Philip Wadler. There's no substitute for linear logic. In 8'th International Workshop on the Mathematical Foundations of Programming Semantics, Oxford, 1992.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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 some of Wadler s earliest attempts to de ne a linear type system for a functional language agged linear types as the exception, rather than the rule [Wad90] although he later reverted to belling the ....

P. Wadler. There's no substitute for linear logic (projector slides). In G. Winskel, editor, Proceedings of the CLICS Workshop (Part I), March 1992, Aarhus, Denmark, May 1992. Available as DAIMI PB-397-I Computer Science Department, Aarhus University.

....In the above mentioned article the ( Gamma R) rule is decorated with the following terms: x 1 : A 1 ; x n : A n u : A 5 The Gentzen style system enjoys the substitution property simply because it is a rule of the system, namely the (Cut) rule. The problem, as pointed out in [Wad91], is as follows: The (Cut) rule together with ( Gamma R) decorated with terms as above) forces a collapse in the categorical model corresponding to the system. The modality is interpreted as a functor, and the two rules together would force to be isomorphic to . The problem is basically ....

Philip Wadler. There's no substitute for linear logic. Manuscript, 1991.

....studies the modeling of impure effects. This will provide the easiness brought by impure effects to the programmer without losing the reference transparency of the model. The backbone of this work will be the study of the linear type system (see [Gir87] Laf88] Ode91] Ode92] Wad90] and [Wad91]) and the monad programming paradigm (see [Wad92a] and [Wad92b] A new semantic description of behaviors based on constraint equations and effect systems using algebraic reconstruction algorithms (see [LG88] and [JG88] will reduce the need for user intervention in the late abstraction ....

P. Wadler. There's no substitute for linear logic. Technical report, Glasgow Univ., December 1991.

....R (0 ) NNILL with these rules (or NILL with similar rules) remains closed under substitution. 6.3. 2 Promotion Early formulations of natural deduction used the following apparently simpler introduction rule for : Gamma ) Q Gamma ) Q (P ) This is the rule to be found in [Avr88] Abr93] [Wad92], originally in [Tro92] Val92] and [RdRR97] 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 ....

P. Wadler. There's No Substitute for Linear Logic. 8th International Workshop on the Mathematical Foundations of Programming Semantics, 1992.

....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 programming [160, 278, 279, 281, 280, 282, 283]. 2.11 Structurally Free Logic A very recent innovation in the proof theory of substructural logics is the advent of structurally free logic. The idea is not new it comes from a 1976 essay by Bob Meyer [171] However, the detailed exposition is new, dating from 1997 [42, 41, 93] The ....

PHILIP WADLER. "There's No Substitute for Linear Logic". In Workshop on Mathematical Foundations of Programming Semantics, Oxford, UK, April 1992. (No proceedings published. Available from http://www.cs.bell-labs.com/who/wadler/topics/linear-logic.html).

.... 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 [79] and Brown and Gurr [30] ....

P. Wadler. There's no substitute for linear logic. Manuscript, December 1991.

....material is of the initial sanity check variety, and one could go much further. 14 We rst validate properties relating typing to substitution and reduction. Typing and reduction are areas where substructural type systems, which are surprisingly delicate, have encountered problems in the past [45, 44, 27, 50], so it is appropriate that they be be explored early. We then spell out how the calculus can be interpreted in any cartesian dcc. 4.1 Substitution and Reduction Before tackling reduction, we need that each of the introduction rules for function types is reversible. This is a property we expect ....

P. Wadler. There's no substitute for linear logic. In Proceedings of the 8th Symposium on Mathematical Foundations of Program Semantics, 1982. Oxford.

....calculi of the intuitionistic systems. In this report we take as our foundation the linear lambda calculus Lin of our earlier work with Odersky, Turner and Wadler [13] which in turn evolved from Wadler s criticisms of early formulations of lambda calculi based on intuitionistic linear logic (ILL) [20, 21] and from Barber and Plotkin s two context formulation of ILL [4] We extend Lin drawing from Rehof and Srensen s treatment Delta of the computational content of classical logic [18] and from Martini and Masini s natural deduction rules for the intensional sum connective O [15] The primary ....

Philip Wadler. There's no substitute for linear logic. In Proc. Eighth Int. Workshop on Mathematical Foundations of Programming Semantics, Oxford (MFPS'92), April 1992.

