Vicent Danos, Jean-Baptiste Joinet, and Harold Schellinx, LKQ and LKT: sequent calculi for second order logic based upon dual linear decompositions of classical implication, Workshop on linear logic (Girard, Lafont, and Regnier, editors), London Mathematical Society Lecture Notes 222, Cambridge University Press, 1995, pp. 211-224.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in Figure 3.7. It is well known that proof search in the intuitionistic logic of these connectives can be focused, in the sense that left introduction rules are only applied to a certain distinguished formula (such focusing is a justification for backchaining in logic programming) Danos et al. [DJS95] present a focused formulation of intuitionistic logic, called ILU. Here, sequents have the form Pi; Gamma Gamma A where Pi denotes a multiset containing at most one formula. If Gamma is a multiset or set of object level formulas, we write b Gammac and d Gammae to be the corresponding ....

....set of meta level formulas resulting from applying the corresponding predicate to all formulas in Gamma. The ILU sequent can then be encoded as the Forum sequent Sigma: Delta; Delta Gamma b Pic; dAe; b Gammac. An encoding of ILU in Forum is presented in Figure 3.8. While the proof system in [DJS95] contains two cut rules, the second of these is implied (linearly) by the first. Proofs in ILU are focused in a sense that the left rules (oe L) and (8 i L) can only be applied to formulas in the left linear context Pi (in Forum, this is the b Deltac formula without the modal prefix) This ....

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Vicent Danos, Jean-Baptiste Joinet, and Harold Schellinx. Lkq and lkt: sequent calculi for second order logic based upon dual linear decompositions of classical implication. In Girard, Lafont, and Regnier, editors, Workshop on Linear Logic, pages 211--224. London Mathematical Society Lecture Notes 222, Cambridge University Press, 1995.

.... the important notion of polarities in logic [9] Parigot s lambda mu calculus [22] which is a natural extension of lambda calculus embodying classical principles, and Danos Joinet Schellinx (DJS) s impressive classification of classical constructivizations within the syntactical framework LK [6, 7]. Decomposing the intuitionistic implication. An important advance in the study of (typed) lambda calculus, was the discovery by Girard [8] that the intuitionistic implication could be decomposed in linear logic (LL) along the now standard equation A B = A ( B yielding the so called Girard s ....

V. Danos, J.-B. Joinet, and H. Schellinx. LKQ and LKT: Sequent calculi for second order logic based upon dual linear decompositions of classical implication. In J.-Y. Girard, Y. Lafont, and L. Regnier, editors, Advances in Linear Logic, volume 222 of London Mathematical Society Lecture Note Series. Cambridge University Press, 1995.

.... 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 ....

V. Danos, J.B. Joinet and H. Schellinx. LKQ and LKT: Sequent calculi for second order logic based on dual linear decompositions of classical implication. In Proceedings of the Workshop on Linear Logic at Cornell

....Linear logic is sometimes described as being resource sensitive because it provides an intrinsic and natural accounting of process states, events, and resources. Linear logic also sheds new light on classical logic and its relationship to intuitionistic logic, see Girard [15, 16] and Danos et al. [11]. An evocative semantic paradigm for linear logic by means of games is proposed by Blass [7] and by Abramsky and Jagadeesan [2] As an intuitive motivation, let us consider reading logical deductions so that assumptions are understood as resources, and conclusions as requirements to be met by ....

V. Danos, J.-B. Joinet, and H. Schellinx. LKQ and LKT: sequent calculi for second order logic based upon dual linear decomposition of classical implication. Manuscript, October 1993.

....is regarded as a constructivization of classical logic. A way to disambiguate classical logic is through translations into linear logic. Such a use of linear logic is exemplified in Girard [10] and it has been developed extensively by V. Danos, J B. Joinet and H. Schellinx, see, e.g. [6, 7]. An inductive decoration strategy for classical sequent calculus is an algorithm to translate LK proofs into proofs in classical linear logic LL such that every logical or structural inference of LK corresponds to a logical 6 inference of LL together with a sequence of exponential inferences. ....

V. Danos, J-B. Joinet and H. Schellinx. LKQ and LKT: Sequent calculi for second order logic based upon linear decomposition of classical implication, in Advances in Linear Logic, J-Y. Girard, Y Lafont and L. Regnier editors, London Mathematical Society Lecture Note Series 222, Cambridge University Press, 1995, pp. 211-224.

