P.N. Benton, G.M. Bierman, V.C.V. de Paiva and J.M.E. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, 1992.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and IMLAL2, which we call light ane categories and hyperdoctrines respectively. The presentation is aligned to the game models of the next chapter, which will be instances of the framework presented here. The results build upon research into categorical semantics of Intuitionistic Linear Logic [19, 86, 20, 76] and second order polymorphism [112, 106, 65, 68, 31] 5.1 Autonomous categories We begin with IMLL whose proofs can be interpreted as morphisms in symmetric monoidal closed categories (also called autonomous or closed) 87] In fact, IMLL cut free proofs (up to congruence) represent morphisms ....

P. N. Benton, G. M. Bierman, V. C. V. de Paiva, and J. M. E. Hyland. Term assignment for intuitionistic linear logic. Technical Report Technical Report 262, Computer Laboratory, University of Cambridge, 1992.

....to exponentials, which, in a general categorical setting, are captured by a strong monoidal functor on the traced category together with some additional structure. Under these conditions on the GoI category is a weak linear category (WLC) i.e. a weakening of a linear category (see [BBPH92]) Moreover, every reflexive object in a WLC gives rise to a linear combinatory algebra (LCA) Following [Abr96] there are two main variants of GoI. In the particlestyle GoI, the tensor on the underlying category is a coproduct and the strong monoidal functor is a countable copower. Girard s ....

.... of f (i.e. see e.g. Sco75] 2 Weak Linear Categories and Linear Graph Models In this section, we discuss Abramsky s construction of an LCA from a weak linear category (WLC) WLCs are the counterpart for linear combinatory algebras of the notion of linear category for linear # models (see [BBPH92]) In particular, we show that the category Rel # with tensor the cartesian product, together with suitable stream based functors turns out to be a WLC. Moreover, the LCAs arising from the WLC Rel # are pointed LGMs. We start by recalling Abramsky s notion of WLC and the construction of an LCA ....

P.Benton, G.Bierman, V. de Paiva, M.Hyland. Term assignment for intuitionistic linear logic, TR 262, Computer Laboratory, Cambridge, 1992.

.... programming syntax and semantics have enjoyed widespread use for over a decade [56] Categorical models have been used to give clean, implementation independent approaches to side effects and state [47, 62, 68, 52] non determinism [53] type disciplines [15, 33] and other logics for computation [7, 63]. The mathematical treatment of some features, such as parametricity and polymorphism, have required categorical tools [57, 21] Logic programming, however, has developed within a different semantic tradition than that of functional or imperative programming. The divide has narrowed in the last ....

....Xi(tv) of sort x must be targeted at x and satisfy 7 t, flu = 7 t, r17rlv as shown in the diagram below (where two triangles fail to commute: t, r o0 and u rlv) X i ( t, x Xi(u ) B pxtz 7[ 1 . p 60 Now we consider [Xi (tv) l(t) which, by definition, is in 1 Im 5e51m 575( 1[7]l)#( B( t, r) v(a) 7) One of the members of this union is that for which ff is the arrow 0 ro , px , since (id v ,TO,to) t, vOxo) 07rl, 071 0 ) TO and we have shown that 07hv Ou. Also observe that 7 is the identity id n. Now recall that we had concluded above (from ....

N. Benton, G. Bierman, V. de Paiva, and J.M.E. Hyland. Term assignment for intuitionistic linear logic. In Proc. CSL '92, 1993.

....a refinement of the usual calculus where the copying and discarding of values is written explicitly in the terms. One of the rules of this system has a deficiency that force to be isomorphic to in any reasonable categorical interpretation. It was in 1992 repaired by the authors of [BBdPH92] (and by the author of this paper) by changing the system in an appropriate way, and by discovering a Natural Deduction style presentation equivalent to the hitherto known Gentzen style presentation of ILL. This work settled the question about how to interpret ILL via the Curry Howard isomorphism. ....

....categorical interpretation. The presence of (Cut) gives us two different interpretations of the same sequent (unless = in a canonical way) In 1992 a new way to decorate the ( Gamma R) rule with terms, together with a Natural Deduction formulation of ILL, was discovered by the authors of [BBdPH92] (and by the author of this paper) The new decoration of ( Gamma R) is as follows: z 1 : A 1 ; z n : A n let z 1 ; z n be x 1 ; x n in u : A The new rule can coexist with (Cut) without collapsing the model, and the derivations that with the old term decoration concluded ....

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N. Benton, G. Bierman, V. de Paiva, and M. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory,

....spaces or certain categories or perhaps games. For ILL we can (amongst other semantics) study natural deduction, which for intuitionistic logic has a long history and is well understood. The natural deduction calculus we primarily consider is that of Benton, Bierman, de Paiva and Hyland from ([BBdPH92], BBdPH93b] BBdPH93a] Bie94] This calculus can be seen in a sequent style in Figure 1.6. We call this calculus NILL. There are several other natural deduction systems for ILL in the literature. Some are perfectly satisfactory alternatives to the one we consider; others less so. We leave ....

