N. Ghani, V. de Paiva, and E. Ritter. Linear explicit substitutions. In Proc. of Westapp'98, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... with vacuous variables permits more programs under type assignment; for example a term such as x: y: x ( w: y) x, which is traditionally considered not linear, can be given the linear type A B (A B) This carries over to the study of explicit substitutions in resource conscious calculi [Ghani et al. 1998] where it might clarify the logical status of the extension operator. ACKNOWLEDGMENTS We would like to thank Roberto Virga, who discovered an error in an earlier version of this paper, and Iliano Cervesato and Carsten Sch urmann for several discussions and comments on a draft of this paper. ....

Ghani, N., Paiva, V. D., and Ritter, E. 1998. Linear explicit substitutions. Tech. Rep. CSR98 -2, University of Birmingham, School of Computer Science. Mar.

....garbage collection. This optimisation attempts to reuse storage used by the arguments of a function for storing the result of that function; clearly this is only legal if there is precisely one reference to the arguments, namely the function s own reference. See for example [Wad90, Wad91, GdPR98] Figure 1 Plan of attack. Haskell source Desugar Core Core to Core transforms Usage inference Code generation C Assembler Usage inference Mogensen [Mog98] presents an extension of the [TWM95a] analysis handling a larger class of data types and adding zero usage annotations. This analysis is ....

Neil Ghani, Valeria de Paiva, and Eike Ritter. Linear explicit substitutions (extended abstract). In Proceedings of WESTAPP'98, 1998.

....important meta theoretic properties: subject reduction (type preservation) confluence, and strong normalization. Subject reduction is more or less obvious from its logical origins; a proof is given in [2] Proofs of confluence and strong normalization for a stronger version of DILL are given in [6]. A very important lemma of subject reduction is the Substitution Property, which states that substitution is a type preserving operation. In DILL, because of its two contexts, there are two versions of this property. We rephrase it in logical terms as cut elimination. 1 Lemma (Substitution ....

N. Ghani, V. de Paiva, and E. Ritter (1998). Linear Explicit Substitutions. Proc. of WESTAPP'98.

....of a bound linear variable as part of the abstraction. In this way a linear redex can be executed in constant time. As a consequence it is possible to have only intuitionistic substitutions for which f u = f . On a technical level, we should mention that the presentation of xDILL found in [11] di ers slightly from that found here. Recall, that one half of the isomorphism between a linear variable of type A and an intuitionistic variable of type A is given by the let substitution jz : A let z be x in x : x : Aj In [11] we permitted such substitutions to be syntactically placed in ....

....we should mention that the presentation of xDILL found in [11] di ers slightly from that found here. Recall, that one half of the isomorphism between a linear variable of type A and an intuitionistic variable of type A is given by the let substitution jz : A let z be x in x : x : Aj In [11], we permitted such substitutions to be syntactically placed in parallel with other substitutions. That is, we replaced the typing rule for let substitutions found in this paper with the more general rule j 2 t : A j hf; t= xi : x : Aj The idea behind this more general rule ....

N. Ghani, V. de Paiva, and E. Ritter. Linear explicit substitutions. In Proc. of Westapp'98, 1998.

....you cannot decide whether it applies by looking at the top of the syntax tree. This means that these rules are unusable for functional programming, and commuting conversions have to be used instead. The ensuing type theories with suitable reduction rules are strongly normalizing and con uent [12]. In this paper we only consider an equational theory, so we do not prove strong normalization or con uence. 3.2 Intuitionistic Linear Logic ILL The type theory ILL, originally presented by Benton, Bierman, de Paiva and Hyland in [4] corrected a lack of substitutivity of the term calculus for ....

N. Ghani, V. de Paiva, and E. Ritter. Linear explicit substitutions. Journal of the IGPL, to appear 2000.

.... intuitionistic arrow type as primitive, as well as rules to introduce it and eliminate it: x : A ; M : A I ; x A :M : A B ; M : A B ; N : A E ; M N : B This sublanguage is already mentioned in [1] We could use a calculus of linear explicit substitutions [6] to show the correctness of the machine presented in this section in a by now well established way [5] Because sharing and update in place make it necessary also to take memory management into account, we will not present such a calculus in this paper but establish the relation between our ....

N. Ghani, V. de Paiva, and E. Ritter. Linear Explicit Substitutions. To appear in Journal of IGPL.

....the need to fill a gap in the conceptual development of the xSLAM project. The xSLAM project is concerned with the design and implementation of abstract machines based on linear logic. For xSLAM we initially developed a linear calculus by adding explicit substitutions to Barber and Plotkin s DILL [GdPR00]. We then considered the categorical models one obtains for both intuitionistic and linear logic with explicit substitutions on the style of Abadi et al. GdPR99] The DILL system [BP97] distinguishes between intuitionistic and linear variables: linear variables are used once during evaluation, ....

N. Ghani, V. de Paiva, and E. Ritter. Linear explicit substitutions. Journal of the IGPL, to appear 2000.

....bound linear variable as part of the abstraction. In this way a linear fi redex can be executed in constant time. As a consequence it is possible to have only intuitionistic substitutions for which f u = f . On a technical level, we should mention that the presentation of xDILL found in [11] differs slightly from that found here. Recall, that one half of the isomorphism between a linear variable of type A and an intuitionistic variable of type A is given by the let substitution Gammajz : A let z be x in x : x : Aj Gamma In [11] we permitted such substitutions to be ....

