Martin Hyland and Valeria de Paiva, Full intuitionistic linear logic, Annals of Pure and Applied Logic 64 (1993), no. 3, 273--291.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....both dual intuitionistic linear logic (DILL) 4] and hereditary Harrop logic underlying linear logic programming [14] The contribution of JILL with respect to these systems is the judgmental account, which gives rise to the new , connectives. Full intuitionistic linear logic (FILL) [16] does not have additives and does not proceed via a judgmental account. It requires either proof terms [7] or occurrence labels [8] in order to formulate the rules for linear implication, which makes it difficult to understand the meanings of the connectives in isolation. On the other hand, ....

....the right hand side to admit more than one true proposition would violate either linearity or the intuitionistic interpretation in our setting. However, multiple conclusions do not necessarily conflict with natural deduction (see, for example, 20] even for intuitionistic [26] and linear logics [16]. Indeed, we can readily incorporate such an approach in our judgmental framework by introducing a new judgment form, C 1 poss C k poss, on the right hand side of a hypothetical judgment. This judgment means that either the hypotheses are contradictory or one of the C i is true. This gives the ....

Martin Hyland and Valeria de Paiva. Full intuitionistic linear logic (extended abstract). Annals of Pure and Applied Logic, 64(3):273--291, 1993.

....system (which we refer to simply as ILL) can be seen in Figure 1.5. Note that, as observed by Schellinx in [Sch94] CLL is not a conservative extension of ILL. The system of Full Intuitionistic Linear Logic (FILL) is therefore of interest CLL is a conservative extension of this system. See [dPH93]) CHAPTER 1. INTRODUCTION AND BACKGROUND 12 P P (ax) Gamma P Delta; P R Gamma; Delta R (subs) I (I I ) Gamma I Delta R Gamma; Delta R (I ) Gamma 1 P 1 : Gamma n P n Gamma 1 ; Gamma n ( I ) Gamma 1 P 1 : Gamma n P n Delta 0 Gamma ....

V.C.V. de Paiva and M. Hyland. Full Intuitionistic Linear Logic. Annals of Pure and Applied Logic, 64(3):273--291, 1993.

.... logic for representing the main programming mechanisms, its interest (as a clear semantics) or diOEculty (how to insert a new operator dedicated to a mechanism into the logic and the impact on proof search ) The second and main one is to show that the Full Intuitionistic Linear Logic (F ILL) [16] seems adequate as logical foundation for a framework to involve object oriented, concurrent and logic programming (following the previously mentioned paradigms) and that to illustrate its ability to represent non determinisms at dioeerent levels (messages, process, proof search) in the context of ....

....process, proof search) in the context of concurrent systems specication. In section 2, we recall the concepts about concurrent computations in linear logic fragments, focusing on the basic ideas inside our computation model based on multiple processes. In section 3, the F ILL fragment is presented [16]. Unlike intuitionistic linear logic (ILL) it includes the multiplication disjunction par, denoted here 2 and it is formalized in a sequent calculus with multiple conclusions and a familiar notation for some kind of parallel process. In section 4, we analyze the dynamics of F ILL through proof ....

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M. Hyland and V. de Paiva. Full intuitionistic linear logic (extended abstract). Annals of Pure and Applied Logic, 64:273291, 1993.

....1991) is usually defined by restricting the right hand side of sequents to an empty or a singleton multiset. However, similarly to the classical case, there exist multi conclusion formulations of ILL with special restriction on some of the rules, e.g. full intuitionistic linear logic (FILL) (Hyland and de Paiva 1993). LL is often referred to as a logic for concurrency (see, e.g. Meseguer 1991; Abramsky 1993) The reason for this can be illustrated by considering the process view of (Kobayashi and Yonezawa 1993) Their approach is based on an interpretation where formulae are viewed as processes and ....

....judgements that take into account the considered operational interpretation. We shall see that the choice between single conclusion sequents, for instance in intuitionistic linear logic (ILL) Hodas and Miller 1994) and multiple conclusion sequents, for instance in LL (Miller 1994) and in FILL (Hyland and de Paiva 1993), has consequences both at the specification and at the proof search level. In a sense, the operational interpretation of sequents makes the difference between theorem proving and logic programming. Both theorem proving and logic programming can be viewed as the process of constructing the proof ....

