Jean-Yves Girard and Yves Lafont, Linear logic and lazy computation, Proceedings of the International Joint Conference on Theory and Practice of Software Development (Pisa, Italy) (Hartmut Ehrig, Robert A. Kowalski, Giorgio Levi, and Ugo Montanari, eds.), SpringerVerlag LNCS 250, March 23--27 1987, pp. 52--66.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....terms under substitutions when commuting conversions are considered. We show that such a bound exists if we only count reductions or consider evaluation to values of closed terms. Key words: linear logic, lambda calculus, complexity, normalization. 1 Introduction Intuitionistic Linear logic [GL87,Laf88] and its associated linear lambda calculus have been used in many ways in the analysis of functional programming languages [Wad90,Laf88] The literature shows di erent lambda calculi associated to Intuitionistic Linear Logic [BBdH93,Ben95,BP97] but they all coincide on Rudimentary Linear Logic, ....

J.Y. Girard and Y. Lafont. Linear logic and lazy computation. In TAPSOFT '87, volume 250 of Lecture Notes in Computer Science, pages 52-66, 1987.

....The linear isomorphism of types corresponds to the isomorphism of objects in free SMC category, and can also be described as the isomorphism of types in the system of lambda calculus which corresponds to intuitionistic multiplicative linear logic. A description of this system can be found in [5] [6], 7] 8] In [7] it was shown that the subsystem of the axiom system above, consisting of the axioms 1) 6) where is understood as times and as 3 linear implication) with the same rules, defines an equivalence relation on types that coincides with the relation of linear isomorphism of ....

G.-Y. Girard, Y. Lafont. Linear logic and lazy computation. In: Proc. TAPSOFT 87 (Pisa), v.2, p.52-66, Lecture Notes in Comp. Sci. v. 250 , 1987.

....and rules. This definition does not differ essentially from the definition of the unlabelled deductive system in [17] though deductive systems for SMC categories can be found in earlier works [11] 13] The calculus may be considered also as the multiplicative intuitionistic linear logic, see [5]. Axioms 9 A A (identity) unit) Structural Rules Gamma A A; Delta B Delta Gamma I Sigma Gamma A Delta; Sigma Gamma A Logical rules Gamma A Delta B A; B; Gamma C ( Omega Gamma ) A; Gamma B ( Gamma , Gamma A B; Delta C Derived objects of ....

G.-Y. Girard, Y. Lafont. Linear logic and lazy computation. In: Proc.TAPSOFT 87 (Pisa), v.2, p.52-66, Lecture Notes in Comp.Sci. v.250 , 1987.

....considered are in fact the internal languages of their respective categories of models, we could derive translations between these type theories as a consequence of relationships between models, if so wished. 3 Linear Type Theories Intuitionistic Linear Logic, introduced by Girard and Lafont [13, 16], has been investigated for its potential applications to functional programming. Several linear type theories, which could have been called the correct typed linear calculus, have been proposed for these applications. We rst present the type theory corresponding to the (non controversial) ....

J.Y. Girard and Y.Lafont. Linear logic and lazy computation. In TAPSOFT '87, volume 250 of Lecture Notes in Computer Science, pages 52-66, 1987.

.... Gamma; A B; Delta C (R L) Gamma A Gamma A B (R R) Gamma B Gamma A B Table 1: A sequent system L for PL oe . With one exception, the rules in Table 1 are the natural generalisations of the rules suggested by Girard for the commutative intuitionistic linear logic, cf. 19] and [20, 21], to the non commutative case, cf [14, 15] Axiom ( is the only exception the expected generalisation is Gamma; Delta A, as in [14, 15] However, the stronger axiom is not valid in our intended interpretation in quasi quantales, cf. 8] Actually, quasi quantales were invented by ....

Girard, J.-Y. and Y. Lafont. Linear Logic and Lazy Computation. In: Proc. TAPSOFT'87 (Pisa), vol. 2, LNCS 250, Springer Verlag, pp 52--66, 1987.

....operations that are supported by games and strategies. It is well known that a symmetric monoidal category can be used as the basis of a categorical semantics of linear logic [See89] Using the Curry Howard isomorphism [How80] we can develop a corresponding calculus, the linear calculus [Abr93, GL87]. Using the implementation of games and strategies, we can make use of the categorical semantics to implement this language. We choose to consider the intuitionistic multiplicative fragment of linear logic only, since this avoids complications with attempting to express the resource duplication ....

J.-Y. Girard and Y. Lafont. Linear logic and lazy computation. In TAPSOFT '87, Volume 2, pages 52--66. Springer LNCS 250, 1987.