....some properties of SILL and discuss possible alternative systems. We also discuss SILL in relation to linear logic programming languages, paying particular attention to Lolli. 6. 1 Natural Deduction The primary natural deduction system we consider is that of Benton, Bierman, de Paiva and Hyland ([BBdPH92], BBdPH93b] BBdPH93a] Bie94] This can be seen in Figure 1.6. We are interested in deductions in normal form and we give the beta reductions and commuting conversions from [Bie94] in order to define normal natural deductions for ILL. With the promotion rule, the discharged assumptions are ....

P.N. Benton, G.M. Bierman, V.C.V. de Paiva, and J.M.E. Hyland. Term Assignment for Intuitionistic Linear Logic. Technical Report 262, Computer Laboratory, Univerity of Cambridge, 1992.

....on encoding the normalisation results (that cutting detours out of proofs ends, and ends in a canonical normal proof ) Second, on the appropriate term encoding of the exponentials of linear logic. Work in this area has not yet reached stability. The work of Benton, Bierman, Hyland and de Paiva [38, 39, 40] shows the difficulty present in the area. Thirdly, on showing that the restrictions on # abstraction in substructural logics has useful parallels in computation where resources may be consumed by computation. Wadler and colleagues show that this kind of term system has connections with functional ....

NICK BENTON, G. M. BIERMAN, J. MARTIN E. HYLAND, AND VALERIA DE PAIVA. "Term Assignment for Intuitionistic Linear Logic". Technical Report 262, Computer Laboratory, University of Cambridge, August 1992.

.... [HO94, HO95] Games naturally incorporate a notion of linearity and there is a strong correspondence with Girard s linear logic [Gir87] The syntax chosen for the calculus associated with linear logic used in this report has been particularly influenced by the work of Wadler [Wad93] and Bierman [BBdPH92]. The standard categorical semantics for linear logic was developed originally by Seely [See89] however, here we only make use of the symmetric monoidal closed category fragment of Seely s semantics. Section 2 explains the concepts of games and gives an implementation of them in a functional ....

....e by e[f=x] By providing these equivalences we do not require a Cut rule in the typing rules of L. We include j equalities, although in an implementation using reduction rules we may never implement them. Considering the equalities as reduction rules, we also require commuting conversions [BBdPH92] that arise from considering cut elimination for the sequent calculus formulation of IMLL. 7.1 SMCC s and the L Calculus The meaning of terms of the IMLL described in the previous section can be used as a categorical semantics for the L calculus. Given a typing judgement of the L calculus, ....

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P.N. Benton, G.M. Bierman, V.C.V. de Paiva, and J.M.E. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, University of Cambridge Computer Laboratory, August 1992. 28

....terms to normal form (i.e. program execution) to normalization of proofs. Since then, many researchers have provided a computational interpretation for intuitionistic linear logic by defining various extensions of the lambda calculus. This chapter contains a brief summary of parts of the work of Benton et al. 1992), who gave one such term assignment system for ILL, the linear term calculus. We will give a slightly modified version of the linear term calculus, using an alternate formulation of the dereliction rule which was mentioned but not used in Benton et al. s paper. Terms of the linear term calculus ....

....terms. Here, I have omitted the author s rules for unity, added my own rules for the quantifiers, slightly modified the form of the promotion rule, and used the alternate formulation of the dereliction rule, but it is obvious that the system is basically the same and that all the results proven in Benton et al. 1992) still apply. CHAPTER 3. TERM ASSIGNMENTS FOR ILL 11 Axiom x : A x : A Cut Gamma e : A Delta; x : A f : B Gamma; Delta f [e=x] B Omega L Gamma; x : A; y : B f : C Gamma; z : A Omega B let z be x Omega y in f : C Omega R Gamma e : A Delta f : B Gamma; ....

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N. Benton, J. Bierman, V. de Paiva, and M. Hyland, Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, 1992.

....of Related Work The rst attempt at nding an internal language for monoidal closed categories was made by Jay [8] Subsequent contributions to the search have largely been in uenced by Linear Logic. Broadly speaking, two main approaches can be distinguished. Benton, Bierman, de Paiva and Hyland [3] have used the standard terms and let constructs to represent proofs of the Intuitionistic Linear Logic. At about the same time, Mackie, Roman and Abramsky have shown in [13] that a similar type theory, in the style of Natural Deduction, can serve as an internal language for autonomous ....

P. N. Benton, G. M. Bierman, V. C. V. de Paiva, and J. M. E. Hyland. Term assignment for intuitionistic linear logic. Technical Report Technical Report 262, Computer Laboratory, University of Cambridge, 1992.

....were made by Jay [12] Subsequent contributions to the search have been influenced by Linear Logic to a varying degree. Broadly speaking, two main approaches can be distinguished. The calculus let construct approach Building on the work of Abramsky in [1] Benton, Bierman, de Paiva and Hyland [4] have used the standard terms and let constructs to represent proofs of the Intuitionistic Linear Logic. At about the same time, Mackie, Roman and Abramsky (we shall call them mra) have shown in [17] that a similar typed calculus can serve as an internal language for autonomous categories. The ....