M. Hyland and V. de Paiva. Full Intuitionistic Linear Logic (extended abstract). Annals of Pure and Applied Logic, 64:273--291, 1993.

.... linear logics and focus on a particular intuitionistic system called (multiplicative) FILL (Full Intuitionistic Linear Logic) that simultaneously embodies features of concurrent logical computations, induced by the par connective and the sequential properties of intuitionistic linear implication [26]. In section 4, we then propose two labelled sequent calculi for (multiplicative) CLL with related characterizations of intuitionistic provability. The first one is a direct extension of the labelled CL sequent calculus to CLL but the provability characterization does not fit well from a logical ....

....but the provability characterization does not fit well from a logical point of view. The second one has a more suitable and useful definition of labels w.r.t. linear logic specificities. Moreover it can be independently considered as a new proof system for FILL without terms or pattern calculus [7, 26] that is more adapted to proof search. As in the classical case, these systems do not give a positive answer to the above mentioned completeness problem. In section 5, we consider a new labelled proof system, based on the notion of proof net and its possible use for proof search. It allows to ....

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M. Hyland and V. de Paiva. Full Intuitionistic Linear Logic (extended abstract). Annals of Pure and Applied Logic, 64:273--291, 1993.

....A is the left unit natural isomorphism associated with the par tensor of a swdc with negation. The above proof net should be considered congruent to the following 8 Note that Girard writes the tensor unit as 1 and the par unit as . 9 The idea is due independently to Hyland and de Paiva; see [11]. 1.2 A Survey of Related Work 11 Gamma; A 0 A Omega B B; Delta which corresponds to the map Gamma A l Gamma1 O OA Delta 0 B Gamma Omega Delta ( l Gamma1 O ) Omega 0 ( OA) Omega B because in a swdc with negation, the diagram O ....

J. M. E. Hyland and V. de Paiva. Full intuitionistic linear logic (extended abstract). Annals of Pure and Applied Logic, 64:273--291, 1993.

....all structure of interest in linear logic is described by linear functors. For example, in light linear logic [G95] the traditional exponential operators are no longer monoidal functors, as pointed out in [G95,KOS97] and so do not form a linear functor. Also, in full intuistionistic linear logic [HP93] the internal hom is not a linear functor. Nonetheless, it should be clear from this paper that the notion of a linear functor is a useful organizational device which does explain basic features of linear logic. The notion of a linearly distributive category with the exponentials was explored in ....

M. Hyland and V. de Paiva \Full intuitionistic linear logic (Extended Abstract)", Annals of Pure and Applied Logic 64, 3 (1993) 273-291.

....for these settings. However, this is not the only interpretation of negation. For example, intuitionistic negation is derived solely from the monoidal closed structure, using the implication (or internal hom for the tensor) This leads one to the Full Intuitionistic Linear Logic (FILL) of (Hyland de Paiva 1993). However, notice that this intuitionistic negation expresses a property which is independent of the par. In (Cockett Seely 1997a) it is pointed out that one way to address this lack of interaction is by altering the interpretation of implication. Given a su ciently strong interaction between ....

M. Hyland and V. de Paiva (1993) Full intuitionistic linear logic (Extended Abstract), Annals of Pure and Applied Logic 64(3) 273-291.

.... and left di#erence in relevant logic [3, page 357] the unsatisfactory attempt to add a dual intensional disjunction to BCK logic [60] the question of cut elimination in non commutative substructural logics [74] the intricacies of cutelimination for full intuitionistic linear logic [41, 12], and the (unsettled ) question of full cut elimination in classical Bi Lambek logic [40] Only recently have we seen attempts to obtain a general picture of these logics using uniform calculi, specifically designed to cater for these structural variations [8, 18, 62, 29, 63, 13, 58, 48, 57, ....

....logics can be deduced from these, although the cut rule is usually not so easy to deduce. I am not aware of a good source for traditional Gentzen systems for substructural logics containing dual logical connectives, but there seems to be some question as to their exact (cut free) formulation [1, 40, 41, 12]. The display calculi presented here hopefully fill this gap. 2 Display Logic Display Logic [8] is a very powerful Gentzen style formulation, due to Nuel Belnap, which generalises Gentzen s notion of structures using multiple, complex, structural connectives. The name comes from a fundamental ....

