J. Maraist, Separating weakening and contraction in a linear lambda calculus, in: Proc. CATS'98, Computing: The Australasian Theory Symposium (Perth, January 1998; unabridged version appears as Technical Report 25/96, Fac. of Computer Science, University of Karlsruhe) 151--165.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... j w M w Gamma Promotion j c M c Gamma Promotion j M Gamma Promotion j let M be w x in M w Gamma Dereliction j let M be c x in M c Gamma Dereliction j let M be x in M Gamma Dereliction 6 A similar calculus has been (independently) considered by Maraist [21]. Further details of this calculus are given in x7. where x is taken from some countable set of variables. Typing judgements are written Gamma; Delta; Sigma; Theta M : OE where Gamma and Theta are multisets of pairs of variables and types, and Delta and Sigma are sets of pairs of ....

.... c M is definitely used. Thus this suggests the following three rules. w M w M M v c M c v M M A detailed study of the operational theory and an investigation of the practical applications of this linear calculus will appear in joint work with A.M. Pitts [13] Maraist [21] has also (independently) considered a linear calculus, related to that given in x5.1. 8 One difference is that he does not have a distinct modality, but rather considers both combinations w c and c w to be equivalent and equal to . Thus typing judgements are still of the ....

J. Maraist. Separating weakening and contraction in a linear lambda calculus. Technical Report 25/96, Department of Computing Science, University of Karlsruhe, 1996.

.... for mapping general reduction sequences from call by name (call by value) Mackie, 1994; Maraist, 1997; Maraist et al. in press) both are sound and complete for mapping standard reduction sequences (Maraist, 1997) It is more difficult to apply these translations to the call by need calculus (Maraist et al. 1998), a more suitable basis for the analysis of lazy functional languages than call by name. Although call by need uses a different notion of reduction than callby name, their observational equivalence theories are the same for convergence both to constants and to function based closures. Unlike ....

....Section 3 with a discussion of various applications and extensions. 1 Background 1. 1 Intuitionistic calculi Figure 1 presents the details of the call by name and call by value systems Name and Val (Church, 1941; Plotkin, 1975) and Figure 2 presents the details of the callby need calculus Need (Maraist et al. 1998). Here and subsequently, we let L; M;N range over terms with a subset of values over which V; W range. We assume the existence of certain base types (which we leave unspecified for now) ranged over by Z, which we leave unspecified. Types, over which A; B range, are either the base type or a ....

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J. Maraist, Separating weakening and contraction in a linear lambda calculus, in: Proc. CATS'98, Computing: The Australasian Theory Symposium (Perth, January 1998; unabridged version appears as Technical Report 25/96, Fac. of Computer Science, University of Karlsruhe) 151--165.

....for our translation is an affine calculus, in which contraction is controlled by the connective but weakening is allowed everywhere. The use of distinct prefixes to control contraction and weakening separately has been studied first by Jacobs [18] for the full logic, and later by Maraist [24,25] for a lambda calculus. We derive the call by need calculus from the call by value calculus in two steps. The first step adds let terms, which enforce sharing, to the call by value calculus. The resulting call by let calculus is observationally equivalent to call by value; the translation, ....

J. Maraist, Separating weakening and contraction a linear lambda calculus, in: Proc. CATS'98, Computing: the Fourth Australian Theory Symposium (Perth, February 1998).

.... result of this section requires more effort; specifically a notion of marked reduction a la Barendregt [5] Marking redexes which are let bindings presents unique difficulties which we first addressed in the context of the call by need system [20, 21] the same techniques apply naturally to SLin [19, 20]. Proposition 12. SLin is confluent. 2.2 Translations into SLin Figure 7 presents the adaptations of the Girard translations to the separated calculus. The translations behave exactly as described in previous work [13, 22] the standard translation allows any function s argument to be duplicated ....

....of the images of Need terms, and a right inverse of , shown in Figure 9. This inverse is sound for mapping typing proofs of mapped terms as well as their reduction sequences, back into Need, which implies completeness of . We enumerate the various cases in the full technical report [19]. Lemma 15. Let S; T ; U and so forth be as in Figure 9. S[x : U ] j S [x : U 1 ] P [x : U ] 1 j P 1 [x : U 1 ] and (D[y] x : U ] 1 j (D 1 [y] x : U 1 ] If E SLin S : A then E Need S : A ; if E SLin P : A then E Need P 1 : A ....

[Article contains additional citation context not shown here]

J. Maraist, Separating weakening and contraction in a linear lambda calculus (unabridged) (Technical Report 25/96, Fakultat fur Informatik, Universitat Karlsruhe, 1996).

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