P. N. Benton, G. M. Bierman, V. C. V. de Paiva, and J. M. E. Hyland. Term assignment for intuitionistic linear logic. Technical Report Technical Report 262, Computer Laboratory, University of Cambridge, 1992.

....Related Work The first attempt at finding an internal language for monoidal closed categories was made by Jay [8] Subsequent contributions to the search have largely been influenced by Linear Logic. Broadly speaking, two main approaches can be distinguished. Benton, Bierman, de Paiva and Hyland [3] have used the standard terms and let constructs to represent proofs of the Intuitionistic Linear Logic. At about the same time, Mackie, Roman and Abramsky have shown in [13] that a similar type theory, in the style of Natural Deduction, can serve as an internal language for autonomous ....

P. N. Benton, G. M. Bierman, V. C. V. de Paiva, and J. M. E. Hyland. Term assignment for intuitionistic linear logic. Technical Report Technical Report 262, Computer Laboratory, University of Cambridge, 1992.

....Related Work The first attempt at finding an internal language for monoidal closed categories was made by Jay [7] Subsequent contributions to the search have largely been influenced by Linear Logic. Broadly speaking, two main approaches can be distinguished. Benton, Bierman, de Paiva and Hyland [3] have used the standard terms and let constructs to represent proofs of the Intuitionistic Linear Logic. At about the same time, Mackie, Roman and Abramsky have shown in [12] that a similar type theory, in the style of Natural Deduction, can serve as an internal language for autonomous ....

P. N. Benton, G. M. Bierman, V. C. V. de Paiva, and J. M. E. Hyland. Term assignment for intuitionistic linear logic. Technical Report Technical Report 262, Computer Laboratory, University of Cambridge, 1992.

....2.7, which entails that there exists a q 2 f1; ng such that t 1in d i = d q . But then c i v c q for every i 2 f1; ng, cf. Theorem 2.6, and we conclude that t 1in c i = c q . 2 3 The category predI a is a model of IAL 3. 1 Definition of categorical models of ILL and IAL In [BBdPH92] proof theoretic considerations are used to derive axioms for a category modelling multiplicative ILL. We take the resulting model as canonical: Definition 3.1 A linear category is a symmetric monoidal closed category (C; I; Omega ; equipped with ffl A symmetric monoidal comonad ( ffi; ....

....to dA being a map from ( A; ffi A ) to ( A Omega A; ffi A Omega ffi A ) m A; A ) Definition 3.2 A model of ILL is a linear category with finite products ( Theta; 1) and finite sums ( 0) A model of IAL is a model of ILL where I = 1. It is easy to see that the arguments found in [BBdPH92] for a linear category can be extended to the full ILL, and to IAL. For example will the presence of a uniquely determined map A I for any object A in a model of IAL enables us to interpret the weakening rule in an appropriate way. 3.2 predI a is a model of IAL Given X D we define pdq X = ....

N. Benton, G. Bierman, V. de Paiva, and M. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, 1992.

....computational point of view. The interest comes from using the notion of Curry Howard Isomorphism (CHI) 7] Indeed, CHI allows to represent derivations of ILL by the slogans functional like terms as deductions of ILL and ILL formulas as types . Proposals of such languages, say ILL , are in [1, 8, 4, 11, 16, 17, 13, 15]. Each ILL comes from computational and or logical investigations of ILL. For every ILL , we call Gamma ILL its type free version, if any. Every Gamma ILL can be viewed, at least, in two ways: 1) as a type free paradigmatic language for resource conscious functional languages to be ....

.... Gamma , defined in [13] as Gamma ILL . Then, we furnish effective tools for looking at Gamma under the points of view (1) and (2) We take Gamma because it is a sort of least generalization of fi , once compared with all other untyped versions of languages appeared in [1, 8, 4, 11, 16, 17, 15]. Indeed, terms of Gamma are built starting from two sets Var and Var of variables. Var contains the variables that can never be duplicated and or erased during the reduction of the terms into which they occur. On the contrary, erasure and duplication can be applied to variables in Var. ....

N. Benton, G. Bierman, V. de Paiva, and M. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, August 1990.

.... ffi ] A : a Gamma[ B : b, C : c, D : d) ffi ) ffi ] A : a [A] Gamma[ B : b, C : c) ffi , D : d) ffi ] A : a Adding [P] to NL gives NLP, a system whose implicit notion of linguistic structure is binary 4 This operator is a compact notation for one used with linear logic by Benton et al. 1992). 5 Such characteristics of structure are reflected in the systems of algebraic semantics that are provided for such logics. Discussion of such issues, however, is beyond the scope of the present paper. branching mobiles (since order is undermined only within the confines of the given ....

Benton, N., Bierman, G., de Paiva, V. & Hyland, M. 1992. `Term assignment for intuitionistic linear logic.' Technical Report, Cambridge University Computer Laboratory.

