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.

....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 ....

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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.

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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, Computer Laboratory, University of Cambridge, 1992.

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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.

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

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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, Computer Laboratory, University of Cambridge, August 1992.

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

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