our proofs. The techniques elaborated in this work have already been found useful in recent works [DL98, DL99] and should be further exploited to prove the properties of other systems based on modality.
Refinement Types for Logical Frameworks
, 2010
"... The logical framework LF and its metalogic Twelf can be used to encode and reason about a wide variety of logics, languages, and other deductive systems in a formal, machinecheckable way. Recent studies have shown that MLlike languages can profitably be extended with a notion of subtyping called r ..."
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The logical framework LF and its metalogic Twelf can be used to encode and reason about a wide variety of logics, languages, and other deductive systems in a formal, machinecheckable way. Recent studies have shown that MLlike languages can profitably be extended with a notion of subtyping called refinement types. A refinement type discipline uses an extra layer of term classification above the usual type system to more accurately capture certain properties of terms. I propose that adding refinement types to LF is both useful and practical. To support the claim, I exhibit an extension of LF with refinement types called LFR, work out important details of its metatheory, delineate a practical algorithm for refinement type reconstruction, and present several case studies that highlight the utility of refinement types for formalized mathematics. In the end I find that refinement types and LF are a match made in heaven: refinements enable many rich new modes of expression, and the simplicity of
Characterizing Strongly Normalizing Terms of a lambdaCalculus with Generalized Applications via Intersection Types
"... An intersection type assignment system for the extension LJ of the untyped lcalculus, introduced by Joachimski and Matthes, is given and proven to characterize the strongly normalizing terms of LJ. Since LJ's generalized applications naturally allow permutative/commuting conversions, this ..."
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An intersection type assignment system for the extension LJ of the untyped lcalculus, introduced by Joachimski and Matthes, is given and proven to characterize the strongly normalizing terms of LJ. Since LJ's generalized applications naturally allow permutative/commuting conversions, this is the first analysis of a term rewrite system with permutative conversions by help of intersection types. Two proofs are given for the fact that the typable terms are strongly normalizing: One by the computability predicates method a la Tait and one showing directly that strongly normalizing typable terms are closed under (generalized) application and substitution. It is also shown that a straightforward extension of the type assignment for lcalculus fails to capture the strongly normalizing terms. Keywords Intersection Types, Strong Normalization, Permutative Conversions, Saturated Sets. 1 Introduction In [5] an extension LJ of lcalculus with generalized applications inspired by vo...
Antisymmetry of higherorder subtyping and equality by subtyping
 Math. Struct. in Comput. Sci
, 2006
"... This paper gives the first proof that the subtyping relation of a higherorder lambda calculus, F ω ≤, is antisymmetric, establishing in the process that the subtyping relation is a partial order—reflexive, transitive, and antisymmetric up to βequality. While a subtyping relation is reflexive and ..."
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This paper gives the first proof that the subtyping relation of a higherorder lambda calculus, F ω ≤, is antisymmetric, establishing in the process that the subtyping relation is a partial order—reflexive, transitive, and antisymmetric up to βequality. While a subtyping relation is reflexive and transitive by definition, antisymmetry is a derived property. The result, which may seem obvious to the nonexpert, is technically challenging, and had been an open problem for almost a decade. In this context, typed operational semantics for subtyping offers a powerful new technology to solve the problem: of particular importance is our extended rule for the wellformedness of types with head variables. The paper also gives a presentation of F ω ≤ without a relation for βequality, apparently the first such, and shows its equivalence with the traditional presentation. 1
Fixed Points of Type Operators and Primitive Recursion
, 2004
"... Abstract. For nested or heterogeneous datatypes, terminating recursion schemes considered so far have been instances of iteration, excluding efficient definitions of fixedpoint unfolding. Two solutions of this problem are proposed: The first one is a system with equirecursive nonstrictly positive ..."
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Abstract. For nested or heterogeneous datatypes, terminating recursion schemes considered so far have been instances of iteration, excluding efficient definitions of fixedpoint unfolding. Two solutions of this problem are proposed: The first one is a system with equirecursive nonstrictly positive type operators of arbitrary finite kinds, where fixedpoint unfolding is computationally invisible due to its treatment on the level of type equality. Positivity is ensured by a polarized kinding system, and strong normalization is proven by a model construction based on saturated sets. The second solution is a formulation of primitive recursion for arbitrary type constructors of any rank. Although without positivity restriction, the second system embeds—even operationally—into the first one. 1
Under consideration for publication in Math. Struct. in Comp. Science Boxed Ambients with Communication Interfaces †
, 2006
"... an ambient calculus with a flexible communication policy. Traditionally, typed ambient calculi have a fixed communication policy determining the kind of information that can be exchanged with a parent ambient, even though mobility changes the parent. BACI lifts that restriction, allowing different c ..."
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an ambient calculus with a flexible communication policy. Traditionally, typed ambient calculi have a fixed communication policy determining the kind of information that can be exchanged with a parent ambient, even though mobility changes the parent. BACI lifts that restriction, allowing different communication policies with different parents during computation. Furthermore, BACI splits communication and mobility by making explicit the channels of communication between ambients. In contrast with other typed ambient calculi where communication policies are global, each ambient in BACI is equipped with a description of the communication policies ruling its information exchange with parent and child ambients. The communication policies of ambients increase when they move: more precisely, when an ambient enters another ambient, the entering ambient and the host ambient can exchange their communication ports and agree on the kind of information to be exchanged. This information is recorded locally in both ambients. We show the typesoundness of BACI, proving that it satisfies the subject reduction property, and we study its behavioural semantics by means of a labelled transition system.
Equality to Equals and Unequals: A Revisit of the Equivalence and Nonequivalence Criteria in ClassLevel Testing of ObjectOriented Software
 IEEE TRANSACTIONS ON SOFTWARE ENGINEERING
, 2013
"... Algebraic specifications have been used in the testing of objectoriented programs and received much attention since the 1990s. It is generally believed that classlevel testing based on algebraic specifications involves two independent aspects: the testing of equivalent and nonequivalent ground ter ..."
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Algebraic specifications have been used in the testing of objectoriented programs and received much attention since the 1990s. It is generally believed that classlevel testing based on algebraic specifications involves two independent aspects: the testing of equivalent and nonequivalent ground terms. Researchers have cited intuitive examples to illustrate the philosophy that even if an implementation satisfies all the requirements specified by the equivalence of ground terms, it may still fail to satisfy some of the requirements specified by the nonequivalence of ground terms. Thus, both the testing of equivalent ground terms and the testing of nonequivalent ground terms have been considered as significant and cannot replace each other. In this paper, we present an innovative finding that, given any canonical specification of a class with proper imports, a complete implementation satisfies all the observationally equivalent ground terms if and only if it satisfies all the observationally nonequivalent ground terms. As a result, these two aspects of software testing cover each other and can therefore replace each other. These findings provide a deeper understanding of software testing based on algebraic specifications, rendering the theory more elegant and complete. We also highlight a couple of important practical implications of our theoretical results.