.... several researchers, that the proof trees are not closed under substitution of deductions for open assumptions (substituting deductions for the assumptions Gamma in an application of introduction leads to a deduction which ends in general not with a correct application of introduction) In [BBHP] it was proposed to generalize the I rule to Delta 1 A 1 ; Delta n A n A 1 ; A n B Delta 1 ; Delta n B In the sequel we shall reserve the designation ILL for this version from [BBHP] Closure under substitution is now taken care of, but for a ....

.... ends in general not with a correct application of introduction) In [BBHP] it was proposed to generalize the I rule to Delta 1 A 1 ; Delta n A n A 1 ; A n B Delta 1 ; Delta n B In the sequel we shall reserve the designation ILL for this version from [BBHP]. Closure under substitution is now taken care of, but for a proof theoretic treatment the new version of the I rule turns out to be somewhat awkward; in a sense, the rule both introduces and eliminates formulas, and there is no direct relation in complexity between B and the formulas A i ; ....

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N. Benton, G. Bierman, J.M.E. Hyland, V.C.V de Paiva, Term assignment for Intuitionistic Linear Logic. Report 262, Computer Laboratory, University of Cambridge, 1992.

....) C and multimaps f 0 : Gamma; A; B; Gamma 0 ) C and linear implication to a natural isomorphism between multimaps g: Gamma; A) B and g 0 : Gamma A GammaffiB. Theorems 1 and 2 are valid for these cases as well. The exponential can be modelled as suggested by Benton et al. [BBdPH92]: we require a comonad on the category of context morphisms with the additional property that every free coalgebra carries naturally the structure of a commutative comonoid in such a way that coalgebra maps are comonoid maps. As already mentioned, the exponentials are difficult to handle in a ....

Nick Benton, Gavin Bierman, Valeria de Paiva, and Martin Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, University of Cambridge Computer Laboratory, August 1992.

....is in a sense the converse of this section, viz. in a monoidal category with a storage cotriple , we may define an action via X ff Y = X Omega Y . The reader might like to verify that this is indeed a contextual category. For definiteness, one could take the setup of either [BCS92] or [BBPH]. As we shall not need this here, we shall leave the details of this verification to such readers. 2 Remark 3.12 (Modules and fibrations) As before it is the case that Context(X) does not have all weighted limits. As before we therefore would like to be assured of a completion in which all ....

.... we can recapture the categorical semantics of tensor and : contextual categories with a contextually strong tensor product whose linear strengths are isomorphisms are very nearly linear categories in the sense of being monoidal categories with a storage operator (as defined more or less in [BBPH] or [BCS92] The diagrams which arise from the assumption of contextual strength include those which make these natural transformations strong, for example: X ff Y 1 ff u R Omega X ff (Y Omega ) X ff Y 1 ff u L Omega X ff ( Omega Y ) R u R Omega fst ....

Benton B.N., G. Bierman, V. de Paiva, M. Hyland "Term assignment for intuitionistic linear logic", M. Bezem and J. F. Groote, eds, Proceedings of the International Conference on Typed Lambda Calculi and Applications, 1992, Springer Lecture Notes in Computer Science 664, 75--90.

....out here in the introduction) as the proofs are more or less straightforward, following the familiar pattern as originally done in (Fox 1976) and as they have no role to play in the present paper. The analogous situation for intuitionistic linear logic has been the object of careful study (e.g. (Benton et al. 1992; Bierman 1995) making our job here much simpler. However, we think that an explicit outline of what coherence conditions are necessary is useful, and so have tried to make the list given here complete, apart from naturality diagrams. Of course, one of the points that we are making is that the ....

....is that this presentation, and its relative simplicity, extends to the proof theory as well, in its categorical version. In connection with cutelimination, it may be worth mentioning that the system presented here (in Table 1) does admit cut elimination, and needs no term assignment system (as in (Benton et al. 1992), for example) to facilitate this. This is because we have no negation, nor implication. We have also studied systems with implication, particularly Lambek s bilinear logic and Hyland and de Paiva s full intuitionistic linear logic, and this will be the object of a sequel to this paper (Cockett ....

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Benton, B.N., Bierman,G., de Paiva, V., Hyland, M. (1992) Term assignment for intuitionistic linear logic (preliminary report). Technical Report 262, University of Cambridge.

....system, Lincoln and Mitchell [LM92] Mackie [Mac91] Wadler [Wad91a] and the authors of this paper in a preceeding work [CGR92] employed approaches that obtain some of the virtues of an ND system for LL. The system used in this paper is based on a proposal of Benton, Bierman, de Paiva, and Hyland [BBdH92] that does satisfy the substitutivity property, even though it lacks some of the desirable properties of the ND presentation of intuitionistic logic (such as freedom from the need to use commuting conversions [GLT89] We refer the reader to their paper for a fuller discussion. 10 Table 2: ....