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M Hyland and V de Paiva. Full intuitionistic linear logic (extended abstract). Annals of Pure and Applied Logic, 64:273--291, 1993.

..... its unit , an exponential ( why not , and a linear negation ( Gamma ) 7 The sequent calculus formulation is given in Figure 7. 7 This is a slightly contentious point. There is a fragment of linear logic, full intutionistic linear logic [25, 10], which has multiple conclusions and these classical connectives and, yet, can still be seen as an intuitionistic fragment. Identity OE Gamma OE Gamma Gamma OE OE; Delta Gamma Cut Gamma; Delta Gamma (t R ) Gamma Gamma t (f L ) Gamma; f Gamma OE Gamma Gamma OE (I L ) ....

J.M.E. Hyland and V.C.V. de Paiva. Full intuitionistic linear logic. Annals of Pure and Applied Logic, 64:273--291, 1993.

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M. Hyland and V. de Paiva. Full intuitionistic linear logic (extended abstract). Annals of Pure and Applied Logic, 64(3):273--291, 1993.

....intuitonistic linear logic. Additive lineales 16 Valeria de Paiva are a sound and complete model for the exponential free fragment of Intuitonistic Linear Logic. Full additive lineales are a sound and complete model for the exponential free fragment of Full Intuitonistic Linear Logic( Hyland and de Paiva (1993)) Classical lineales are a sound and complete model for the exponential free fragment of Classical Linear Logic. 6.1 Adding Modalities To add the modality (or exponential) is not dicult. De nition 16 Given a (full, additive or classical) lineale L call it a (full, additive or classical ....

....not a basic one. This does not seem sensible for the applications that we have in mind. The notions of IL(CL) algebras are the closest ones in the literature to our purposes, but using them would make it dicult to discuss the variant of Linear Logic called Full Intuitionistic Linear Logic (Hyland and de Paiva (1993)) For the system Full Intuitionistic Linear Logicwe have a sequent calculus of multiple conclusions, hence a notion of . ....

Hyland, M. and V. de Paiva. 1993. Full intuitionistic linear logic (extended abstract). Annals of Pure and Applied Logic, 64.

....Full Intuitionistic Linear Logic (FILL) The formal system devised to carry out the proof also seems of independent interest and we mention some of its possible applications at the end of the note. The system FILL is a variant of (multiplicative) Linear Logic proposed by M. Hyland and V. de Paiva [HdP93] whose logical connectives are all independent, that is, they are not interderivable, as they are in (multiplicative) Classical Linear Logic (CLL) This is analogous to the situation concerning the relationship between Intuitionistic Logic (IL) and Classical Logic (CL) In CL all the connectives ....

....Given the similarity between CLL and FILL, and the appealing analogy between FILL and IL, and CLL and CL, one might think that proving cut elimination for FILL would be a trivial matter. If this seems the case in this note, it is because the presentation of FILL has had a long gestation period. In [HdP93] a term assignment system was proposed to handle the dependency condition of our definition 2.3. There was a small mistake in that paper, concerning the side condition on the rule for the par connective. The mistake is corrected here and, independently, in [Bel95, Bie95] The notion of ....

M. Hyland and V. de Paiva. Full intuitionistic linear logic (extended abstract). Annals of Pure and Applied Logic, 64, 1993.

....a number of areas which need to be covered in the future. Clearly we need to consider the additive connectives. We should also like to consider quantifiers within this framework. Especially we should like to consider some of the many variants of Intuitionistic Linear Logic that have been proposed [10, 9, 8]. Acknowledgements We should like to thank Andy Pitts and two anonymous referees for detailed comments on this work. This paper was prepared using Paul Taylor s T E X macros. ....

Martin Hyland and Valeria de Paiva. Full intuitionistic linear logic. Unpublished manuscript, 1992.