.... [22] It may also be used as a basis for some inductive proofs about derivations in LJ or natural deductions in NJ (e.g. those in [12] where the strong eliminability of cut, rather than just the admissibility, is required; and, we anticipate, those in [5] and [15] Herbelin called (following [6]) his calculus LJT , a name we avoid in case of confusion with that in the first author s [9] we call its cut free fragment MJ because it is intermediate between [13] Gentzen s cut free LJ and NJ. We apologise to Herbelin for not adopting his nomenclature. We use MJ cut for the extension ....

Danos, V., J. B. Joinet and H. Schellinx: "LKQ and LKT: Sequent calculi for second order logic based upon dual linear decompositions of classical implication", in "Advances in Linear Logic" (Proceedings of the Cornell Workshop on Linear Logic, edited by J.-- Y. Girard, Y. Lafont and L. Regnier), Cambridge University Press (1995), 211--224.

.... are in bijective correspondence with cut free sequent calculus proofs in a suitable restriction of Gentzen s LJ (or LK) 8] Danos, Joinet, and Schellinx identi ed the same restriction of LK and called it LKT as part of a thorough investigation of linear logic encodings of classical proofs [5, 6]. Having gained through this correspondence the naturalness that was making the natural deduction usually preferred in practice, there was no reason any longer not to systematically study calculus through sequent calculus rather than through the traditional Curry Howard correspondence with ....

.... type commands, while the sequents typing terms (contexts) 1 Danos has also recognized the relevance of these three categories in [4] where he extends the work of Ogata [16] on the relation between LKQ and call by value CPS translations (LKQ is the other natural restriction of LK considered in [5, 6]) are of the form A j ( j A ) The symbol j serves to single out a distinguished conclusion output (hypothesis input) which stands for where the computation will continue ( where it happened before ) In the rest of this introduction, we o er as a prologue a simple justi cation of ....

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V. Danos, J.-B. Joinet, H. Schellinx, LKQ and LKT: sequent calculi for second order logic based upon dual linear decompositions of classical implication, in Advances in Linear Logic, 211-224, Cambridge University Press (1995).

.... terms and cut free proofs. For this purpose, we restrict the use of the I L rule in order to forbid the second proof. The calculus we obtain has two kind of sequents. We call it LJT, since it appears as the intuitionistic fragment of a calculus called LKT and defined by Danos, Joinet, Schellinx [2]. A sequent of LJT has either the form Gamma ; A or the form Gamma ; A B. In both cases, Gamma is defined as a set of named formulas. The semi colon delimits a place on its right. A uniform notation for sequents of LJT is the following one: Gamma ; Pi B where Pi is a notation to say ....

....; A B C I L Gamma; A; B Gamma ; A B I R Remarks: 1) With these rules, the first proof above is not directly a proof in the restriction: the axiom rule of LJ has to be encoded in the restriction by an axiom rule followed by a contraction rule. 2)This calculus appears also in Danos et al. [2] with a slight difference in the treatment of structural rules. Like its classical version LKT, it has been considered by Danos et al. for its good behaviour w.r.t. embedding into linear logic. The calculus LJT appears also as a fragment of ILU, the intuitionistic neutral fragment of unified logic ....

V. Danos, J-B. Joinet, H. Schellinx: "LKQ and LKT: Sequent calculi for second order logic based upon dual linear decompositions of classical implication", in Proceedings of the Workshop on Linear Logic, Cornell, edited by J-Y. Girard, Y. Lafont, L. R'egnier, 1993.

....the want of structural rule for positive formulas. This induces a denotational semantics for classical logic. However, we easily see that there are many choices (using the two conjunctions and the two exponentials) when we want to interpret classical conjunction, similarly for disjunction, see [8], this volume. However, we can restrict our attention to choices enjoying an optimal amount of denotational isomorphisms. This is the reason behind the tables shown on next page. It is easily seen that in terms of isomorphisms, negation is involutive, conjunction is commutative and associative, ....

V. Danos, J.-B. Joinet, and H. Schellinx. LKQ and LKT : sequent calculi for second order logic based upon dual linear decompositions of classical implication. In this volume, 1995.

.... [22] It may also be used as a basis for some inductive proofs about derivations in LJ or natural deductions in NJ (e.g. those in [12] where the strong eliminability of cut, rather than just the admissibility, is required; and, we anticipate, those in [5] and [15] Herbelin called (following [6]) his calculus LJT , a name we avoid in case of confusion with that in the first author s [9] we call its cut free fragment MJ because it is intermediate between [13] Gentzen s cut free LJ and NJ. We apologise to Herbelin for not adopting his nomenclature. We use MJ cut for the extension ....

Danos, V., J. B. Joinet and H. Schellinx: "LKQ and LKT: Sequent calculi for second order logic based upon dual linear decompositions of classical implication", in "Advances in Linear Logic" (Proceedings of the Cornell Workshop on Linear Logic, edited by J.--Y. Girard, Y. Lafont and L. Regnier), Cambridge University Press (1995), 211--224.

