G. Barthe, J. Hatcli#, and M. Srensen, A notion of a classical pure type system, Electronic Notes in Theoretical Computer Science, vol. 6 (1997), Proceedings of MFPS'97. (S. Brookes and M. Mislove, editors).  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... typed calculi, see, e.g. 27, 28, 8] and applied to the compilation and optimization of typed languages, see, e.g. 19, 44] Grin s discovery initiated a series of studies on the computational content of classical proofs where CPS translations are a frequently employed tool, see, e.g. [13, 33, 38, 39, 34, 12, 4, 42, 24, 25, 35, 36, 6]. Inductive and coinductive types, see, e.g. 31, 29, 20, 15, 40] are syntactic representations for initial algebras (such as natural numbers and lists) resp. nal coalgebras (such as conatural numbers and streams) in typed calculi. Despite being pervasive in the type theoretical literature ....

G. Barthe, J. Hatcli, and M. H. Srensen. A notion of classical pure type system (preliminary version). In S. Brookes and M. Mislove, eds., Proc. of 13th Ann. Conf. on Math. Found. of Programming Semantics, MFPS'97, vol. 6 of Elect. Notes in Theor. Comp. Sci.. Elsevier, 1997.

.... typed calculi, see, e.g. 27, 28, 8] and applied to the compilation and optimization of typed languages, see, e.g. 19, 44] Grin s discovery initiated a series of studies on the computational content of classical proofs where CPS translations are a frequently employed tool, see, e.g. [13, 33, 38, 39, 34, 12, 4, 42, 24, 25, 35, 36, 6]. Inductive and coinductive types, see, e.g. 31, 29, 20, 15, 40] are syntactic representations for initial algebras (such as natural numbers and lists) resp. nal coalgebras (such as conatural numbers and streams) in typed calculi. Despite being pervasive in the type theoretical literature ....

G. Barthe, J. Hatcli, and M. H. Srensen. A notion of classical pure type system (preliminary version). In S. Brookes and M. Mislove, eds., Proc. of 13th Ann. Conf. on Math. Found. of Programming Semantics, MFPS'97, vol. 6 of Elect. Notes in Theor. Comp. Sci.. Elsevier, 1997.

....Curry Howard terms denote nothing but derivations in Int itself and thus yield a circular provability semantics for the latter. Note that the Curry Howard isomorphism does not specify Int; decent calculi can be built for a variety of logics including proper fragments of Int, classical logic ([19], 81] 82] Abstract computational and functional semantics for Int which did not address the issue of the original BHK semantics for Int were also studied in [64] 87] and many other papers (cf. 17] 20] 96] Kuznetsov Muravitsky Goldblatt semantics for Int is based on a ....

G. Barthe, J. Hatcliff, and M. Srensen, A notion of a classical pure type system, Electronic Notes in Theoretical Computer Science, vol. 6 (1997), Proceedings of MFPS'97. (S. Brookes and M. Mislove, editors).

....non dependent logical pure type systems extended with a variant of the operator, and shows that the extended systems is strongly normalizing if the same holds for the underlying pure type systemusing CPS translations from the extended systems to the underlying pure type systems. The authors [7] study a more general notion of classical pure type system, whereby most logical pure type systems, including those with dependent types, give rise to a corresponding classical pure type system. In order to study embeddings of classical pure type systems into pure type systems, in general, the ....

....those with dependent types, give rise to a corresponding classical pure type system. In order to study embeddings of classical pure type systems into pure type systems, in general, the need arises for more general CPS translations, and preliminary versions of such translations in fact appeared in [7]. 1.4. DiOEculties with dependent types Recall Plotkin s translation in Figure 1. The translation is dened by induction over the structure (or alternatively, the size) of terms. This method of denition scales up to richer languages that are either untyped [18, 58] or typed using a iCurry stylej ....

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G. Barthe, J. Hatclioe, and M.H. S#rensen. A notion of classical pure type system. In Proceedings of MFPS'97, volume 6 of Electronic Notes in Theoretical Computer Science, 1997.

....if the theory of explicit substitutions can be extended among other to control like operators; 2. control like operators and explicit substitutions both have applications in theorem proving and proof theory. See e.g. 24] for applications of control like operators in theorem proving and [5, 14, 24] for applications of control like operator in proof theory. The former are used in classical theorem proving and the latter to represent incomplete proofs. By studying explicit substitutions with control operators, we lay the foundations for a classical theorem prover with the ability to handle ....

....When considered as a translation on typed terms, the translation corresponds to Kolmogorov s double negation translation. Also note that the explicit CPS translation yields a CPS translation : from pure Delta terms to pure terms in the obvious way; this translation is proved correct in [5]. Theorem 23 (Correctness of CPS translation) 1. For every two terms M;N , M = fi oe N ) M = fioe N 2. For every judgement ( Gamma; M;A) Gamma exp M : A ) h[ Gamma ]i exp M : h[A]i Proof. The first item is proved in three steps: 1. prove by induction on the structure of the terms that ....

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G. Barthe, J. Hatcliff, and M.H. Sørensen. A notion of classical pure type system. In Proceedings of MFPS'97, volume 6 of Electronic Notes in Theoretical Computer Science, 1997. To appear.