.... propositions are encoded by terms in the grammar M : x j (x : s: M) j (M M) j (store M where x 1 = M 1 ; x n = M n ) j (fetch M) j (share x; y as M in M) j (dispose M before M) Our notation here essentially corresponds to that in [CGR92, LM92] modulo incorporating adjustments from [BBdH92]. The store operation, store M where x 1 = M 1 ; x n = M n ) binds the variables x 1 ; x n in the expression M and the share operation (share x; y as M in N) binds the variables x and y in N . The notation for store can be somewhat unwieldy when writing programs, but most ....

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N. Benton, G. Bierman, V. de Paiva, and M. Hyland. Term assignment for intuitionistic linear logic. Announced on the Types electronic mailing list, 1992.

....Many researchers proposed pairs of natural deductions for ILL and functional languages for the same purposes as ours. Such proposals can be grouped into two classes. The first one uses ILL as an independent logical system, i.e. the assumptions of the deductions of ILL are only linear formulas. [1, 13, 3, 16] belong to this group. 1, 13, 16] are briefly over viewed in Appendix B, while [3] will be recalled with much more detail in Section 3. The second group uses ILL as a subsystem of the Unity of Logic [8] i.e. the assumptions of the judgments of ILL contain both linear and intuitionistic formulas ....

....the same purposes as ours. Such proposals can be grouped into two classes. The first one uses ILL as an independent logical system, i.e. the assumptions of the deductions of ILL are only linear formulas. 1, 13, 3, 16] belong to this group. 1, 13, 16] are briefly over viewed in Appendix B, while [3] will be recalled with much more detail in Section 3. The second group uses ILL as a subsystem of the Unity of Logic [8] i.e. the assumptions of the judgments of ILL contain both linear and intuitionistic formulas [20, 19, 4] This proposal belongs to the first class. 3] is the reference ....

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N. Benton, G. Bierman, V. de Paiva, and M. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, August 1990.

....described in this paper. In fact, in order for derivations to be first class objects, a proof system so that every formula is annotated with a term supporting its validity must be considered. This complex subject, deeply interwingled with linear type theory, has recently received some attention [3, 24], but a definitive answer has not been found yet. In particular, the treatment of the quantifiers, a point that is crucial, has not been investigated yet. The application of the techniques devised in this document to other net models, in particular to elementary nets and condition event nets is ....

N. Benton, G. Bierman, V. de Paiva, M. Hyland: "Term Assignment for Intuitionistic Linear Logic", Technical Report 262, Computer Laboratory, University of Cambridge, 1992.

.... using the inference rules (3) and the additional Sigma clauses shown following (with the obvious implicit extensions of the directional lambda system, and of Buszkowski s semantics) Labelling of [fflI] inferences is via pairing, and that of [fflE] inferences uses an operator adapted from Benton et al. 1992), where a term [b=vfflw] a implicitly represents the substitution of b for v w in a. This rule is used in (4) 3) B : v] C : w] A : a BfflC : b fflE A : b=vfflw] a A : a B : b fflI AfflB : ha; bi ( Sigma.6) Sigma(ha; bi) Sigma(a) Delta Sigma(b) where FV(a) FV(b) Sigma.7) ....

Benton, N., Bierman, G., de Paiva, V. & Hyland, M. 1992, `Term assignment for intuitionistic linear logic.' Technical Report, Cambridge University Computer Laboratory.

....extended to the untyped case (with reflexive types) the fi v calculus is related to the linear intuitionistic logic, where the modality characterizes the values. It turns out that a suitable class of categories for interpreting the fi v calculus is a restriction of the one defined in [2] for interpreting the multiplicative and exponential fragment of intuitionistic linear logic, equipped with a suitable retraction, and having enough values . The retraction is D . T (D Gammaffi D) where D is the object representing the domain of interpretation, T is a suitable functor and ....

....by one (or more) element playing the r ole of the undefined element. An easy example is the set of strict functions from a cpo D to a cpo D 0 , which is isomorphic to the space of partial function from D n f D g to D 0 n f D 0 g. For interpreting the language T we know, from [10] and [2], that we need a category C with the following features: ffl it must be closed monoidal symmetric w.r.t. a tensor product Omega , in order to interpret contexts as tensor product and correctly interpret rules (Id) Gammaffi I) Gammaffi E) and (exc) ffl it must have a comonad (T ; ffi; ....

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N. Benton, G. Bierman, V. de Paiva, and M. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, August 1990.

....important work is the one by Abramsky [1] He describes a linear machine more faithful in spirit to the SECD machine of Landin. Wadler has also done work in this area, introducing the distinction between linear short term memory and non linear garbage collected memory [152, 153] Other works are [31, 111]. Actually implemented linear functional languages are Linear ML (Chirimar, Gunter and Riecke [45, 46] Lilac (Mackie [117] and an implementation of Lisp [90] Work in a rather different vein has been done on optimal lambda reduction. An optimal evaluator should utilize all the term sharing ....

N. Benton, G. Bierman, V. de Paiva, and J. Hyland. Term assignment for intuitionistic linear logic. Manuscript, Sept. 1992.