..... its neutral element , the negation and to led to consider a multi conclusion representation. Then 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 ....

....rule applied to a mono conclusion sequent gives the same result than the ILL corresponding rule. Let us note that if we consider a version of CLL including the rules for the connective then the ILL proof will be identical to the CLL proof. Works on ILL have been developed from linear calculus [6, 8, 24], but the lack of the . connective seems more motivated by technical reasons and then we could ask for an intuitionistic fragment of linear logic including it. In this context, the semantics of . ....

J.Y. Girard and Y. Lafont. Linear logic and lazy computation. In TAPSOFT 87, LNCS 250, pages 52--66, Pisa, Italia, March 1987.

....of logical system that includes an explicit representation of proofs. With this in mind, we have considered a categorical generalization of the notion of consequence relation whereby proofs become morphisms in a consequence category satisfying some weak closure properties (as in linear categories [GL87] It seems difficult, however, to develop the notions of structured presentation and structured search in such a way that a witness to the fact that a sentence is a consequence of a structured presentation may be extracted. The difficulty seems to lie in the fact that structured presentations ....

J.-Y. Girard and Y. Lafont. Linear logic and lazy computation. In H. Ehrig, R. Kowalski, G. Levi, and U. Montanari, editors, TAPSOFT '87: Colloquium on Functional and Logic Programming and Specifications, pages 52--66, Pisa, Italy, March 1987.

....the MILL cut rule) are interderivable. However, this precise match is obtained by paring down both linear logic and our logic. We can go further and draw a connection with full intuitionistic linear logic, both syntactically and semantically. First, syntactically, intuitionistic linear logic (ILL) [8,11,5] can be embedded in our logic by the mapping: This mapping is such that the rules of ILL can be derived within our logic. In particular, we can derive the strong rules for #that correspond to an interpretation of as a maximal fixpoint, by some non trivial use of our rules. So, 1 , ....

Girard, J-Y., Lafont, Y.: Linear Logic and Lazy Computation. TAPSOFT 87, LNCS 250 vol 2, 53-66, Springer, 1987.

....to describe properties of nets. Linear logic is a logic discovered by J Y. Girard [28] It can be thought of as a resource conscious logic and has thus recently gained considerable interest from the computer science community. Applications include implementation models for functional languages [34, 47, 63], the study of process algebras [3, 2, 1] logic programming [4, 14, 40] planning in AI [50] and net theory [5, 9, 10, 11, 8, 24, 39, 49] The list is by no means meant to be comprehensive because the field is advancing very rapidly. The applications relevant to our purposes are the applications ....

Girard, J.-Y. and Lafont, Y. Linear logic and lazy computation. TAPSOFT '87. Springer LNCS 250, 1987, pp. 52--66.

....In precise technical terms, the target of our translation is an intuitionistic version of mall, presented by two sided sequents with at most one consequent formula. Similar intuitionistic versions of various fragment of linear logic are considered in relationship to computer science, e.g. in [14, 23, 6, 11, 16, 1, 24]. Apart from the foundational interest, we believe that the result of this paper, which is theoretical in nature, contributes to the understanding of the role of linear logic as an expressive and natural framework for describing control structure of logic programs. This logic programming ....

....to be the translation of the rightmost branch of the iil proof. The left main branch of the proof progresses by applying the GammaffiL rule. Here there is a choice to be made in the way we split the context Sigma 0 among 2 Our arguments also apply to the sequent calculus given on p. 53 of [14] without the 0 rule. I p p Sigma A A; Gamma Delta Sigma; Gamma Delta Cut Omega L Sigma; A; B Delta Sigma; A Omega B) Delta Sigma A Gamma B Sigma; Gamma (A Omega B) Omega R GammaffiL Sigma A Gamma; B Delta Sigma; Gamma; A GammaffiB) ....

J.-Y. Girard and Y. Lafont. Linear logic and lazy computation. In TAPSOFT '87, Volume 2, pages 52--66. Springer LNCS 250, 1987.

.... ( Dere a) a ) Dere a ) a) weak : Arr (Bang a) Unit weak = Arr ( Weak, Unit) Weak,Unit) cont : Arr (Bang a) Ten (Bang a) Bang a) cont = Arr ( Cont x y) Ten x y ) Cont x y ) Ten x y) To construct of course values, we use Girard s make combinator [GL87]: make : Arr a b Arr a Unit Arr a (Ten a a) Arr a (Bang b) make d w c = Arr ( u case u of (a, Dere b ) let (a ,b) app d (a,b ) in (a ,Dere b) a, Weak) let (a , Unit) app w (a, Unit) in (a , Weak) a, Cont x y ) let (a , Ten a1 a2) ....

