M. Parigot. Classical proofs as programs. In Computational logic and proof theory, volume 713 of Lect. Notes Comput. Sci, pages 263--276. Springer-Verlag, 1993.  Home/Search   Document Not in Database   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 ....

M. Parigot. Classical proofs as programs. In G. Gottlob, A. Leitsch, and D. Mundici, eds., Proc. of 3rd Kurt Godel Colloquium, KGC'93, vol. 713 of Lecture Notes in Computer Science, pp. 263-276. Springer-Verlag, 1993.

.... 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 ....

M. Parigot. Classical proofs as programs. In G. Gottlob, A. Leitsch, and D. Mundici, eds., Proc. of 3rd Kurt Godel Colloquium, KGC'93, vol. 713 of Lecture Notes in Computer Science, pp. 263-276. Springer-Verlag, 1993.

.... d associer un programme, sous la forme d un # terme, a toute preuve intuitionniste, formalisee dans le calcul des predicats du second ordre (voir, par exemple [3] Cette correspondance a ete etendue, assez recemment, a la logique classique moyennant une extension convenable du # calcul (voir [1,4,5,6]) Chaque theoreme formalise en logique du second ordre correspond donc a une specification de programme. Il se pose alors le probleme, en general tout a fait non trivial, de trouver la specification associ ee a un theoreme donne ; autrement dit, de determiner le comportement operationnel commun ....

....t # t # et u # u # # tu # t # u # et #xt # #xt # ) et telle que : c)tu # (c)#k(t#z(k) z)u)u; c)#k t # t si k n apparat pas dans t; k)tu # (k)t et (k # ) k)t # (k)t lorsque k, k # sont des continuations. Ces notions sont proches de celles definies par M. Parigot dans [5,6] pour sa construction du # calcul. 2. Le probleme de la specification. Soit maintenant L 0 le langage 0, s, ## . Le theoreme de completude peut s ecrire comme une formule du second ordre dans ce langage : #x #M[Mod # (M) #Mx] # #J[Ded(J) # Jx] Dans Mod # (M) et dans Ded(J) on a ....

M. Parigot. Classical proofs as programs. In Proc. KGC'93, LNCS vol. 713, p. 263-276 (1993).

....coroutine. Related work The extension of the well known formulas as types paradigm to classical logic has been widely investigated by T. G. Griffin [10] C. R. Murthy [14] F. Barbanera and S. Berardi [1] N. J. Rehof and M. H. Srensen [23] P. De Groote [6] J. L. Krivine [13] and M. Parigot [18, 19] : H. Nakano, Y. Kameyama and M. Sato [16, 15, 17, 11, 12, 26] have proposed various logical frameworks that are intended to provide a type system for a lexical variant of the catch throw mechanism used in functional languages such as Lisp. Moreover, H. Nakano has shown that it is possible to ....

....Nakano has shown that it is possible to restrict the catch throw mechanism in order to stay in an intuitionistic (propositional) framework. In this paper, we generalize H. Nakano s result in several ways: ffl We use M. Parigot s calculus and its type system, the Classical Natural deduction (CND) [18, 19], which is confluent and strongly normalizing in the second order framework [20] ffl We deal with the first and second order frameworks. ffl We consider a type system a la Curry, which allows us to rephrase the above restriction on pure (i.e. untyped) terms, and not only on typed terms as in ....

M. Parigot. Classical proofs as programs. In Computationnal logic and theory, volume 713 of LNCS, pages 263--276. Springer-Verlag, 1993.

....if we want to define some calculus whose type system corresponds to subtractive logic, we need to restrict the abstraction (i.e. the right introduction rule of implication) of some classical calculus. A forthcoming paper will be devoted to this subject (where we will use M. Parigot s calculus [17, 18] whose computational properties are better known than those of A. Filinski s symmetrical calculus) ....

M. Parigot. Classical proofs as programs. In Computationnal logic and theory, volume 713 of LNCS, pages 263--276. Springer-Verlag, 1993.

....The link between classical logic and functional languages has been established few years ago by Griffin in [Gri90] where Felleisen s generic control operator is given the type : A A. This correspondence is however more difficult to establish, in this wider setting. This is explained in [Par93], where Parigot advocates the interest of his lambda mu calculus in this area, and also the difficulties encountered. This calculus is an extension of the calculus, and shares the same properties of confluence and strong normalization when this point makes sense. It provides the computational ....