....usual calculus where copying and discarding of values is written explicitly in the terms. One of the rules of this system has the property that it forces to be isomorphic to in any reasonable categorical interpretation, as pointed out in [Wad91] In 1992 this was remedied by the authors of [BBdPH92] (and by the author of this paper) by changing the rule in an appropriate way, and by discovering a natural deduction formulation equivalent to the Gentzen style formulation of ILL (the hitherto known natural deduction formulation, Mac91] did not possess that property) This work settled the ....

....to the Gentzen style formulation of ILL (the hitherto known natural deduction formulation, Mac91] did not possess that property) This work settled the question about how to interpret ILL via a Curry Howard isomorphism. The Curry Howard interpretation of ILL, the linear calculus, is in [BBdPH92] given a sound categorical interpretation. The [Gir87] paper introduced the Girard Translation which embeds IL into ILL. This translation works at the level of formulas as well as at the level of proofs. The Girard Translation at the level of proofs induce an embedding of terms of the calculus ....

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N. Benton, G. Bierman, V. de Paiva, and M. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, 1992.

....Intuitionistic Linear Logic. 1 Introduction This paper introduces a functional language R enjoying two features at once. R is type free: it has the same computational power as the calculus fi [4] R is also resource aware: it refines fi computationally like the typed typable calculi T R [2, 18, 25, 5, 26, 19, 22, 6, 24], used to encode Intuitionistic Linear Logic [11] by Curry Howard Isomorphism [13] refine the simply typed typable calculus fi T . The terms of R , like those in [2] have a less debatable computational meaning than the terms of fi: some of them are naturally joined at an eager evaluation ....

....to R . Finally, we have to develop some tools for reasoning about the theory that the virtual machine induces on R . Results. We define a type free resource aware functional language R . The syntax of R derives from the syntax of the typable resource aware functional language B , introduced in [5] and recalled in Fig. 1. Then, we introduce the rewriting system R for R . R has to be either callby value, or call by name. This because we are shifting from the typed language B to an untyped one, and, like when moving from fi T to fi, call by name x : A B x : A (Id) Gamma B M : A ....

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N. Benton, G. Bierman, V. de Paiva, and M. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, August 1990.

....M 2 j 1 M j 2 M j hM 1 ; M 2 i j x:M 1 j let be M 1 in M 2 j promote M 1 ; Delta Delta Delta ; M n for x 1 ; Delta Delta Delta ; x n in M j j derelict(M) j copy z as x; y in M j discard M 1 in M 2 The typing rules for this extended language are in Figure 8. They are taken from [BBdPH92]. The Promotion and Dereliction rules arise because, under certain circumstances, it is necessary to change the type of a term from x: to x: and vice versa. Intuitively, CHAPTER 5. ADDING 69 Rules from Figure 2, plus Delta 1 e 1 : 1 Delta Delta Delta Delta n e n : n x 1 : ....

....of a term t. For both the base and inductive cases, there is a further case analysis, considering each of the rules that could have been used in the reduction of the term, t. It is quite similar to the proof of Proposition 5, from Chapter 3. The rules for have been considered elsewhere [BBdPH92]. The only other new rule is the one for block. Case: t is reduced by (derelict f) alloc 0) k F Omega n block f (k 1) F n Here, it is necessary to show that Delta imp L 0 n: nat if Delta imp L block f : nat. Since (derelict f) alloc 0) reduces to Omega n via a strictly ....

P.N. Benton, G.M. Bierman, V.C.V. de Paiva, and J.M.E. Hyland. Term assignment for intuitionistic linear logic. Technical report, University of Cambridge, Computer Laboratory, August 1992. Report 262.

....are significant connections between the two, despite their apparent differences. The intuitionistic fragment of linear logic (ILL) may be modelled in a linear model a symmetric monoidal closed category with a comonad which satisfies some extra conditions relating it to the monoidal structure [6]. Moggi s computational metalanguage may be modelled in a monad model a cartesian closed category with a monad T satisfying some different conditions relating it to the cartesian structure. The situations are tantalisingly close to dual: is almost, but not quite, entirely like T . Benton has ....

....corresponds to both a logic and an associated term calculus. The logic associated with monad models and the computational metalanguage is an intuitionistic modal logic, dubbed CL logic in [5] Associated with linear models and ILL are several proposals for linear term calculi, such as those of [6], 18] 11] Here we choose to work with the calculus of [6] Corresponding to adjoint models are the LNL term calculus (here referred to as the adjoint calculus) and LNL logic of [4] Girard proposed two mappings of intuitionistic logic (or, equivalently, the simply typed lambda calculus) into ....

[Article contains additional citation context not shown here]

P. N. Benton, G. M. Bierman, J. M. E. Hyland, and V. C. V. de Paiva. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, Aug. 1992.