....in the usual way. Arr a c Arr b d Arr (With a b) With c d) f g = pair (left f) right g) The similarities between the operations on a and the operations on aNb is reminiscent of the equation a = 1 N affl a N a that appears in some variants of linear logic [GL87]. Indeed, the above is an isomorphism in the model presented here. The existence of pair, left, and right means that the arrow c Gamma aNb is isomorphic to the pair of arrows (c Gamma a; c Gamma b) 8.2 Top The unit for N is . Just as N is represented by a sum of two alternatives (Inl and ....

J.-Y. Girard and Y. Lafont. Linear logic and lazy computation. In H. Ehrig, R. Kowalski, G. Levi, and U. Montanari, editors, Proceedings of the International Joint Conference on Theory and Practice of Software Development, volume 2, pages 5266, Pisa, Italy, March 1987. SpringerVerlag LNCS 250.

....which is based on Linear Logic rather than Intuitionistic Logic, has been proposed as a means of adding resource control into functional programming. All the work done so far on linear functional programming is based on Intuitionistic Linear Logic, the fragment considered by Lafont and Girard in [GL87]. For this fragment, since derivations have a single conclusion, it is easier to formulate a Natural Deduction version, and hence the similarity with standard functional programming is clearer. To extend this kind of approach to the system of FILL seems very promising. Because FILL is a multiple ....

J.-Y. Girard and Y. Lafont. Linear logic and lazy computation. In Proceedings of TAPSOFT '87, LNCS, volume 250. Springer-Verlag, 1987.

....isomorphism, or formulas as types principle [How80] Through this isomorphism, intuitionistic proofs of propositions may be viewed as functional programs, and logical propositions may viewed as types. A similar use of linear logic has been initiated by Girard and Lafont, and Abramsky [GL87, Abr90]. In [GL87] a linear calculus was developed which effectively determines reduction order, while explicitly marking the points where contraction and weakening are used. Abramsky further defined a type inference system, here called seq, which is discussed in Section 3.3. Abramsky went on to ....

....or formulas as types principle [How80] Through this isomorphism, intuitionistic proofs of propositions may be viewed as functional programs, and logical propositions may viewed as types. A similar use of linear logic has been initiated by Girard and Lafont, and Abramsky [GL87, Abr90] In [GL87], a linear calculus was developed which effectively determines reduction order, while explicitly marking the points where contraction and weakening are used. Abramsky further defined a type inference system, here called seq, which is discussed in Section 3.3. Abramsky went on to generalize this ....

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J.-Y. Girard and Y. Lafont. Linear logic and lazy computation. In TAPSOFT '87, Volume 2, pages 52--66. Springer LNCS 250, 1987.

....one developed by Lafont and the other by Holmstrom. In his thesis [17] Lafont describes the implementation of a small functional programming language based on the linear calculus. However, instead of attempting to apply what he calls the brutish compilation scheme of his published papers [8, 16] to an ordinary functional language, he designs his own linear functional language, called LIVE. This language exposes the programmer to the full rigours of the linear calculus, and only a few small types (such as integers) are permitted to escape the linearity constraint. LIVE is implemented ....

J.-Y. Girard and Y. Lafont. Linear logic and lazy computation. In Proceedings of the International Joint Conference on Theory and Practice of Software Development (TAPSOFT'87), pages 52--66. Springer-Verlag, March 1987. LNCS 250.

....derivability. This technique has been used in proof search in the prover of the system PRIZ, developped in Tallinn institute of cybernetics in the 80 es (Volozh, Matskin, Mints and Tyugu 1983) Intuitionistic linear logic without exponentials and additive conjunction is considered (taking system (Girard and Lafont 1987)) It turns out to be possible to use the idea from PRIZ without too radical modifications, and at the same time to give a framework for possible modifications. It was a pleasant surprise, that exponentials do not appear when this modified technique is applied to a system of linear logic without ....

.... Gamma T Gamma A Gamma A Phi B ( Phil) Gamma B Gamma A Phi B ( Phir) Gamma; A C Gamma; B C Gamma; A Phi B C ( Phi ) Gamma; F A Gamma; A B Gamma A Gamma , B ( Gamma , Gamma A Delta; B C Gamma; Delta; A Gamma , B C ( Gamma , cf. (Girard and Lafont 1987)) As in the ordinary intuitionistic logic, the rule of substitution of formulas for variables is admissible. We shall use the folloving properties of Linear Intuitionistic Logic: Proposition 2.1. Cut elimination property. Let S be a sequent. If S is derivable in Linear Intuitionistic Logic, ....