.... in which garbage collection was replaced by explicit duplication operations based on linear logic [21] More recent work has attempted to find a linear logical basis for many optimizations in (lazy) functional programming language implementations by concentrating on linear logic as a type system [1, 15, 39, 40, 25, 8, 29, 41]. Other applications include analyzing the control structure of logic programs [7] generalized logic programming [4, 16] and natural language processing [23] A natural characterization of polynomial time computations can be given in a bounded version of linear logic [13] obtained by limiting ....

....the resource sensitivity of linear logic to encode difficult problems in even the propositional fragment of linear logic. Current work is progressing to exploit the unique features of linear logic for use as a type system to study computational complexity [13] and compiler optimization techniques [40, 8, 29, 41, 34, 25], as well as uses in logic programming [16, 3, 4] natural language processing [24, 38] and concurrency [5, 30, 35] These recent contributions are developing linear logic from a theoretical curiosity into a tool that already has practical use within mainstream computer science. ....

P. Wadler. There's no substitute for linear logic. Draft, 1991.

....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 some of Wadler s earliest attempts to define a linear type system for a functional language flagged linear types as the exception, rather than the rule [Wad90] although he later reverted to belling ....

P. Wadler. There's no substitute for linear logic (projector slides). In G. Winskel, editor, Proceedings of the CLICS Workshop (Part I), March 1992, Aarhus, Denmark, May 1992. Available as DAIMI PB-397-I Computer Science Department, Aarhus University.

....variables are terms of m , the hypotheses we formulate about them are assertions concerning bags of resources. Then x : may be read as x is one of the resources of a bag satisfying . Since our typing system as opposed to the various linear term calculi one finds in the literature ([3,4,15,16,21]) does not record the various manipulations of the hypothesis like weakenings, contractions, derelictions, as terms constructions, we shall factorize these manipulations into just one rule. For this purpose, we write Gamma AE Delta whenever the hypothesis Delta results from ....

....characters, except that is replaced by a set of propositional variables possibly subject to universal quantification , and that we use the exponential OE. There are by now some papers, by Abramsky [3] Benton et al. 4] Lafont [12,14] Lincoln and Mitchell [15] Mackie [16] Wadler [21], among others, which investigate the possible use of intuitionistic Linear Logic in functional programming. The common expectation is that Linear Logic, as a logic of resources, could help in analysing and solving implementation problems regarding storage management and evaluation strategies. ....

Ph. Wadler, There's no substitute for Linear Logic, draft (1991).

....rules should be direct. Other researchers have independently formulated similar typing rules, although none we know of incorporate a rule of the form of the SR rule of nat. Lafont, Girard, Abramsky, and others have studied systems very similar to seq [GL87, Abr90] In recent unpublished notes [Abr91, Wad91b] and an MS thesis [Mac91] systems close to nat2 have been studied. Walder also discusses alternative rules for SR and the implications of syntaxless Subst rule in the context of a nat like system. O Hearn discusses similar issues from a different viewpoint [O H91] We take this parallel ....

....and polymorphism. Abramsky has described the implementation of a linear SECD machine further studied by Mackie [Abr90, Mac91] and went on to generalize the linear calculus to one based on classical linear logic and described an implementation based on the chemical abstract machine [BB90] Wadler [Wad91b] has also described several implementation issues regard ing the linear calculus. He points out the importance of ( A) being isomorphic to A (which is true in our operational model) and suggests several extensions, including, for example, arrays, let with read only access, the removal of ....

P. Wadler. There's no substitute for linear logic. Draft, 1991.

....term language is essentially a refinement of the 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 ....

....the rules are decorated with terms. The above mentioned rule for recursion in the extended ILL is appropriate for a Gentzen style formulation, but we lose the Substitution Property if we add it to a natural deduction formulation of ILL. This problem is similar to the problem with the ( R ) rule, [Wad91]. The solution is similar too; we generalise the rule for recursion to the one given in Appendix D. The above mentioned reduction rule for recursion in the extended ILL in Gentzen style corresponds to the one given in the natural deduction formulation, the linear rec calculus. Recursion in the ....

