M. Parigot. Strong normalization for second order classical natural deduction. J. Symb. Log., 62(4):1461--1479, 1997.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....method. We conclude how our approach confronts to the Selinger s work on co control category[24] in the last section 6. 2 Background In this section, we recall necessary definitions and notations for our presentation. Basically, we follow the notion of indexed logical system according to Parigot[18]. It first appeared in Zucker s pioneering work[26] 2.1 Indexed Logical Systems In the following, we use the word derivation, instead of proof, for a tree of derivation rules. Formulas are that of second order propositional logic constructed from #. We use A, B, C, for formulas and X, ....

Michel Parigot. Strong normalization for second order classical natural deduction. In Proceedings, Eighth Annual IEEE Symposium on Logic in Computer Science, pages 39--46, Montreal, Canada, 19--23 June 1993. IEEE Computer Society Press.

....Girard s LC[5] of which the negative fragment is LKT) by means of CPS calculi with intuitionistic extract method. 2 2 Background In this section, we recall necessary definitions and notations for our presentation. Basically, we follow the notion of indexed logical system according to Parigot[14]. It was firstly appeared in Zucker s pioneering work[19] 2.1 Indexed Logical Systems In the following, we use the word derivation, instead of proof, for a tree of derivation rules. Formulas are that of second order propositional logic constructed from #. We use A, B, C, for formulas and ....

Michel Parigot. Strong normalization for second order classical natural deduction. In Proceedings, Eighth Annual IEEE Symposium on Logic in Computer Science, pages 39--46, Montreal, Canada, 19--23 June 1993. IEEE Computer Society Press.

....correspondence with natural deduction style intuitionistic logic. Parigot extends this idea to a classical logic. Its computational interpretation is a natural extension of call by name (CBN) calculus, called calculus. Moreover its reduction relation is known to be Strongly Normalizing (SN) [23] and Church Rosser(CR) 22] What is LKT : LKT is the other solution to the same problem in cut elimination of Gentzen s sequent style classical logic: LK. Gentzen s Hauptsatz states that any LK proof with cuts can be reduced by cut elimination into a cut free proof. Numerous cut elimination ....

....view of cut elimination as computation . As for the CPS translation for calculus, the pioneering one is De Groote[6] 2 Background In this section, we recall necessary definitions and notations for our presentation. Basically, we follow the notion of indexed logical system according to Parigot[23]. However we postpone the introduction of derivation rules for both CND and LKT. They are introduced together with its term assignment judgment in the following section. 2.1 Indexed Logical Systems In the following, we use the word derivation, instead of proof, for a tree of derivation rules. ....

Michel Parigot. Strong normalization for second order classical natural deduction. In Proceedings, Eighth Annual IEEE Symposium on Logic in Computer Science, pages 39--46, Montreal, Canada, 19--23 June 1993. IEEE Computer Society Press.

....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, Strong normalization for the second order classical natural deduction. In Proceedings of the eight annual IEEE symposium on Logic in Computer Science, LICS'93.

....: 0 0 ) c : 0 0 ) 4 Reducibility Sets The reducibility method comes from Tait [28] who used it to prove the normalization of G odel s system T. It has been extended to system F by Girard [15] and to second order calculus (classical system F) by Parigot [24]. Strong normalization with explicit substitution raises several problems. In a calculus without operator of explicit substitution, a term can only interact against its arguments. In the presence of explicit substitutions, a term can also interact with a substitution applied to it. The idea of ....

M. Parigot, Strong normalization for second order classical natural deduction, Proceedings of 8 th Annual IEEE Symposium on Logic in Computer Science (LICS'93), IEEE Computer Society Press, 39-46 (1993).

....a semantical storage operator for A. Nevertheless, there are a few questions that arise from this result: ffl Is the notion of extension necessary (this is discussed in the section 3) ffl Our theorem uses a particular form of Godel translation. We could generalize it to any usual translation [15] as long as we leave atomic formulas bound by a negative second order quantifier unchanged (this does not matter for 8 positive types ) However is the theorem still true for symmetric Godel translation ffl If T : A A then T can store any term typable of type A. But is T a storage ....

M. Parigot. Strong normalization for second order classical natural deduction. In Logic in Computer Sciences, pages 39--46, 1993.