....encountered. This calculus is an extension of the calculus, and shares the same properties of confluence and strong normalization when this point makes sense. It provides the computational interpretation for classical proofs developed in a natural deduction system with multiple conclusions [Par91, Par93]. Actually, Mu comes from the introduction of a new kind of variables, introduced precisely for dealing with the labeling of the different formulae on the right side of a judgment. We do not go into full details, but insist on the fact that this system is strongly justified from the logical ....

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M. Parigot, Classical Proofs as Programs. In G. Gotlod, A. Leitsch and D. Mundici eds., Proceedings of the third Kurt Godel Colloquium KGC'93. LNCS 733.

....in logic and computer science. It says that normalization of proof in intuitionistic logic corresponds to fi reduction of calculus[2, 3] After the work of Griffin[1] there have been many works to extend the isomorphism for classical logic and to understand computational meaning of classical logic[8 11, 14]. Most of these research uses sequent calculus with multiple conclusion. However, in sequent calculus, we need a new concept of list of formulae in right side which is not logical. It is regarded as disjunction in semantics, but it contains the axiom ff which means that any formula can be ....

M. Parigot: "Classical Proofs as Programs", Lecture Notes in Computer Science, 713, pp. 263 -- 276, 1993.

....to LK. Despite of the restrictions, soundness and completeness w.r.t. classical provability is still retained in LKT. LKT is classical logic in this sence. What is Classical Natural Deduction : The other constructive classical logic is classical natural deduction (CND) presented by Parigot [9]. Church s calculus is widely accepted as the logical basis of functional programming. It is also well known that typed calculus has Curry Howard correspondence with intuitionistic natural deduction. Parigot extends this idea to a classical logic: CND. Its computational interpretation is a ....

....with intuitionistic natural deduction. Parigot extends this idea to a classical logic: CND. Its computational interpretation is a natural extension of call by name (CBN) calculus, called calculus. The calculus, equiped with so called structural reduction rule, is known to be SN and CR [9]. This exactly means the normalization procedure (in the sence of Parigot) of CND is SN and CR. Therefore CND can also be considered as a constructive classical logic. Hereafter we refer to Parigot s calculus by n in order to put stress on the fact that it is CBN. Our Work: In this paper, we ....

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Michel Parigot. Classical proofs as programs. In 3rd Kurt Godel Colloquium, pages 263--276. Springer-Verlag LNCS 713, 1993.

.... of the equivalence is realized by the equivalence: 1 ) 2 ) and ( 3 ) are realized by the equivalence; push push) and (push pop) are realized by the equivalence; for the (pop pop) equation we rst show the following equivalence which corresponds to the (S 3 ) rule in [10]: 0 : 0 ] x: t : x: 0 : 0 ] x: t : x by ( x: 0 : 0 ] x: t)x : by ( x: 0 : 0 ] t : by ( x: 0 : t[ 0 = by ( where the reduction also substitutes the other sub terms [ 0 ....

Michel Parigot. Classical proofs as programs. In Proceedings of Kurt Godel Colloquium, volume 713 of Lecture Notes in Computer Science, pages 263-276. Springer, 1993.

....of a proof. As it is well known (see, e.g. Goto, 1979; Martin Lof, 1982] such a uniform mechanism can be defined for constructive calculi enjoining a Normalization Theorem or a Cut elimination Theorem; also suitable fragments of classical calculi can be considered (see, e.g. Murthy, 1990; Parigot, 1993]) In this paper we will analyze two kinds of calculi from whose proofs the information can be extracted in a uniform way. The first one corresponds to our definition of uniformly constructive calculus, which aims to characterize uniform extraction methods also for constructive systems neither ....

Parigot, M. (1993). Classical proofs as programs. In Computational Logic and Proof Theory.

.... a connection providing automatic assistance for program veri cation, for instance with Nuprl [79] and Coq [120] Such work is motivated by the possibility of applying classical tools in constructive proofs developments but also from the classical side of considering classical proofs as programs [126] and identifying executable subsets of classical theory to execute speci cations as prototypes. The paper by Caldwell, Gent and Underwood, in this volume, presents a ne analysis of the problem from extraction to veri cation and back , using the classical language of PVS [118] and the type theory ....

M. Parigot. Classical proofs as programs. In Computational Logic and Proof Theory, KGC'93, LNCS 713, pages 263-276, Brno, Czech. Rep., August 1993.