....on the calculus are actually consequences of the proof theory of the logic. We have also extended the class of interesting constructive logics for which there is a perfect three way correspondence between logic, term calculus and categorical models. This is part of an ongoing project of ours, see (Benton et al. 1992; Bierman de Paiva, 1996) In fact, there is a close relationship between CLlogic and intuitionistic linear logic. Any linear category (model for intuitionistic linear logic, see (Benton et al. 1992; Bierman, 1995) gives rise to a CL model as a subcategory of the category of algebras for the ....

....logic, term calculus and categorical models. This is part of an ongoing project of ours, see (Benton et al. 1992; Bierman de Paiva, 1996) In fact, there is a close relationship between CLlogic and intuitionistic linear logic. Any linear category (model for intuitionistic linear logic, see (Benton et al. 1992; Bierman, 1995) gives rise to a CL model as a subcategory of the category of algebras for the comonad. Whilst this is interesting, not all CL models arise in this way because the monad part of the 16 Benton, Bierman and de Paiva model is always a commutative strong monad. More discussion ....

Benton, P.N., Bierman, G.M., de Paiva, V.C.V., & Hyland, J.M.E. (1992). Term assignment for intuitionistic linear logic. Tech. rept. 262. Computer Laboratory, University of Cambridge.

.... b[a=x] B Proof. By induction on the derivation Delta; x : A b : B. ut As one would expect there is an exact equivalence between the natural deduction and sequent calculus formulations (indeed the substitution property is essential for this) The details of this equivalence are given in [2]. 4 Cut Elimination In this section we consider cut elimination for the sequent calculus formulation of MELL, extended or decorated with terms. Suppose that a derivation in the term assignment system of Fig. 2 contains a cut: x : A x : A Gamma; x : A e : B ( Gammaffi I ) Gamma x: A:e ....

Nick Benton, Gavin Bierman, Valeria de Paiva, and Martin Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, August 1992.

...., which we have not so far given explicitly. We shall use the notation that the application of the Cut rule in the following proof is called a (D 1 ; D 2 ) cut. D 1 Gamma e : A D 2 Delta; x : A f : B Cut Gamma; Delta f [e=x] B As in our previous work in intuitionistic linear logic [1], we have principal cuts and secondary cuts as well as insignificant cuts. 5.1 Principal Cuts Principal cuts are the ones where we introduce on the right and on the left of a cut the formula that is being cut. The new case is Gamma e : A TR Gamma val(e) TA Delta; x : A f : TB TL ....

....category of cpos (not necessarily with a bottom) with continuous maps and the lifting monad. Note how the tensorial strength is needed in the interpretation of the T elimination rule. This is an instance of a general phenomenon which arises in modelling many different logics (see, for example, [1, 3, 4]) the interpretations of logical connectives have to behave well with respect to the (tensor) product which is used to represent the multicategorical structure implied by the comma on the left of sequents. This means firstly that there has to be some extra categorical structure relating the ....

[Article contains additional citation context not shown here]

P.N. Benton, G.M. Bierman, V.C.V. de Paiva, and J.M.E. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, August 1992.

.... Gamma is taken to represent the multiset A 1 ; An . The natural deduction presentation proved harder to formalize and early proposals [1, 15] failed to have the vital property of closure under substitution. A natural deduction system which has this property was given by Benton et al. [4] and is given below. A x ] Delta Delta Delta B ( Gammaffi I ) x A GammaffiB Delta Delta Delta A GammaffiB Delta Delta Delta A ( Gammaffi E ) B (I I ) I Delta Delta Delta A Delta Delta Delta I (I E ) A Delta Delta Delta A Delta Delta Delta B ( Omega I ....

....conversions , which arise from consideration of the subformula property, as well as those suggested by the process of cut elimination for the sequent calculus formulation. 1 For the purposes of this paper these need not be considered here. The interested reader is again referred to other work [6, 4]. 1 In fact there are other term equalities due to the interaction between our formulation of the Promotion rule and the fact that we are suppressing the Exchange rule. 2 Two Categorical Models The fundamental idea of a categorical treatment of proof theory is that propositions should be ....

[Article contains additional citation context not shown here]

P.N. Benton, G.M. Bierman, V.C.V. de Paiva, and J.M.E. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, August 1992.

....are significant connections between the two, despite their apparent differences. The intuitionistic fragment of linear logic (ILL) may be modelled in a linear model a symmetric monoidal closed category with a comonad which satisfies some extra conditions relating it to the monoidal structure [6]. Moggi s computational metalanguage may be modelled in a monad model a cartesian closed category with a monad T satisfying some different conditions relating it to the cartesian structure. The situations are tantalisingly close to dual: is almost, but not quite, entirely like T . Benton has ....

....corresponds to both a logic and an associated term calculus. The logic associated with monad models and the computational metalanguage is an intuitionistic modal logic, dubbed CL logic in [5] Associated with linear models and ILL are several proposals for linear term calculi, such as those of [6], 18] 11] Here we choose to work with the calculus of [6] Corresponding to adjoint models are the LNL term calculus (here referred to as the adjoint calculus) and LNL logic of [4] Girard proposed two mappings of intuitionistic logic (or, equivalently, the simply typed lambda calculus) into ....