....the confluence of many reduction systems. However, the method does not work for calculus. In fact, the diamond property does not hold for the formulation of parallel reduction in [12] So the proof of confluence is not so trivial as it seems to be. The calculus is known to be strongly normalizing[13] and weak ChurchRosser. For notions of deduction, these two properties yield confluency[2] But type free calculus is not strongly normalizing. For instance, the untypable term (x:xx) x:xx) dose not have normal form. The correct proof of confluency of type free calculus is never published as ....

M. Parigot: "Strong Normalization for Second Order Classical Natural Deduction", Proc. 8th Annual IEEE Symposium on Logic in Computer Science, pp. 39--46, 1993.

....functions. A number of implementations of this idea have been done, including Nuprl [11] Oyster and CLAM [8] and Coq [14] In [27] a computational interpretation of classical natural deduction is discussed, in which lambda terms may be extracted from proofs; this investigation is continued in [28]. For other papers dealing with computational interpretations of classical logic see [19] 15] 25] and [26] These systems typically supply a computational interpretation to all classical proofs; in contrast, our system uses classical logic for the part of the proof that does not affect the ....

M. Parigot. Strong normalization for second order classical natural deduction. In Proceedings of the Eighth Annual IEEE Symposium on Logic in Computer Science, pages 39--46, Montreal, Canada, June 19-23 1993.

....of LK c=t . 7 2.3 Reduction Rules of LK c=t Defining the notion of 1 step reduction in LK c=t is relatively more difficult than in LK c=t . We first define the third form of substitution a[b= 3 u B C ] which was left undefined. This substitution is close to one in Parigot s calculus[12]. It is defined only when b has type B. Intuitively, a[b B = 3 u B C ] replaces all the subterms in the form throw(u B C ; c) where u B C is free in a, by throw(u C ; apply(c; b) For brevity, we use the same name u for the tag variable after the substitution even if its type is ....

Parigot, M.: "Strong Normalization for Second Order Classical Natural Deduction ", Proc. 8th Annual IEEE Symposium on Logic in Computer Science, pp. 39-46, 1993.

....the condition of stability by reduction, the reducibility and admissibility conditions are usually those required for a set theoretic interpretation of types. There are many possible reducibility candidates such as those of Girard [GLT88] the saturated sets of Tait [Tai67] or those of Parigot [Par93] to which we can add the condition of stability by reduction without loosing those of admissibility. A survey is [Gal90] In the following, RC will refer to any admissible set of reducibility candidates. Definition 5.5 (Interpretation of the kinds) Geu95] The interpretation of the kinds is ....

M. Parigot. Strong normalization for second order classical natural deduction. In Proceedings 8th Annual IEEE Symp. on Logic in Computer Science, LICS'93, Montreal, Canada, 19--23 June 1993, pages 39--46. IEEE Computer Society Press, 1993.

.... of isomorphism which corresponds to our synonymy , and hence avoids the problem of non associativity of cuts that comes with the notion of isomorphism used in this paper ( 12] 5 The same result in calculus We now re contextualize our result in the frame of typed calculus (see [8, 9, 10] for de nitions) 5.1 Embedding typed calculus into LK p Terms in this calculus denote deductions in Parigot s Classical Natural Deduction (CND for short) restricted to the multiplicative implication and universal quanti ers of rst and second order. This natural deduction is embeddable ....

Parigot, M. (1993) Strong normalization for second order classical natural deduction. In: Logic in Computer Science, pp. 39-46. IEEE Computer Society Press. LICS 1993.

....at this point. Clearly, further work is needed to clarify the connection between Hyland and Ong s approach and ours. We have checked that in all proofs of reducibility that we are aware of, except for a recent paper by McAllester, Kucan, and Otth [19] and a recent paper by Michel Parigot [21], the conditions on sets of realizers are sheaf conditions. 9 One simply needs to change slightly the definition of Cov. However, the pre applicative structures defined in this paper are not always general enough to carry out these proofs (for example, in the case of untyped terms and typing ....

M. Parigot. Strong normalization for second-order classical natural deduction. In Eighth Annual IEEE Symposium on Logic In Computer Science, pages 39--46. IEEE, 1993.

.... we see their approach as much less fundamental and too restrictive (it only seems to deal with strong normalization) it would be interesting to understand how this method relates to the method presented in this paper or in Gallier [7] The papers by Hyland and Ong [11] and by Michel Parigot [16], also present proofs of strong normalization, using new variants of the reducibility method. The technical details are very different, and we are unable to make a precise comparison at this point. Clearly, further work is needed to clarify the connection between these approaches and ours. ....