Girard, G.-Y., Lafont, Y.(1987) Linear logic and lazy computation. In Proc.TAPSOFT 87 (Pisa), v.2, p.52-66, LNCS v.250.

....can be either specified explicitly by the programmer, giving the compiler richer information for storage management, or inferred by a smart type inference mechanism, thus using linear calculus as a lower level intermediate code. Some early work in this direction has been done by Girard and Lafont [72] and Holmstrom [88] Lafont introduced a Linear Abstract Machine [99] which implements a functional language without the use of garbage collection (but with much copying of terms) An important work is the one by Abramsky [1] He describes a linear machine more faithful in spirit to the SECD ....

J.-Y. Girard and Y. Lafont. Linear logic and lazy computation. In H. Ehrig, R. Kowalski, G. Levi, and U. Montanari, editors, Theory and Practice of Software Development (TAPSOFT'87), Vol. 2, number 250 in LNCS, pages 52--66, Mar. 1987.

....is full classical linear logic, which includes the additive operations. These correspond to requiring that the category C have products and coproducts. If C is autonomous, one of these will imply the other by de Morgan duality. There is also Girard s notion of intuitionistic linear logic [GL87], which omits linear negation and par this corresponds to merely requiring that C be 1 In other papers we have used the notation for the unit for Omega , and Phi instead of . ....

Girard, J.-Y. and Y. Lafont "Linear logic and lazy computation", in tapsoft'87, 2, Lecture Notes in Computer Science 250, Springer-Verlag, Berlin, Heidelberg, New York, 1987, 52--66.

....linear isomorphism of types corresponds to the isomorphism of objects in free SMC category, and can be also described as the isomorphism of types in the system of lambda calculus which corresponds to intuitionistic multiplicative linear logic. A description of this system can be found in [5] [6], 7] 8] In [7] it was shown that the subsystem of the axiom system above, consisting of the axioms 1) 6) where is understood as times and as linear implication) with the same rules, defines an equivalence relation on types that coincides with the relation of linear isomorphism of types. ....

G.-Y. Girard, Y. Lafont. Linear logic and lazy computation. In: Proc.TAPSOFT 87 (Pisa), v.2, p.52-66, Lecture Notes in Comp.Sci. 250 (1987).

....between appropriate definable functors as a consequence of a more general Full Completeness Theorem below. 7. 1 Interpreting Omega ; Gammaffi We shall first work in the theory of symmetric monoidal closed ( smc) categories without units, equivalently in intuitionistic MLL without units [20, 9]. Thus formulas are built from atoms, using the connectives Omega ; Gammaffi. Following the lead of functorial polymorphism (loc cit) we interpret formulas as multivariant functors over an smc category C, using the following functorial operations on n ary multivariant functors F; G : C op ) ....

J.Y. Girard, Y. Lafont, Linear Logic and Lazy Computation, in: Springer Lecture Notes in Computer Science 250, (1988)

....with the nice feature that rewriting techniques may be applied to it successfully. This process of transformation of the data provided by the theoretical considerations follows a definite pattern that is described in Section 2.3. Intuitionistic Linear Logic was introduced by Girard and Lafont in [4]. The basic assumption of Linear Logic is that one should be able to have a logical control of the resources available for a derivation. This resource sensitiveness of Linear Logic is its main claim to applicability, and applications have sprung in all areas of Computer Science (for an ....

J.-Y. Girard, Y. Lafont, `Linear Logic and Lazy Computation', Proc. TAPSOFT 87, Lecture Notes in Computer Science 250, Vol. 2, Springer-Verlag, 1987, pp. 52--66.

....as derivability is concerned, and dual to . 3.1 A sequent system for LPoe A sequent style presentation of the logic is given in Table 2. With one exception, the rules in Table 2 are the natural generalisations of the rules given by Girard for the commutative intuitionistic linear logic, cf. [12, 13, 14], to the non commutative case, cf [7, 8] The exceptional axiom is ( Its expected generalisation is Gamma; Delta A, as in [7, 8] However, the stronger axiom is not valid in our intended interpretation in quasi quantales, cf. 4] as described in section 4. Embedding the usual logic into ....

Girard, J.-Y. and Y. Lafont. Linear Logic and Lazy Computation. In: Proc. TAPSOFT'87 (Pisa), vol. 2, LNCS 250, Springer Verlag, 1987, pp 52--66.