....(intuitionistic) Natural Deduction. Parigot extends this idea to a classical logic: the Classical Natural Deduction (CND) Its computational interpretation is a natural extension of call by name (CBN) calculus, called calculus. The calculus, viewed as a reduction system, is also SN and CR [15]. Otherwise said, CND is also a constructive classical logic in natural deduction style. Hereafter we refer to Parigot s system by n . 1.2 Our Work We show how LKT and CND are related. That is, the normalization for CND can be simulated by tq protocol for LKT(simulation theorem) The Strongly ....

....Table 1. Original Derivation Rules for LKT 2 t for LKT 2.1 Indexed Logical Systems In this subsection, we will explain our formulation of logical systems. We mainly handle three logical systems in this paper: LK by Gentzen [6] LKT by DJS [2] and Classical Natural Deduction CND by Parigot[15]. First, we quote the original derivation rules for LKT from [2] in Table 1. is the entailment sign of the calculus. We use rhs and lhs for the right hand side and left hand side of the entailment sign, respectively. From the view point of [3] LKT is the fragment of LK j where all formulas ....

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Michel Parigot. Classical proofs as programs. In 3rd Kurt Godel Colloquium, pages 263--276. Springer-Verlag LNCS 713, 1993.

....computing by proof and programmingby proof. Computing by proof is concerned with computations, the object of functional programming languages such as LISP or SML. Programming is mainly concerned with actions such as sorting a disk file. In this case, classical program synthesizing systems such as [7, 14, 16, 19, 21, 22, 23] are not adapted because they rely on more or less classical logics and produce expressions. Even if imperative execution can be modeled with functions, it is not realistic. Logics and actions. Classical and even intuitionistic logics are also not adapted for processing actions unless we use the ....

PARIGOT, M. Classical proofs as programs. In KGC'93, Third K. Godel Colloquium, Computational logic and Proof theory, Brno (1993).

....to functions of any arity. See how natural the formulation of the theorem is, 0 being, just by virtue of its concluding sequent, a correct converter, converting classical integers to linear integers. The reader should compare this to the LJ based approaches to this same conversion problem in [22, 23, 31]. Another possibility would be the use of a calculus enabling the fusion of intuitionistic and classical reasoning (cf. 12] as in [26] Quite surprisingly we even get a stronger result. Suppose one adds to classical logic the juxtaposition rule, then the same analysis shows that 0 will now ....

Parigot, M.(1993) Classical proofs as programs. In: Gottlob, G., Leitsch, A., and Mundici, D., editors, Computational Logic and Proof Theory, pp. 263--277. Springer Verlag. LNCS 713.

....negation [20] They use a control operator akin to Felleisen s C to extract the computational content of classical proofs of Sigma 0 1 sentences. On the proof theoretic side, Parigot has introduced the calculus, an algorithmic interpretation of cut elimination in classical natural deduction [16, 17, 18]. From a computer science point of view, the iclassicalj constructs of the calculus may be interpreted in terms of labels and jumps [7] Independently of Parigot, Rehof and S#rensen have developped a calculus ( Delta ) reminiscent of the calculus [21] They use applications of the form (xM ) ....

M. Parigot. Classical proofs as programs. In G. Gottlod, A. Leitsch, and D. Mundici, editors, Proceedings of the third Kurt G#del colloquium KGC'93, pages 263276. Lecture Notes in Computer Science, 713, Springer Verlag, 1993.

....and the reduction system approach (CPS) Otherwise said, we found new, strict Curry Howard isomorphism between Gentzen style classical logic and programs. As a slogan, it can be said as classical proofs as programs and cut elimination as computation . Particularly Classical Natural Deduction (CND)[20] style programs and its computation are interpreted by LKT and LKQ proofs and cut elimination. We also show that Plotkin s CPS translation, in fact, can be understood as the translation from ND terms to CND terms. LKT and LKQ are variations of Gentzen s original system LK. Confluency is ....

Michel Parigot. Classical proofs as programs. In 3rd Kurt Godel Colloquium, pages 263--276. Springer-Verlag LNCS 713, 1993.

....of applications of k. Notice also that the same relationship holds between the solution in F 2 and ;oe Delta in the case of sums. 7 Related Work Upon completion of this work two lines of closely related work was brought to our attention. M. Parigot has in a series of papers [Par91] Par92] [Par93a], Par93b] studied a so called calculus. This system incorporates both first order classical logic and second order logic, presented as a so called free deduction system comprising features of both natural deduction and sequent calculus. Church Rosser and Strong Normalization theorems are proved ....

Michel Parigot. Classical Proofs as Programs. In Lecture Notes in Computer Science 713. 1993.

