P. D. Lincoln and J. C. Mitchell. Operational aspects of linear lambda calculus. In Proc. 7th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, pages 235-- 246. IEEE Computer Society Press, Los Alamitos, California, June 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....type system. Our notion of expansion does not depend on the type derivation. It is a de nition by induction on the terms which can be used as a basis to more speci c term transformation processes. The linear calculus was also studied in a di erent context by the linear logic community [Abr93, LM92, Wad90, BBdH92]. In these works the linear calculus is de ned as the computational interpretation of linear logic and the problem of transforming non linear terms into linear ones is not a central issue. 3 Types Here we describe the type systems used in the rest of the paper. 3.1 Curry Types The Curry ....

P. Lincoln and J. Mitchell. Operational aspects of linear lambda calculus. In 7th Symposium on Logic in Computer Science, 1992.

....CLEAN and the calculus to be presented here (the LNL term calculus) but also by the fact that other researchers looking at practical implementations of linear languages have come up with systems which include aspects of the LNL term calculus. For example, Lincoln and Mitchell s linear variant [LM92] of Fairbairn and Wray s three instruction machine [FW87] divides memory into two spaces corresponding to linear and non linear objects. Similarly, Wadler s active and passive type system [Wad92] separates linear from non linear types in an interesting way. It should also be mentioned that ....

P. Lincoln and J. C. Mitchell. Operational aspects of linear lambda calculus. In Proceedings of the 7th Annual Symposium on Logic in Computer Science. IEEE, 1992.

....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 ILL (or, ....

P. Lincoln and J. Mitchell. Operational aspects of linear lambda calculus. In Proceedings of the Seventh Symposium on Logic in Computer Science, Santa Cruz. IEEE Press, June 1992.

....interpretation of linear logic and the proofs of memory management invariants. For instance, a compiler cannot make optimisations solely on the basis of linear type information, since extra information is required to ensure that a value has the single pointer property. Lincoln and Mitchell [LM92] describe an implementation of a linear language which allocates non linear values in a garbage collected heap and linear values in a separate area which is never garbage collected. However, they do not specify exactly how they evaluate non linear values, so it is unclear whether they can avoid ....

Patrick Lincoln and John Mitchell. Operational aspects of linear lambda calculus. In IEEE Symposium on Logic in Computer Science. IEEE Computer Society, 1992.

....is associated with a sequent 1 This work was supported by ARC grant A49230989. style proof in linear logic. The computational procedures for these proof expansions are inherently parallel in nature and provide a clear dichotomy between lazy and eager evaluation procedures. Lincoln and Mitchell [9], Benton et al. 3] and Ronchi della Rocca and Roversi[10] have all developed a term calculus for the intuitionistic fragment of propositional linear logic. However the intuitionistic fragment does not capture the essential nature of Girard s linear negation operator ( where (ff ) ....

P.Lincoln and J. Mitchell, Operational aspects of linear lambda calculus, in Proc. IEEE Symposium on Logic in Computer Science, pp 235--246, 1992.

....Unfortunately, natural deduction with this rule is not closed under substitution. This is a fairly fundamental property from a computational point of view, and so another formulation is desirable. The system we have already described above is closed under substitution, as is the system NAT in [LM92] (this system is similar to the one we discuss, in particular, it has the same rule for promotion) The promotion rule for ILL suggested so far is still a rather strange looking rule. It is an introduction rule, yet looks more like an elimination rule. It has the form taken in order to make the ....

P. Lincoln and J. Mitchell. Operational Aspects of Linear Lambda Calculus. In Seventh Annual Symposium of Logic in Computer Science, pages 235--246. IEEE Computer Society Press, 1992.

....Future Work This paper draws together two di erent branches of type theory designed for managing computer resources. Research on linear types originated with Girard s linear logic [13] and Reynolds syntactic control of interference [29] Linear type systems were later studied by many researchers [19, 37, 1, 20, 6, 36, 42, 15]. Type and e ect systems were introduced by Gi ord and Lucassen [12] and they too have been explored by many others [17, 32, 34, 24] More recently, a number of new linear type systems, or more generally, substructural type theories, have been developed. For example, Kobayashi s quasi linear ....