....what the cube and the domain free cube are to the domain full and domain free approach, respectively see Figure 1. For the sake of completeness, we also study the relationship between the type assignment cube and the two other cubes (Section 6) The paper is an extended and updated version of (Barthe and S rensen, 1997). 2 Domain free pure type systems Domain free pure type systems are generated from specifications, just like pure type systems are see (Barendregt, 1992) Specifications are triples expressing certain abstract dependencies. Definition 1 A specification is a triple S = S; A; R) where 1. S is ....

....assignment system, and used the former correspondence to study a number of applications. Despite failing to have decidable type checking, domain free pure type systems provide an attractive alternative to pure type systems and have been used in several theoretical studies (Barthe et al. 1996; Barthe et al. 1997; S rensen, 1997) and implementations (Magnusson, 1994; Pfenning, 1994) Acknowledgments. The authors are grateful to J. Hatcliff and T. Coquand for their interest in this work, to R. Pollack for discussions on type checking for domain free pure type systems, and to J. Wells for suggesting to use ....

Barthe, G., Hatcliff, J., & Sørensen, M.H. (1997). A notion of classical pure type system.

....in the sense that the generalised function space Pi x : A: B is its only type constructor. The minimality of CC imposes strict limits to its applicability and has given rise to a spate of proposals to include new term type constructors: algebraic types [3, 7] fixpoints [1] control operators [6] and inductive types [21] to mention only the examples considered in this paper some proposals are actually concerned with the more general setting of Pure Type Systems [4] While most of these calculi, which we call Generalised Calculi of Constructions (GCCs) are known to enjoy ....

....in S nat . Proof. The interpretation of S nat in cPICC validates 0 = Nat 1. Since cPICC is consistent, we conclude. 4. 4 Classical logic Classical typed calculi, i.e. typed calculi enriched with control like structures, have received much attention since Griffin s seminal work [18] In [6], J. Hatcliff, M.H. S rensen and the author propose a notion of Classical Pure Type System that allows for dependent types. Our last application is concerned with one specific Classical Pure Type System, namely the Classical Calculus of Constructions S cl for which the technique of this paper ....

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G. Barthe, J. Hatcliff, and M.H. Sørensen. A notion of classical pure type system. In Proceedings of MFPS'97, volume 6 of Electronic Notes in Theoretical Computer Science. Elsevier, 1997.

....of 4 Griffin s discovery was followed by several lines of work on classical logic, control operators, and the Curry Howard isomorphism some initiated independently of his work. It is not possible here to explain the aims and achievements of the individual lines of work, but see e.g. [7] for more references. Gamma M : oe implies h Gammai hMi : hoei. This shows that logical consistency of minimal, propositional logic implies consistency of classical propositional logic. Indeed, assume classical propositional logic were inconsistent, i.e. M : for some M . Then also hMi ....

....non dependent logical pure type systems extended with a variant of the operator, and shows that the extended systems is strongly normalizing if the same holds for the underlying pure type system using CPS translations from the extended systems to the underlying pure type systems. The authors [7] study a more general notion of classical pure type system, whereby most logical pure type systems, including those with dependent types, give rise to a corresponding classical pure type system. In order to study embeddings of classical pure type systems into pure type systems, in general, the ....

[Article contains additional citation context not shown here]

G. Barthe, J. Hatcliff, and M.H. Sørensen. A notion of classical pure type system. In Proceedings of MFPS'97, volume 6 of Electronic Notes in Theoretical Computer Science, 1997.

....inconsistent too. 4 Griffin s discovery was followed by several lines of work on classical logic, control operators, and the Curry Howard isomorphism some initiated independently of his work. It is not possible here to explain the aims and achievements of the individual lines of work, but see [5]. hxi = k:x k hx:Mi = k:k (x:hMi) hM 1 M 2 i = k:hM 1 i (y:y hM 2 i k) h x:Mi = k: hMifx : h:h (j:i:j k)g) z:z Figure 4: Call by name CPS translation of The embedding property can also be used to relate normalization in classical logic with normalization in minimal logic and to extract ....

....non dependent logical pure type systems extended with a variant of the operator, and shows that the extended systems is strongly normalizing if the same holds for the underlying pure type system using CPS translations from the extended systems to the underlying pure type systems. The authors [5] study a more general notion of classical pure type system, whereby most logical pure type systems, including those with dependent types, give rise to a corresponding classical pure type system. In order to study embeddings of classical pure type systems into pure type systems, in general, the ....

[Article contains additional citation context not shown here]

G. Barthe, J. Hatcliff, and M.H. Sørensen. A notion of classical pure type system. In Proceedings of MFPS'97, volume 6 of Electronic Notes in Theoretical Computer Science, 1997.

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G. Barthe, J. Hatcli#, and M. Srensen, A notion of a classical pure type system, Electronic Notes in Theoretical Computer Science, vol. 6 (1997), Proceedings of MFPS'97. (S. Brookes and M. Mislove, editors).

No context found.

Gilles Barthe, John Hatcliff, and Morten Heine Srensen. A notion of classical pure type systems. Unpublished manuscript, October 1997. Cited on page 111.

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Gilles Barthe, John Hatcliff, and Morten Heine Srensen. A notion of classical pure type systems. In S. Brookes, M. Main, A. Melton, and M. Mislove, editors, Mathematical foundations of programming semantics, volume 6 of Electronic Notes in Computer Science. Elsevier, 1997. Cited on page 110.

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