....with (intuitionistic) Natural Deduction. Parigot extends this idea to a classical logic: the Classical Natural Deduction (CND) Its computational interpretation is a natural extension of calculus, called calculus, which is Strongly Normalizing (SN) as well as of ChurchRosser (CR) [16, 17, 18, 19]. Hereafter we refer to Parigot s system by n because this is callby name(CBN) calculus as we show in this paper. Proof theory: There is also a long line of proof theoretical approaches to understanding deconstructive Gentzen style classical logic, i.e. classical logic equipped with SN and ....

....which is not true in their system. 2 t for LKT 2.1 Indexed Logical Systems In this subsection, we will explain our formulation of logical systems. We mainly handle three logical systems in this paper: the LK by Gentzen [5] the LKTg by DJS [2] and the Classical Natural Deduction CND by Parigot[18]. First, we quote the original derivation rules for the LKT from [2] in Table 1. is the entailment sign of the calculus. We use rhs and lhs for the right hand side and left hand side of the entailment sign, respectively. From the view point of [3] LKT is the fragment of LK j where all ....

[Article contains additional citation context not shown here]

Michel Parigot. Classical proofs as programs. In 3rd Kurt Godel Colloquium, pages 263-- 276. Springer-Verlag LNCS 713, 1993.

.... a connection providing automatic assistance for program verification, for instance with Nuprl [79] and Coq [120] Such work is motivated by the possibility of applying classical tools in constructive proofs developments but also from the classical side of considering classical proofs as programs [126] and identifying executable subsets of classical theory to execute specifications as prototypes. The paper by Caldwell, Gent and Underwood, in this volume, presents a fine analysis of the problem from extraction to verification and back , using the classical language of PVS [118] and the type ....

M. Parigot. Classical proofs as programs. In Computational Logic and Proof Theory, KGC'93, LNCS 713, pages 263--276, Brno, Czech. Rep., August 1993.

....for LC . It is because he didn t consider the relation with the computation rules and cut elimination procedure. Incidentally, Murthy s paper also includes good references for classical control calculi which we omit in this paper. calculus: In addition to CPS calculus, Parigot s calculus [20, 21, 22, 23] is also considered as a standard for classical proofs as programs approach on classical natural deduction (CND) DeGroote [6] revealed the relation between CBN CPS and calculus. Moreover, Ong and Stewart (OS) introduce a call by value (CBV) version of calculus in [18] in which they ....

....( as logical connectives, since this will be sufficient for us to explain our subject. 2 Decorations and Constrictive Morphisms 2.1 Notations In this paper, we entirely use the indexed formula version of logical system. We use five logical system. LK and LJ by Gentzen [8] CND by Parigot [21, 22, 23] and LKT LKQ by DJS [4] We choose our notation to adapt that of DJS [5] Parigot[22] and Ong and Stewart[18] Formulas are that of first order propositional logic constructed from . We don t use different logical symbol between each logical system as it is obvious from the context. Indexed ....

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Michel Parigot. Classical proofs as programs. In 3rd Kurt Godel Colloquium, pages 263--276. Springer-Verlag LNCS 713, 1993.

....Tait and Martin Lof parallel reduction method gives an essentially straightforward proof of confluence. Strong normalization is harder. We use a candidates of reducibility method but require an induction hypothesis stronger than that in [30] in order to handle the i arg rule. Remark 2. 6 In [29] Parigot observed that, in contrast to the Curry Howard representation theory of intuitionistic logic, with respect to his cbn calculus, numerals are not uniquely represented by normal forms. For instance both the Church numeral xf:f(fx) and the term xf : ff: ff] f (f fi: ff] f(fx) denote 2. ....

M. Parigot. Classical proofs as programs. In Proc. 3rd Kurt Godel Colloquium, pages 263--276. Springer-Verlag, 1993. LNCS Vol. 713.

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M. Parigot. Classical proofs as programs. In Computational logic and proof theory, volume 713 of Lect. Notes Comput. Sci, pages 263--276. Springer-Verlag, 1993.

No context found.

M. Parigot. Classical proofs as programs. In Computational logic and proof theory, volume 713 of Lect. Notes Comput. Sci, pages 263--276. Springer-Verlag, 1993.

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Parigot, M. Classical proofs as programs. In KGC'93, Third K. Godel Colloquium, Computational logic and Proof theory, Brno (1993).

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M. Parigot, Classical proofs as programs, in G. Gottlod, A. Letsch and D. Mundici eds., KGS'93, LNCS 713, p.263-276, 1993.

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