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20
Higherorder Unification via Explicit Substitutions (Extended Abstract)
 Proceedings of LICS'95
, 1995
"... Higherorder unification is equational unification for βηconversion. But it is not firstorder equational unification, as substitution has to avoid capture. In this paper higherorder unification is reduced to firstorder equational unification in a suitable theory: the λσcal ..."
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Cited by 103 (13 self)
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Higherorder unification is equational unification for βηconversion. But it is not firstorder equational unification, as substitution has to avoid capture. In this paper higherorder unification is reduced to firstorder equational unification in a suitable theory: the λσcalculus of explicit substitutions.
A New Deconstructive Logic: Linear Logic
, 1995
"... The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different a ..."
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Cited by 102 (11 self)
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The main concern of this paper is the design of a noetherian and confluent normalization for LK 2 (that is, classical second order predicate logic presented as a sequent calculus). The method we present is powerful: since it allows us to recover as fragments formalisms as seemingly different as Girard's LC and Parigot's , FD ([9, 11, 27, 31]), delineates other viable systems as well, and gives means to extend the Krivine/Leivant paradigm of `programmingwithproofs' ([22, 23]) to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for nonadditive proof nets, to be precise) using appropriate embeddings (socalled decorations); it is unifying: it organizes known solutions in a simple pattern that makes apparent the how and why of their making. A comparison of our method to that of embedding LK into LJ (intuitionistic sequent calculus) brings to the fore the latter's defects for these `deconstructi...
Strong Stability and the Incompleteness of Stable Models for λCalculus
 ANNALS OF PURE AND APPLIED LOGIC
, 1999
"... We prove that the class of stable models is incomplete with respect to pure λcalculus. More precisely, we show that no stable model has the same theory as the strongly stable version of Park's model. This incompleteness proof can be adapted to the continuous case, giving an incompleteness proof for ..."
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Cited by 20 (0 self)
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We prove that the class of stable models is incomplete with respect to pure λcalculus. More precisely, we show that no stable model has the same theory as the strongly stable version of Park's model. This incompleteness proof can be adapted to the continuous case, giving an incompleteness proof for this case which is much simpler than the original proof by Honsell an Ronchi della Rocca. Moreover, we isolate a very simple finite set, F , of equations and inequations, which has neither a stable nor a continuous model, and which is included in Th(P fs ) and in T
Intersection types for explicit substitutions
, 2003
"... We present a new system of intersection types for a compositionfree calculus of explicit substitutions with a rule for garbage collection, and show that it characterizes those terms which are strongly normalizing. This system extends previous work on the natural generalization of the classical inte ..."
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Cited by 17 (6 self)
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We present a new system of intersection types for a compositionfree calculus of explicit substitutions with a rule for garbage collection, and show that it characterizes those terms which are strongly normalizing. This system extends previous work on the natural generalization of the classical intersection types system, which characterized head normalization and weak normalization, but was not complete for strong normalization. An important role is played by the notion of available variable in a term, which is a generalization of the classical notion of free variable.
Completeness and Partial Soundness Results for Intersection & Union Typing for λµ ˜µ
 Annals of Pure and Applied Logic
"... This paper studies intersection and union type assignment for the calculus λµ ˜µ [17], a proofterm syntax for Gentzen’s classical sequent calculus, with the aim of defining a typebased semantics, via setting up a system that is closed under conversion. We will start by investigating what the minima ..."
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Cited by 6 (6 self)
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This paper studies intersection and union type assignment for the calculus λµ ˜µ [17], a proofterm syntax for Gentzen’s classical sequent calculus, with the aim of defining a typebased semantics, via setting up a system that is closed under conversion. We will start by investigating what the minimal requirements are for a system for λµ ˜µ to be closed under subject expansion; this coincides with System M ∩ ∪ , the notion defined in [19]; however, we show that this system is not closed under subject reduction, so our goal cannot be achieved. We will then show that System M ∩ ∪ is also not closed under subjectexpansion, but can recover from this by presenting System M C as an extension of M ∩ ∪ (by adding typing rules) and showing that it satisfies subject expansion; it still lacks subject reduction. We show how to restrict M ∩ ∪ so that it satisfies subjectreduction as well by limiting the applicability to type assignment rules, but only when limiting reduction to (confluent) callbyname or callbyvalue reduction M ∩ ∪ ; in restricting the system, we sacrifice subject expansion. These results combined show that a sound and complete intersection and union type assignment system cannot be defined for λµ ˜µ with respect to full reduction.
On Automating The Extraction Of Programs From Proofs Using Product Types
, 2002
"... We investigate an automated program synthesis system based on the paradigm of programming by proofs. To automatically extract a term that computes a recursive function given by a set of equations the system must nd a formal proof of the totality of the given function. Because of the particular log ..."
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Cited by 4 (1 self)
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We investigate an automated program synthesis system based on the paradigm of programming by proofs. To automatically extract a term that computes a recursive function given by a set of equations the system must nd a formal proof of the totality of the given function. Because of the particular logical framework, usually such approaches make it dicult to use techniques such as those in rewriting theory. We overcome this diculty for the automated system that we consider by exploiting product types. As a consequence, this would enable the incorporation of termination techniques used in other areas while still extracting programs.
A general method to prove the normalization theorem for first and second order typed λcalculi
 Mathematical Structures in Computer Science
, 1999
"... and second order typed λcalculi ..."
Computational isomorphisms in classical logic (Extended Abstract)
, 1996
"... We prove that any pair of derivations, without structural rules, of F ` G and G ` F , where F , G are firstorder formulas `without any qualities', in a constrained classical sequent calculus LK j p , define a computational isomorphism up to an equivalence on derivations based upon reversibility p ..."
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Cited by 3 (0 self)
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We prove that any pair of derivations, without structural rules, of F ` G and G ` F , where F , G are firstorder formulas `without any qualities', in a constrained classical sequent calculus LK j p , define a computational isomorphism up to an equivalence on derivations based upon reversibility properties of logical rules. This result gives a rationale behind the success of Girard's denotational semantics for classical logic, in which all standard `linear' boolean equations are satisfied.
Computational isomorphisms in classical logic
 THEORETICAL COMPUTER SCIENCE
, 2003
"... All standard ‘linear’ boolean equations are shown to be computationally realized within a suitable classical sequent calculus LKη p. Specifically, LKη p can be equipped with a cutelimination compatible equivalence on derivations based upon reversibility properties of logical rules. So that any pair ..."
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Cited by 2 (1 self)
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All standard ‘linear’ boolean equations are shown to be computationally realized within a suitable classical sequent calculus LKη p. Specifically, LKη p can be equipped with a cutelimination compatible equivalence on derivations based upon reversibility properties of logical rules. So that any pair of derivations, without structural rules, of F ⇒ G and G ⇒ F, where F, G are firstorder formulas ‘without any qualities’, defines a computational isomorphism.
Collections, Sets and Types
, 1995
"... We give a first order formulation of Church's type theory in which types are mere sets. This formulation is obtained by replacing λcalculus by a language of combinators (skolemized comprehension schemes), introducing a distinction between propositions and their contents, relativizing quantifiers an ..."
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Cited by 1 (0 self)
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We give a first order formulation of Church's type theory in which types are mere sets. This formulation is obtained by replacing λcalculus by a language of combinators (skolemized comprehension schemes), introducing a distinction between propositions and their contents, relativizing quantifiers and at last replacing typing predicates by membership to some sets. The theory obtained this way has both a type theoretical flavor and a set theoretical one. Like set theory, it is a first order theory, and it uses only one notion of collection. Like type theory, it gives an explicit notation for objects, a primitive notion of function and propositions are objects.