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Collapsing Partial Combinatory Algebras
 HigherOrder Algebra, Logic, and Term Rewriting
, 1996
"... Partial combinatory algebras occur regularly in the literature as a framework for an abstract formulation of computation theory or recursion theory. In this paper we develop some general theory concerning homomorphic images (or collapses) of pca's, obtained by identification of elements in a pca. We ..."
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Cited by 12 (2 self)
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Partial combinatory algebras occur regularly in the literature as a framework for an abstract formulation of computation theory or recursion theory. In this paper we develop some general theory concerning homomorphic images (or collapses) of pca's, obtained by identification of elements in a pca. We establish several facts concerning final collapses (maximal identification of elements). `En passant' we find another example of a pca that cannot be extended to a total one. 1
A Notion of Classical Pure Type System
 Proc. of 13th Ann. Conf. on Math. Found. of Programming Semantics, MFPS'97
, 1997
"... We present a notion of classical pure type system, which extends the formalism of pure type system with a double negation operator. 1 Introduction It is an old idea that proofs in formal logics are certain functions and objects. The BrowerHeytingKolmogorov (BHK) interpretation [15,51,40], in the ..."
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Cited by 7 (5 self)
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We present a notion of classical pure type system, which extends the formalism of pure type system with a double negation operator. 1 Introduction It is an old idea that proofs in formal logics are certain functions and objects. The BrowerHeytingKolmogorov (BHK) interpretation [15,51,40], in the form stated by Heyting [40], states that a proof of an implication P ! Q is a "construction " which transforms any proof of P into a proof of Q. This idea was formalized independently by Kleene's realizability interpretation [46,47] in which proofs of intuitionistic number theory are interpreted as numbers, by the CurryHoward (CH) isomorphism [21,43] in which proofs of intuitionistic implicational propositional logic are interpreted as simply typed terms, and by the LambekLawvere (LL) isomorphism [52,55] in which proofs of intuitionistic positive propositional logic are interpreted as morphisms in a cartesian closed category. In the latter cases, the interpretations have an inverse, in th...
Verifying Properties of Module Construction in Type Theory
 In Proc. MFCS'93, volume 711 of LNCS
, 1993
"... This paper presents a comparison between algebraic specificationsinthelarge and a type theoretical formulation of modular specifications, called deliverables. It is shown that the laws of module algebra can be translated to laws about deliverables which can be proved correct in type theory. The a ..."
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Cited by 6 (1 self)
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This paper presents a comparison between algebraic specificationsinthelarge and a type theoretical formulation of modular specifications, called deliverables. It is shown that the laws of module algebra can be translated to laws about deliverables which can be proved correct in type theory. The adequacy of the Extended Calculus of Constructions as a possible implementation of type theory is discussed and it is explained how the reformulation of the laws is influenced by this choice.
Synthetic Domain Theory in Type Theory: Another Logic of Computable Functions
 In Proceedings of TPHOL
, 1996
"... Abstract. We will present a Logic of Computable Functions based on the idea of Synthetic Domain Theory such that all functions are automatically continuous. Its implementation in the Lego proofchecker – the logic is formalized on top of the Extended Calculus of Constructions – has two main advantag ..."
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Cited by 3 (0 self)
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Abstract. We will present a Logic of Computable Functions based on the idea of Synthetic Domain Theory such that all functions are automatically continuous. Its implementation in the Lego proofchecker – the logic is formalized on top of the Extended Calculus of Constructions – has two main advantages. First, one gets machine checked proofs verifying that the chosen logical presentation of Synthetic Domain Theory is correct. Second, it gives rise to a LCFlike theory for verification of functional programs where continuity proofs are obsolete. Because of the powerful type theory even modular programs and specifications can be coded such that one gets a prototype setting for modular software verification and development. 1
Abstract
"... Pure Type Systems are a good way to factorize the questions of metatheory about a large family of type systems. They have been introduced as a generalization of Barendregt’s λcube, an abstraction of several type systems like the Simply Typed λCalculus, System F or the Calculus of Constructions. O ..."
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Pure Type Systems are a good way to factorize the questions of metatheory about a large family of type systems. They have been introduced as a generalization of Barendregt’s λcube, an abstraction of several type systems like the Simply Typed λCalculus, System F or the Calculus of Constructions. One critical detail of the Pure Type Systems is their conversion rule that allows to do computation at the level of types. Traditionally, Pure Type Systems are presented in a natural deduction style, and use an untyped notion of conversion. Through the years, several presentations of the Pure Type Systems have been used, with subtle differences like sequent calculus instead of natural deduction, or the use of a typed conversion instead of the untyped original one. The question to know whereas the latter choice leads to equivalent systems has been first asked by Geuvers in the early 90’s, and the answer was only known for particular subclasses of Pure Type Systems. The main contribution of this dissertation is to finally provide a final and positive answer to this question by proving
DOI 10.1007/s1070101296548 Coreflections in Algebraic Quantum Logic
"... Abstract Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular posets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so ..."
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Abstract Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular posets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so via a combination of totalization (of partially defined operations) and transfer of structure via coreflections.
unknown title
, 2011
"... Pure Type Systems are a good way to factorize the questions of metatheory about a large family of type systems. They have been introduced as a generalization of Barendregt’s λcube, an abstraction of several type systems like the Simply Typed λCalculus, System F or the Calculus of Constructions. O ..."
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Pure Type Systems are a good way to factorize the questions of metatheory about a large family of type systems. They have been introduced as a generalization of Barendregt’s λcube, an abstraction of several type systems like the Simply Typed λCalculus, System F or the Calculus of Constructions. One critical detail of the Pure Type Systems is their conversion rule that allows to do computation at the level of types. Traditionally, Pure Type Systems are presented in a natural deduction style, and use an untyped notion of conversion. Through the years, several presentations of the Pure Type Systems have been used, with subtle differences like sequent calculus instead of natural deduction, or the use of a typed conversion instead of the untyped original one. The question to know whereas the latter choice leads to equivalent systems has been first asked by Geuvers in the early 90’s, and the answer was only known for particular subclasses of Pure Type Systems. The main contribution of this dissertation is to finally provide a final and positive answer to this question by proving
Under consideration for publication in J. Functional Programming 1 Pure Type System conversion is always typable
, 2011
"... Pure Type Systems are usually described in two different ways, one that uses an external notion of computation like betareduction, and one that relies on a typed judgment of equality, directly in the typing system. For a long time, the question was open to know whether both presentations described ..."
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Pure Type Systems are usually described in two different ways, one that uses an external notion of computation like betareduction, and one that relies on a typed judgment of equality, directly in the typing system. For a long time, the question was open to know whether both presentations described the same theory. A first step towards this equivalence has been made by Adams for a particular class of Pure Type Systems (PTS) called functional. Then, his result has been relaxed to all semifull PTSs in previous work. In this paper, we finally give a positive answer to the general question, and prove that equivalence holds for any Pure Type System. 1