M. Parigot. Strong normalization for second-order classical natural deduction. In Eighth Annual IEEE Symposium on Logic In Computer Science, pages 39--46. IEEE, 1993.

....normalization and confluence. We follow the original proof of Tait [30] The key point is to define a subset of strongly normalising terms, the so called reducible terms with additional closure properties which make it possible to show that a term is reducible. Parigot s proof for the calculus [20] uses Girard s reducibility candidates because he presents an impredicative second order calculus. as we consider a first order calculus only, we can define the set of reducible terms by induction over the type structure. This proof has been presented in [26] but it is reproduced here for ....

M. Parigot. Strong normalization for second order classical natural deduction. In: Proc. LICS 93 , 39--47, IEEE Computer Soc. Press, 1993.

.... 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 ([10, 12, 29, 33]) delineates other viable systems as well, and gives means to extend the Krivine Leivant paradigm of programming with proofs ( 24, 25] to classical logic; it is painless: since we reduce strong normalization and confluence to the same properties for linear logic (for non additive proof nets, ....

Parigot, M. (1993) Strong normalization for second order classical natural deduction. In: Logic in Computer Science, pp. 39--46. IEEE Computer Society Press. LICS 1993.

.... we see their approach as much less fundamental and too restrictive (it only seems to deal with strong normalization) it would be interesting to understand how this method relates to the method presented in this paper or in Gallier [7] The papers by Hyland and Ong [11] and by Michel Parigot [16], also present proofs of strong normalization, using new variants of the reducibility method. The technical details are very different, and we are unable to make a precise comparison at this point. Clearly, further work is needed to clarify the connection between these approaches and ours. ....

M. Parigot. Strong normalization for second-order classical natural deduction. In Eighth Annual IEEE Symposium on Logic In Computer Science, pages 39--46. IEEE, 1993.

....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. Strong normalization for second order classical natural deduction. In Proceedings of the eighth annual IEEE symposium on logic in computer science, pages 3946, 1993.

....ordinary substitution. While this is sucient for the study of equality in [18] we are interested in strong normalization. Hence, also the call by value and call by name formulations in [2] which are possible with ordinary substitution do not suce. Second order calculus is strongly normalizing [13]. 1 It amounts to a calculus notation for classical natural deduction in the style of Prawitz [16] 4 4 Extension of F by Iteration on Stabilization If : is provable, then is called stable. We consider an extension F ] of system F by a least stable supertype ] for any type . ....

Michel Parigot. Strong normalization for second order classical natural deduction. In Proceedings, Eighth Annual IEEE Symposium on Logic in Computer Science, pages 39-46, Montreal, Canada, 1993. IEEE Computer Society Press.

....ordinary substitution. While this is sucient for the study of equality in [16] we are interested in strong normalization. Hence, also the call by value and call by name formulations in [2] which are possible with ordinary substitution do not suce. Second order calculus is strongly normalizing [13]. 4 Extension of F by Iteration on Stabilization If : is provable, then is called stable. We consider an extension of system F by a least stable supertype ] for any type . This is expressed as follows: We 4 add a type constant for falsity (and set : and for every type ....

Michel Parigot. Strong normalization for second order classical natural deduction. In Proceedings, Eighth Annual IEEE Symposium on Logic in Computer Science, pages 39-46, Montreal, Canada, 1993. IEEE Computer Society Press.

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M. Parigot. Strong normalization for second order classical natural deduction. J. Symb. Log., 62(4):1461--1479, 1997.

No context found.

M. Parigot. Strong normalization for second order classical natural deduction. J. Symb. Log., 62(4):1461--1479, 1997.

No context found.

M. Parigot. Strong normalization for second order classical natural deduction. In 8th Annual IEEE Symposium on Logic in Computer Science, pages 39--46, Montreal, Canada, June 1993. IEEE Computer Society Press.

No context found.

M. Parigot. Strong normalization for second order classical natural deduction. In 8th Annual IEEE Symposium on Logic in Computer Science, pages 39--46, Montreal, Canada, June 1993. IEEE Computer Society Press.

No context found.

M. Parigot. Strong normalization for second order classical natural deduction. in Logic in Computer Science, pp. 39-46. IEEE Computer Society Press. Proceedings of the Eight Annual Symposium LICS, Montr'eal, June 19-23, 1992.

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M. Parigot. Strong normalization for second order classical natural deduction. In Proceedings of the eighth annual IEEE symposium on logic in computer science, pages 39--46, 1993.

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