....Full Intuitionistic Linear Logic (FILL) The formal system devised to carry out the proof also seems of independent interest and we mention some of its possible applications at the end of the note. The system FILL is a variant of (multiplicative) Linear Logic proposed by M. Hyland and V. de Paiva [HdP93] whose logical connectives are all independent, that is, they are not interderivable, as they are in (multiplicative) Classical Linear Logic (CLL) This is analogous to the situation concerning the relationship between Intuitionistic Logic (IL) and Classical Logic (CL) In CL all the connectives ....

....Given the similarity between CLL and FILL, and the appealing analogy between FILL and IL, and CLL and CL, one might think that proving cut elimination for FILL would be a trivial matter. If this seems the case in this note, it is because the presentation of FILL has had a long gestation period. In [HdP93] a term assignment system was proposed to handle the dependency condition of our definition 2.3. There was a small mistake in that paper, concerning the side condition on the rule for the par connective. The mistake is corrected here and, independently, in [Bel95, Bie95] The notion of ....

M. Hyland and V. de Paiva. Full intuitionistic linear logic (extended abstract). Annals of Pure and Applied Logic, 64, 1993.

..... and the modality , called by Girard why not . We are here concerned with Full Intuitionistic Linear Logic (FILL) a variant of Linear Logic, introduced by Hyland and de Paiva [HdP93] with the aim of giving a syntactic reconstruction of their previous semantical work on Dialectica categories [dP88] Dialectica categories came out of Hyland s insight that an internal categorical description of Godel s Dialectica Interpretation was possible, but instead of providing a model of ....

....negation A is defined as A ( and it is not an involution. Thus we have A A but not vice versa, as is the case in Classical Linear Logic. Moreover, we can extend the intuitive BrouwerHeyting Kolmogorov interpretation to deal with proofs in FILL. The syntactical reconstruction in [HdP93] was motivated by Schellinx s observation that the original system did not satisfy cut elimination [Sch91] But the presentation in [HdP93] had a mistake and the system as presented, still did not enjoy the cut elimination property. In this paper which is a revised version of the technical ....

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M. Hyland and V. de Paiva. Full intuitionistic linear logic (extended abstract) . Annals of Pure and Applied Logic, 64, 1993.

....ffl The links between the process of cut elimination and proof normalisation still appear to require further study. Certainly the work of Zucker [31] and Pottinger [24] need to be considered in this new linear framework. ffl Many variants of Intuitionistic Linear Logic have been proposed [18, 3, 17, 14]. Clearly these need to be considered in the light of this work. Details of term calculi and various resource logics will be discussed in [4] ffl It has been postulated that computation of Intuitionistic Linear Logic terms should give insight into possible optimisations of lambda calculus. This ....

Martin Hyland and Valeria de Paiva. Full intuitionistic linear logic. In Proceedings of CLICS Workshop, pages 547--570, March 1992. Available as Aarhus University Technical Report DAIMI PB 397-II.

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Martin Hyland and Valeria de Paiva, Full intuitionistic linear logic, Annals of Pure and Applied Logic 64 (1993), no. 3, 273--291.

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Martin Hyland and Valeria de Paiva, Full intuitionistic linear logic, Annals of Pure and Applied Logic 64 (1993), no. 3, 273--291.

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Martin Hyland and Valeria de Paiva, Full intuitionistic linear logic, Annals of Pure and Applied Logic, vol. 64 (1993), no. 3, pp. 273--291.

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J. Martin E. Hyland and Valeria de Paiva. Full intuitionistic linear logic (extended abstract). Annals of Pure and Applied Logic, 64(3):273--291, 1993.

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J. Martin E. Hyland and Valeria de Paiva. Full intuitionistic linear logic (extended abstract). Annals of Pure and Applied Logic, 64(3):273--291, 1993.

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Martin Hyland and Valeria de Paiva, Full intuitionistic linear logic, Annals of Pure and Applied Logic 64 (1993), no. 3, 273--291.

No context found.

Martin Hyland and Valeria de Paiva, Full intuitionistic linear logic, Annals of Pure and Applied Logic 64 (1993), no. 3, 273--291.

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Martin Hyland and Valeria de Paiva, Full intuitionistic linear logic, Annals of Pure and Applied Logic, vol. 64 (1993), no. 3, pp. 273--291.

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