Patrick Lincoln and John Mitchell. Operational aspects of linear lambda calculus. In IEEE Symposium on Logic in Computer Science. IEEE Computer Society, 1992.

....about time consumption of recursive programs involving lists and trees. Their language is a standard one and no optimisation due to heap space reuse is taken into account. The relationship between linear types and garbage collection has been recognised as early as 87 by Lafont [15] see also [11, 1, 25, 17] and [4] for a similar approach not based on the syntax of linear logic. But again, due to the absence of # types, these systems do not provide in place update but merely deallocate a linear argument immediately after its use. This e#ect, however, is already achieved by traditional reference ....

P. Lincoln and J. Mitchell. Operational aspects of linear lambda calculus. In Proc. LICS 1992, IEEE, 1992.

....is used elsewhere in the program. There are several possibilities to solve this problem. The easiest one is to give up static checking and separate the environments into intuitionistic and linear environments. Whenever a linear environment is accessed, update inplace is possible, otherwise not [LM92]. Another approach was taken by Hofmann [Hof00] He introduces a special type 3, which intuitively models a location of memory. He modi es the types of the constructor and selector of lists to take memory usage into account. His calculus is ane: memory cells may not be used (weakening) but cannot ....

Patrick Lincoln and John Mitchell. Operational aspects of linear lambda calculus. In Proceedings, Seventh Annual IEEE Symposium on Logic in Computer Science, pages 235-246, Santa Cruz, California, June 1992. IEEE Computer Society Press.

....linear logic. A rst one is based on formulas as types and proofs as programs paradigms in which propositions are interpreted as types, proofs as programs and proof normalization processes as computations. Here, main works have been devoted to term assignment for intuitionistic linear logic (ILL) [6, 23] or classical LL [1] including proposals of concurrent functional programming languages. A second way is based on the formulas as states and proofs as computations paradigms [27] where the connections with Petri nets and linear logic have been investigated. The correspondence between ILL and Petri ....

....There are several major approaches to model concurrent computation using fragments of linear logic. One is the approach based on formulas as types and proofsas programs paradigms in which propositions are interpreted as types, proofs as programs and proof normalization process as computation [1, 6, 23] with proposals of concurrent functional programming languages. Another approach is the one based on formulas as states and proofs as computations [22, 27] where the connections with Petri nets and Linear Logic have been investigated. It is also investigated in a rather dioeerent way in the ....

P. Lincoln and J. Mitchell. Operational aspects of linear lambda calculus. In 7th IEEE Symposium on Logic in Computer Science, pages 235246, Santa-Cruz, California, 1992.

....also new completeness results for ILL. 1 Introduction Linear logic (denoted LL) 8] is a powerful and expressive logic with connections to a variety of topics in computer science as logic programming, concurrency or functional programming [2] In this context, Intuitionistic Linear Logic (ILL) [4,14] and some of its sub fragments are often used as the underlying logic of logical frameworks. There exists di erent semantics of ILL based on phases spaces [8] quantales [20] and Petri nets [15] The completeness for ILL with respect to Petri nets as a model has been studied in [6] For instance, ....

....or refutation search in these logics. 2 Intuitionistic Linear Logic Linear logic (LL) and its intuitionistic fragment are used in several areas of computer science. Some works tried to nd optimizations in functional programming language implementations by using linear logic as a type system [1,14]. Other applications deal with concurrent logic programming [3,10,11] From the speci cation point of view, ILL provides a natural and simple encoding of Petri net reachability [15] The sequent calculus of the propositional multiplicative and additive fragment of ILL is presented in gure 1. For ....

P. Lincoln and J. Mitchell. Operational aspects of linear lambda calculus. In 7th IEEE Symposium on Logic in Computer Science, pages 235-246, Santa-Cruz, California, 1992.