[Article contains additional citation context not shown here]

P. N. Benton, G. M. Bierman, J. M. E. Hyland, and V. C. V. de Paiva. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, Aug. 1992.

....systems; in natural deduction resulting from normalisation of proofs and in the sequent calculus from the process of eliminating cuts. Our systems have important properties which previous proposals lack, namely closure under substitution and subject reduction. In the full version of this paper [2] we give a general categorical model for our calculus. We also show how the term assignment system can be alternatively suggested by our semantics. This paper represents preliminary work; much more remains. Given our term calculus it is our hope that this refined setting should shed new light on ....

Nick Benton, Gavin Bierman, Valeria de Paiva, and Martin Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, August 1992.

.... applicability, and applications have sprung in all areas of Computer Science (for an introduction to and some applications of Linear Logic see [8] Since (multiplicative exponential) Intuitionistic Linear Logic is a refinement of Intuitionistic Logic, it makes sense to develop a linear calculus [1] and a system of linear combinators. Our main concern here is with a calculus of (linear) categorical combinators that models Intuitionistic Linear Logic faithfully and efficiently. Our categorical combinators and their relationship to Category Theory and Intuitionistic Linear Logic are explained ....

N. Benton, G. Bierman, V. de Paiva, M. Hyland, `Term assignment for intuitionistic linear logic', Technical Report No. 262, Computer Laboratory, University of Cambridge, August 1992.

.... 2 I 2 (The semantic braces, Delta Delta Delta ] serve to remind that all the assumptions must be modal and discharged. It is easy to see that this rule satisfies the property of closure under substitution. This formulation was first used for the modality of ILL by Benton et al. [7] and for the necessity modality of the modal logic IS4, by Bierman and de Paiva [12] Subsequently, a number of other presentations of the modality have been proposed for ILL; most notably Barber s dual context formulation [3] and Benton s dual system formulation [5] All three presentations will ....

P.N. Benton, G.M. Bierman, V.C.V. de Paiva, and J.M.E. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, August 1992.

....a single underlying pressure The intuitionistic fragment of Girard s formalism may be modeled in a symmetric monoidal closed category with a comonad and some addditional structure. An early model was sketched by Seely [See89] and a later model detailed by Benton, Bierman, de Paiva, and Hyland [BBdPH92], dubbed the Gang of Four . The two models are similar, and their relation has been detailed by Bierman [Bie94] Meanwhile, Moggi s formalism may be modelled in a cartesian closed category with a monad T and some different additional structure. The situations are tantalisingly close to dual: ....

....computational lambda calculus. There are a plethora of candidates for the linear calculus, including proposals of Lafont [Laf88] Holmstrom [Hol88] Wadler [Wad90, Wad91] Abramsky [Abr93] Mackie [Mac94a] Lincoln and Mitchell [LM92] Troelstra [Tro92] Benton, Bierman, dePaiva, and Hyland [BBdPH92, Bie94], and della Rocca and Roversi [RR94] This version of this paper uses the syntax of [BBdPH92] Girard and Moggi proposed various embeddings of lambda calculus into their systems. Girard proposed two mappings of lambda calculus into linear logic, labeled ffi and [Gir87] It later turned out that ....

[Article contains additional citation context not shown here]

N. Benton, G. Bierman, V. de Paiva, and M. Hyland, Term assignment for intuitionistic linear logic. Technical report 262, Computer Laboratory, University of Cambridge, August 1992.

No context found.

P.N. Benton, G.M. Bierman, V.C.V. de Paiva and J.M.E. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, 1992.

No context found.

P.Benton, G.Bierman, V. de Paiva, M.Hyland. Term assignment for intuitionistic linear logic, TR 262, Computer Laboratory, Cambridge, 1992.

No context found.

P.Benton, G.Bierman, V. de Paiva, M.Hyland. Term assignment for intuitionistic linear logic, TR 262, Computer Laboratory, Cambridge, 1992.

No context found.

Benton, N., Bierman, G., de Paiva, V. & Hyland, M. 1992. `Term assignment for intuitionistic linear logic.' Technical Report, Cambridge University Computer Laboratory.

No context found.

P. N. Benton, G. M. Bierman, V. C. V. de Paiva, and J. M. E. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, August 1992.

No context found.

Nick Benton, Gavin Bierman, Valeria de Paiva & Martin Hyland. 1992. `Term Assignment for Intuitionistic Linear Logic.' Tech. Report 262, Cambridge University Computer Lab.

No context found.

N. Benton, G. Bierman, V. de Paiva, and M. Hyland. Term assignment for intuitionistic linear logic. Technical Report 262, Computer Laboratory, University of Cambridge, August 1990.

No context found.

Benton P.N., G. Bierman, V. de Paiva, M. Hyland "Term assignment for intuitionistic linear logic", M. Bezem and J. F. Groote, eds, Proceedings of the International Conference on Typed Lambda Calculi and Applications, 1992, Springer Lecture Notes in Computer Science 664, 75--90.

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