....mention that the presentation of xDILL found in [11] differs slightly from that found here. Recall, that one half of the isomorphism between a linear variable of type A and an intuitionistic variable of type A is given by the let substitution Gammajz : A let z be x in x : x : Aj Gamma In [11], we permitted such substitutions to be syntactically placed in parallel with other substitutions. That is, we replaced the typing rule for let substitutions found in this paper with the more general rule Gammaj Delta 1 f : Gamma 0 j Delta 0 Gammaj Delta 2 t : A Gammaj Delta hf; ....

N. Ghani, V. de Paiva, and E. Ritter. Linear explicit substitutions. In Proc. of Westapp'98, 1998.

....As linear calculi have grown in popularity, so has the need for solid and e cient support for their implementation. A linear adaptation of explicit substitution techniques is a prime candidate. The authors of this paper have separately explored this possibility in two distinct settings: In [6], Ghani, de Paiva, and Ritter have designed the language xDILL, geared towards the implementation of functional languages. It is based on Barber and Plotkin s DILL (Dual Intuitionistic Linear Logic) 2] and is characterized, among other things, by variables of two di erent kinds: linear variables ....

....arbitrarily many times. The extra information about usage of linear variables makes it possible to apply various optimizations like update in place of aggregate data structures such as arrays, or savings in memory allocation. This signi cantly in uenced the design decisions of the calculus in [6]. On the other hand, Cervesato and Pfenning have based their implementation of the linear logical framework LLF [4] on a form of linear explicit substitution, although they did not thoroughly investigate its meta theory. LLF is a close relative of DILL (for example, both distinguish linear and ....

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Neil Ghani, Valeria de Paiva, and Eike Ritter. Linear explicit substitutions. In Proceedings of the First International Workshop on Explicit Substitutions: Theory and Applications to Programs and Proofs | WESTAPP'98, Tsukuba, Japan, March 1998.

....the design and construction of linear abstract machines. The normalisation and confluence results proved in this paper are the basic results required for a sound theoretical foundation for our our abstract machines. On a technical level, we should mention that the presentation of xDILL found in [9] differs slightly from that found here. Recall, that one half of the isomorphism between a linear variable of type A and an intuitionistic variable of type A is given by the compound substitution Gammajz : A let z be x in x : x : Aj Gamma In [9] we permitted such substitutions to be ....

....that the presentation of xDILL found in [9] differs slightly from that found here. Recall, that one half of the isomorphism between a linear variable of type A and an intuitionistic variable of type A is given by the compound substitution Gammajz : A let z be x in x : x : Aj Gamma In [9], we permitted such substitutions to be syntactically placed in parallel with other substitutions. That is, we replaced the typing rule for compound substitutions found in this paper with the more general rule Gammaj Delta 1 f : Gamma 0 j Delta 0 Gammaj Delta 2 t : A Gammaj Delta ....

N. Ghani, V. de Paiva, and E. Ritter. Linear explicit substitutions. In Proc. of Westapp'98, 1998. Full version submitted for publication.

....logic by replacing the modality by the 2 modality. We inherit the semantic foundations of Barber and Plotkin s calculus, and add to this a reduction calculus, for which we prove several syntactic properties required for developing an abstract machine. The second step follows the approach of [8] in adding explicit substitutions to the calculus to provide the basis for an abstract machine. The addition is based on categorical techniques. Following [17] the categorical approach straightforwardly allows us to carry across, from the earlier calculus without explicit substitutions, proofs of ....

N. Ghani, V. de Paiva, and E. Ritter. Linear explicit substitutions. In Proc. of Westapp'98, 1998.

....and hence our methodology requires linear analogues of the modifications described above. In particular, we want a linear calculus extended with explicit substitutions, a categorical model for the calculus and a CurryHoward relationship between them. The calculus appears in a companion paper [GdPR98] and some of the rewriting properties of (a fragment of) the system are described in [NdPR94] In this paper we concentrate on the more refined categorical models for the linear calculus extended with explicit substitutions. Indexed categories cannot be used as models of linear calculi of ....

.... calculus with explicit substitutions in a L category. We use as an underlying linear calculus a calculus developed by Barber called DILL [BP97] because it incorporates the semantic separation of linear and non linear contexts directly into the syntax we therefore call our calculus xDILL [GdPR98]. 6 creation of comonoids 7 in the obvious componentwise sense Of course, this choice is merely a matter of convenience and the connection between the syntax and its categorical semantics could also be established for Bierman s version of the linear calculus. We present the typing ....

N. Ghani, V. de Paiva, and E. Ritter. Linear explicit substitutions. In This reader. 1998.

....to implement linear logic and we require linear analogues of the modifications described above. In particular, we want a linear calculus extended with explicit substitutions, a categorical model for the calculus and a Curry Howard relationship between them. The calculus appears in full in [10] and this paper concentrates on the more refined categorical models for the linear calculus extended with explicit substitutions. Indexed categories cannot be used as models of linear calculi of explicit substitution as they are an inherently non linear structure. Asking that the fibres form a ....

....5.1 xDILL A Linear Calculus of Explicit Substitutions We now extend the monoidal oe calculus with types and prove that L categories form a sound and complete class of models for this calculus. Underlying our extended calculus is Barber s DILL [3] hence we call our calculus xDILL [10]. We use DILL because it incorporates the semantic separation of linear and nonlinear contexts directly within the syntax although we could have started from Bierman s linear calculus. Formally, the types of xDILL are base types, unit, function, tensor and types and the raw expressions are t ....

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N. Ghani, V. de Paiva, and E. Ritter. Linear explicit substitutions. In Proc. of Westapp'98, 1998. Full version submitted for publication.

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