....and a dual to . 3.1 A sequent system for LP = oe A sequent style presentation of the logic is given in Table 2 in the appendix. With one exception, the rules in Table 2 are the natural generalisations of the rules given by Girard for the commutative intuitionistic linear logic, cf. [10, 11, 12], to the non commutative case, cf [7] The exceptional axiom is ( Its expected generalisation is Gamma; Delta A, as in [7] However, the stronger axiom is not valid in our intended interpretation in quasi quantales as described in section 4 and in. 4] Embedding the usual logic into LP ....

Girard, J.-Y. and Y. Lafont. Linear logic and lazy computation. In: Proc. TAPSOFT '87 (Pisa), vol. 2, LNCS 250, Springer Verlag, 1987, pp 52--66.

....Functional Programming and Sequent Calculus to motivate the introduction of Linear Logic. In particular, we consider a variant of Lambda Calculus based on Sequent Calculus and we point out the problems due to structural rules. Linear Logic itself is not presented here: we refer to [Girard87] [Girafont], Giraflor] Lafont88a] and [Lafont88b] A clear introduction to Proof Theory and Lambda Calculus is also available in [Giraflor] 1 From Natural Deduction to Functional Programming 1.1 Rules of Natural Deduction Natural Deduction is a formal system for (constructively) proving a logical ....

J.Y. Girard & Y. Lafont, Linear Logic and Lazy Computation, TAPSOFT '87, vol. 2, LNCS 250 (Springer-Verlag, Pisa) 52-66.

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Jean-Yves Girard and Yves Lafont, Linear logic and lazy computation, Proceedings of the International Joint Conference on Theory and Practice of Software Development (Pisa, Italy) (Hartmut Ehrig, Robert A. Kowalski, Giorgio Levi, and Ugo Montanari, eds.), SpringerVerlag LNCS 250, March 23--27 1987, pp. 52--66.

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Jean-Yves Girard and Yves Lafont, Linear logic and lazy computation, Proceedings of the International Joint Conference on Theory and Practice of Software Development (Pisa, Italy) (Hartmut Ehrig, Robert A. Kowalski, Giorgio Levi, and Ugo Montanari, eds.), SpringerVerlag LNCS 250, March 23--27 1987, pp. 52--66.

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Jean-Yves Girard and Yves Lafont, Linear logic and lazy computation, Proceedings of the international joint conference on theory and practice of software development (Pisa, Italy) (Hartmut Ehrig, Robert A. Kowalski, Giorgio Levi, and Ugo Montanari, editors), Springer-Verlag LNCS 250, March 23--27 1987, pp. 52--66.

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Jean-Yves Girard and Yves Lafont. Linear logic and lazy computation. In Proceedings of TAPSOFT '87, vol 2, volume 250 of Lecture Notes in Computer Science, pages 52--66, 1987.

No context found.

Jean-Yves Girard and Yves Lafont, Linear logic and lazy computation, Proceedings of the International Joint Conference on Theory and Practice of Software Development (Pisa, Italy) (Hartmut Ehrig, Robert A. Kowalski, Giorgio Levi, and Ugo Montanari, eds.), SpringerVerlag LNCS 250, March 23--27 1987, pp. 52--66.

No context found.

Jean-Yves Girard and Yves Lafont. Linear Logic and Lazy Computation. In Proc. TAPSOFT 87 (Pisa), vol. 2, pages 52--66. Springer-Verlag (LNCS 250), 1987.

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Jean-Yves Girard and Yves Lafont, Linear logic and lazy computation, Proceedings of the International Joint Conference on Theory and Practice of Software Development (Pisa, Italy) (Hartmut Ehrig, Robert A. Kowalski, Giorgio Levi, and Ugo Montanari, eds.), SpringerVerlag LNCS 250, March 23--27 1987, pp. 52--66.

No context found.

Jean-Yves Girard and Yves Lafont, Linear logic and lazy computation, Proceedings of the international joint conference on theory and practice of software development (Pisa, Italy) (Hartmut Ehrig, Robert A. Kowalski, Giorgio Levi, and Ugo Montanari, editors), Springer-Verlag LNCS 250, March 23--27 1987, pp. 52--66.

No context found.

G.-Y. Girard, Y. Lafont. Linear logic and lazy computation. In: Proc.TAPSOFT 87 (Pisa), v.2, p.52-66, Lecture Notes in Comp.Sci. v.250 , 1987.

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Girard, Jean-Yves and Lafont, Y., Linear logic and lazy computation, manuscript, (1986).

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GL86 Girard, Jean-Yves and Lafont, Y. Linear logic and lazy computation, manuscript, (1986).

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