.... possible application in functional programming is in optimization of copying in lazy functional programming language implementation ( singlethreadedness ) studied by Guzm an and Hudak [58] Recent topics involve linear lambda calculus and memory allocation, investigated by Lincoln and Mitchell [70], Chirimar et al. 33, 34] Wadler [90] Mackie [77] and Benton et al. 24] A strong relationship of the multiplicative fragment of linear logic to Petri nets has been demonstrated by Gehlot and Gunter [57, 44, 43] Asperti et al. 12, 14] Engberg and Winskel [41] Marti Oliet and Meseguer ....

P. Lincoln and J. Mitchell. Operational aspects of linear lambda calculus. In Proc. 7-th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, pages 235--246. IEEE Computer Society Press, Los Alamitos, California, June 1992.

....the grammar of formulae in ILL is : p j j j j j , where p is an atomic formula or a constant , 0 or 1. In [24] Girard and Lafont illustrate the interest of linear logic, compared to IL to represent types of functions and linear terms have been defined in this context [27]. Therefore, the connective is a strict while the connective is a lazy . In this setting, the 8 LABELLED DEDUCTION real meaning of the . connective is not clear with this interpretation of proofs as functions. The relationship between CLL and ....

P. Lincoln and J. Mitchell. Operational aspects of linear lambda calculus. In 7th IEEE Symposium on Logic in Computer Science, pages 235--246, Santa-Cruz, California, 1992.

....about time consumption of recursive programs involving lists and trees. Their language is a standard one and no optimisation due to heap space reuse is taken into account. The relationship between linear types and garbage collection has been recognised as early as 87 by Lafont [14] see also [10, 1, 21, 16]. But again, due to the absence of # types, these systems do not provide in place update but merely deallocate a linear argument immediately after its use. This e#ect, however, is already achieved by traditional reference counting which may be the reason why linear functional programming hasn t ....

P. Lincoln and J. Mitchell. Operational aspects of linear lambda calculus. In Proc. LICS 1992, IEEE, 1992.

....for ILL. 1 Introduction Linear logic (denoted LL) 7] is a powerful and expressive logic with connections to a variety of topics in computer science as logic programming, concurrency or functional programming. Works have been devoted to term assignment for Intuitionistic Linear Logic (ILL) [2,11] and classical LL [1] with proposals of linear lambda calculi having important properties as subjectreduction or substitution property. From natural deduction and sequent calculus proof systems of ILL (see appendix A for the ILL sequent calculus) we can investigate the problems of type inference ....

....LL [1] with proposals of linear lambda calculi having important properties as subjectreduction or substitution property. From natural deduction and sequent calculus proof systems of ILL (see appendix A for the ILL sequent calculus) we can investigate the problems of type inference and type safety [11] but also apply the programming with proofs paradigm with possible mechanized program extraction from proofs. In this case, proof search corresponds to c fl1998 Published by Elsevier Science B. V. Larchey Wendling and Galmiche program synthesis and proof normalisation to computation. The ....

P. Lincoln and J. Mitchell. Operational aspects of linear lambda calculus. In 7th IEEE Symposium on Logic in Computer Science, pages 235--246, Santa-Cruz, California, 1992.

.... calculus. In brief, there have been various attempts to describe a linear calculus. The rst linear calculus proposed by Abramsky (1993) had a problem; it did not satisfy substitution in general. This problem was also present in the version in the book Troelstra (1992) and is discussed in Lincoln and Mitchell (1992). The problem was noted by many people, but especially made clear in the Wadler article There is no substitute for linear logic. The problem was corrected in many di erent ways, N. Benton and Paiva (1993) Brauner (1996) Barber and Plotkin (1997) Troelstra (1995) etc. The calculus in Benton et ....

Lincoln, P. and J. Mitchell. 1992. Operational aspects of linear lambda calculus. In Proceedings, Seventh Annual IEEE Symposium on Logic in Computer Science, pages 235-246, Santa Cruz, California. IEEE Computer Society Press.