P. Wadler. There's no substitute for linear logic. Manuscript, 1991.

....have come up with systems which include aspects of the LNL term calculus. For example, Lincoln and Mitchell s linear variant [10] of the three instruction machine divides memory into two spaces corresponding to linear and non linear objects. Similarly, Wadler s active and passive type system [14] separates linear from non linear types. Jacobs [9] has also described how a sequent calculus inspired by CLEAN s uniqueness types may be interpreted using the linear categories of [4] under some extra simplifying assumptions. From a more logical point of view, there has recently been much ....

P. Wadler. There's no substitute for linear logic (projector slides). In G. Winskel, editor, Proceedings of the CLICS Workshop (Part I), March 1992, Aarhus, Denmark, May 1992. Available as DAIMI PB-397-I Computer Science Department, Aarhus University.

....term assignment system for Girard s Intuitionistic Linear Logic [10] Previous approaches have simply annotated the sequent calculus formulation with terms and have given little or no justification for their choice. A poor choice can have serious consequences. An example discovered by Phil Wadler [29] is that the substitution lemma does not hold for the term assignment system corresponding to an intuitive natural deduction formulation of Intuitionistic Linear Logic: a consequence is that such a system is too weak to provide a proof theory for linear logic. We have approached the formulation of ....

....show that the interpretation in C is independent of the derivation. It is laborious but not essentially difficult to prove this directly; however the 2 This assumption has the effect that in the categorical model, which we shall consider later, the comonad is idempotent : a point noted by Wadler [29]. f [let x be in e=y] let x be in f [e=y] f [let z be x Omega y in g=w] let z be x Omega y in f [g=w] f [discard z in e=y] discard z in f [e=y] f [copy z as x; y in e=w] copy z as x; y in f [e=w] Figure 3: Naturality Equations result also follows easily from a consideration of the ....

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Philip Wadler. There's no substitute for linear logic. Draft Paper, December 1991.

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P. Wadler, There's no substitute for linear logic. Presented at Workshop on Mathematical Foundations of Programming Language Semantics, Oxford, April 1992.

....it would be wrong to burden him with too much blame. His syntax is coherent with the operational model he uses. In Brookes et al. editors, Mathematical Foundations of Programming Semantics, New Orleans, April 1993, Springer Verlag LNCS 802. This difficulty was spotted previously by myself [Wad92]. Other researchers have not only observed the problem, but also proposed a solution in the form of a syntax that boxes the Promotion rule, in much the same way that boxes are used in proof nets. Notable in this regard is the work of Benton, Bierman, de Paiva, and Hyland [BBdPH92] which ....

....The proof is by examination of overlapping rules. All of the variations of Abramsky s syntax cited above suffer from this problem in one form or another. In a natural deduction system, this problem reveals itself in a failure of the Substitution Lemma: substitution does not commute with Promotion [Wad92]. The same difficulty is at the root of problems that Lincoln and Mitchell [LM92] and Chirimar, Gunter, and Riecke [CGR92] encountered with Subject Reduction theorems, forcing them to be restricted in various ways. One way to fix the problems is to restrict the class of categorical models. In an ....

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P. Wadler, There's no substitute for linear logic. Presented at Workshop on Mathematical Foundations of Programming Language Semantics, Oxford, April 1992.

.... The particular formulation of linear logic presented here is based on Girard s Logic of Unity, a refinement of linear logic [7] This overcomes some technical problems with other presentations of linear logic, some of which are discussed by Benton, Bierman, de Paiva, and Hyland [2] and Wadler [23, 24]. Much of the insight for this work comes from categorical models of linear logic [19, 15] The particular system presented here was suggested to the author by Girard, and a similar system has been suggested by Plotkin. For further background on traditional logic see the wonderful introduction by ....

P. Wadler, There's no substitute for linear logic. Presented at Workshop on Mathematical Foundations of Programming Language Semantics, Oxford, April 1992.

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Philip Wadler. There's no substitute for linear logic. In 8'th International Workshop on the Mathematical Foundations of Programming Semantics, Oxford, 1992.

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Philip Wadler. There's no substitute for linear logic. In 8'th International Workshop on the Mathematical Foundations of Programming Semantics, Oxford, 1992.

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Philip Wadler. There's no substitute for linear logic. In 8th International Workshop on the Mathematical Foundations of Programming Semantics, 1992.

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Philip Wadler. There's no substitute for linear logic. In 8th International Workshop on the Mathematical Foundations of Programming Semantics, 1992.

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