....at least one normalisation strategy exists, in the course of which this specific subproof will be duplicated or erased. But this is not the case: even when logically necessary , exponentials can be computationally superfluous . The paper is largely self contained (though some acquaintance with Danos et al. 1993b) will be useful) In fact, this is the first in a series of papers, all of which make use of linear logic as a proof theoretical tool. Though for purposes of exposition in this paper we limit ourselves to the implicational fragment of intuitionistic sequent calculus, the notions and techniques ....

....they were already marked at an earlier stage. A glance at the conclusion of the decorated proof will tell which (sub)formulas have been subjected to structural manipulations in the original derivation. 3 Linear decorations of this type are studied in detail in Joinet(1993) Schellinx(1994) and Danos et al. 1993a) 3 Decorated formulas A formula is decorated by prefixing its subformulas with strings of linear modalities. Clearly the number of distinct decorations is infinite. However, if we call any (possibly empty) sequence M : m 1 m 2 : m n with m i 2 f ; g a modality , then it is not very ....

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Danos, V., Joinet, J.-B., and Schellinx, H. (1993a). LKQ and LKT: Sequent calculi for second order logic based upon dual linear decompositions of classical implication. Pr'epublication 51, Equipe de Logique Math'ematique, Universit'e Paris VII. To appear in the Proceedings of the Workshop on Linear Logic, Cornell University, June 1993.

....2 (CND) of [32] thus extending the Curry Howard isomorphism between terms and proofs in intuitionistic natural deduction to terms and proofs in classical natural deduction. Our own approach is based upon earlier work on the linear decoration of derivations in classical sequent calculus (see [17, 38, 5]) There we observed that restrictions on classical derivations, comparable to those in Girard s LC, are induced by certain modal translations of classical into linear logic. Thus we defined calculi (we called them LKT and LKQ) that are complete for classical provability, and moreover ....

....5.6. The restriction defining LK j , when made to bear upon proofs of LJ t (the intuitionistic t sequent calculus, i.e. at most one formula at the rhs and all formulas coloured t) results in a system that is very close to natural deduction, a system that was referred to as ILU in e.g. [3, 5], or LJT in [14] It is not a bad intuition to think of LK j as a classical natural deduction. 5.3 Stability In the following we will investigate the effect of tq reductions on main active interspaces (cf. section 2.1) having a specific, simple, structure, namely having the form BA = oe 0 ; oe ....

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Danos, V., Joinet, J.-B., and Schellinx, H. (1995) LKQ and LKT: Sequent calculi for second order logic based upon dual linear decompositions of classical implication. In: Girard, J.-Y., Lafont, Y. and Regnier, L., editors, Advances in Linear Logic, pp. 211--224. Cambridge University Press.

....embeddings of intuitionistic logic and classical logic into linear logic (LL) The intuitionistic embeddings, e.g. mapping A B to A ( B or A ( B, allow to study calculus directly inside nets. In the same way, the classical embeddings, e.g. mapping A B to A ( B or A ( B (see [DJS94]) allow to study classical calculi (for example, the calculus of Parigot [Par92] A second reason for choosing nets is that they provide a syntax that allows easy description of geometric entities like paths. Everything here works for LL proof nets as well as for any known kind of (pure) ....

Vincent Danos, Jean-Baptiste Joinet, and Harold Schellinx. LKQ and LKT: Sequent calculi for second order logic based upon dual linear decompositions of classical implication. In this volume, 1994.

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Vicent Danos, Jean-Baptiste Joinet, and Harold Schellinx, LKQ and LKT: sequent calculi for second order logic based upon dual linear decompositions of classical implication, Workshop on linear logic (Girard, Lafont, and Regnier, editors), London Mathematical Society Lecture Notes 222, Cambridge University Press, 1995, pp. 211-224.

No context found.

Vincent Danos, Jean-Baptiste Joinet, and Harold Schellinx. LKQ and LKT: sequent calculi for second order logic based upon dual linear decompositions of classical implication. In Girard, Lafont, and Regnier, editors, Workshop on Linear Logic, pages 211--224. London Mathematical Society Lecture Notes 222, Cambridge University Press, 1995.

No context found.

Danos, V., J.B.Joinet and H. Schellinx: "LKQ and LKT: Sequent calculi for second order logic based upon dual linear decompositions of classical implication", in "Advances in Linear Logic" (Proceedings of the Cornell Workshop on Linear Logic, edited by J.Y-Girard, Y. Lafont and L. Regnier), Cambridge University Press (1995), pp 211--224.

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