.... (A) While this system allows one to estimate the required heap and stack size it does not perform in place update either (and cannot due to the absence of linear types) The relationship between linear types and garbage collection has been recognised as early as 87 by Lafont [13] see also [9, 1, 20, 15]. But again, due to the absence of 3 types, these systems do not provide in place update but merely deallocate a linear argument immediately after its use. This effect, however, is already achieved by traditional reference counting which may be the reason why linear functional programming hasn t ....

P. Lincoln and J. Mitchell. Operational aspects of linear lambda calculus. In Proc. LICS 1992, IEEE, 1992.

....linear type do not necessarily have one pointer to them. Wadler s language, though, is slightly different than ours he considers terms without the linear constructs of delay, fetch, dispose, and share, and attempts to infer the position of these connectives in raw terms. Lincoln and Mitchell [19], on the other hand, use almost exactly the same language as ours with slightly different typing rules. They also define an abstract machine with two heaps, one for objects with only one pointer to them, and one for objects with possibly more than one pointer to them. As with our work, Lincoln ....

Patrick Lincoln and John C. Mitchell. Operational aspects of linear lambda calculus. In Proceedings, Seventh Annual IEEE Symposium on Logic in Computer Science, 1992. To appear.

....is the syntax of Abramsky [Abr90] and there is the semantics of Seely [See89] Each of these has become a standard. Abramsky was inspired by the earlier work of Lafont [Laf88] and Holmstrom [Hol88] and in turn inspired related systems by Chirimar, Gunter, and Riecke [CGR92] Lincoln and Mitchell [LM92], Mackie [Mac91] Troelstra [Tro92] and Wadler [Wad90, Wad91] Seely provided a categorical model, that subsumes other models such as coherence spaces [Gir87] event spaces [Pra91] games [LS91] and the Geometry of Interaction [AJ92] Unfortunately, Abramsky s syntax is incoherent with Seely s ....

....(A Gammaffi B) Delta u : A Id y : B y : B Gammaffi L Delta; f : A Gammaffi B) f u) B Cut Gamma; Delta (t u) B Note the central role played here by Cut. Sequent and natural deduction versions of linear calculus are presented and shown equivalent by Lincoln and Mitchell [LM92]. Various mixtures of the two systems have been used by various researchers [BBdPH92, CGR92, Wad90, Wad91] Id x : A x : A Exchange Gamma; x : A; y : B ; Delta t : C Gamma; y : B ; x : A; Delta t : C Cut Gamma t : A x : A; Delta u : B Gamma; Delta u[t=x ] B Omega ....

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P. Lincoln and J. Mitchell, Operational aspects of linear lambda calculus. In 7'th Symposium on Logic in Computer Science, IEEE Press, Santa Cruz, California, June 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. ....

P. Lincoln and J. Mitchell. Operational aspects of linear lambda calculus. In Proceedings of Symposium on Logic in Computer Science, pages 235 -- 246, June 1992.

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P. D. Lincoln and J. C. Mitchell. Operational aspects of linear lambda calculus. In Proc. 7th Annual IEEE Symposium on Logic in Computer Science, Santa Cruz, California, pages 235-- 246. IEEE Computer Society Press, Los Alamitos, California, June 1992.

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P. Lincoln and J. Mitchell. Operational aspects of linear lambda calculus. In Proc. 7th IEEE Symp. on Logic in Computer Science, 1992.

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P. Lincoln and J. Mitchell, Operational aspects of linear lambda calculus. In 7'th Symposium on Logic in Computer Science, IEEE Press, Santa Cruz, California, June 1992.

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P. Lincoln and J. Mitchell, Operational aspects of linear lambda calculus. In 7'th Symposium on Logic in Computer Science, IEEE Press, Santa Cruz, California, June 1992.

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Patrick Lincoln and John Mitchell. Operational aspects of linear lambda calculus. In Proceedings, Seventh Annual IEEE Symposium on Logic in Computer Science, pages 235--246, Santa Cruz, California, June 1992. IEEE Press.

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