Yves Lafont. The Linear Abstract Machine. Theoretical Computer Science, 59:157--180, 1988.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....this chapter it will be su#cient if one understands how graphs, specializations, reductions, sums and pushouts are defined 39 furthermore, one should be aware of the algebraic laws stated in section 4.1.4. We do not in any way claim our approach to be original, in particular the approach in [Rao84] is rather close to ours. However, the development (especially in section 4.3) is given a new particular twist in order to suit our (later) purposes. The graphs in section 4.1 have all been singlelabeled; in section 4.2 multilabeled graphs are introduced enabling one to express e.g. that ....

.... Some notation: we say that a morphism m from G 1 to G 2 respects a node n i# the following holds : m(n) is the same kind of node (active passive virtual) as n; if n is passive then 2 (m(n) 1 (n) hence also Ar 2 (m(n) Ar 1 (n) and if n is active or passive then for all [Rao84] uses the terminology that m is a morphism at n i# the last two conditions hold, i.e. if labels and successors are preserved. 45 ## ## ## ## ## ## T T T l l l l T T T T f v f v v r 1 r 2 Figure 4.2: Two reductions. Ar(n) m(S 1 (n, i) 2 (m(n) i) ....

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Jean Claude Raoult. On graph rewritings. Theoretical Computer Science, 32:1--24, 1984.

....think that LTAL can serve as an important core for more realistic systems. Of course, the idea of employing linearity is not new many researchers have proposed linear languages and implementation techniques for implementing functional languages without garbage collection or using bounded space [12, 4, 8, 11]. But none of these approaches carry type information all the way down to a realistic assembly language as we do. Recently, Aspinall and Compagnoni [2] have developed Heap Bounded Assembly Language (HBAL) a variant of TAL that employs linearity to guarantee nite heap usage with direct memory ....

....functions like filter and map cannot be expressed in LFPL. 7. RELATED WORK The idea of using linearity to implement functional languages without garbage collection has a long history. Here we focus on work that is closely related to or strongly in uenced ours. Lafont s Linear Abstract Machine [12] was an early approach showing how to translate linear functional programs to the instructions of a linear abstract machine that recycled memory directly. Wadler [27] was an early proponent of using linearity to support imperative features in purely functional languages, and Wakeling and Runciman ....

Yves Lafont. The linear abstract machine. Theoretical Computer Science, 59:157-180, 1988.

....(e.g. fold and unfold) are sucient to do some programming tasks, but are by no means complete. 6. 2 Related Work Our type system builds upon foundational work by other groups on syntactic control of interference [31] linear logic [13] and linear type systems in functional programming languages [20, 42, 1, 15, 3, 8, 40]. Our research also has much in common with e orts to de ne program logics for reasoning about aliasing [6, 9, 26, 32, 17] In particular, if we view propositions as types, there are striking similarities with recent work by Reynolds [32] who builds on earlier research by Burstall [6] Reynolds ....

Yves Lafont. The linear abstract machine. Theoretical Computer Science, 59:157{ 180, 1988.

....can avoid garbage collection for all linear values. We give a formal proof that, in the case where our operational interpretation of linear logic satisfies the strong single pointer property, it is possible to allocate linear values in a separate area which need never be garbage collected. Lafont [Laf88] and Abramsky [Abr92] both describe abstract machines for intuitionistic linear logic. Both their abstract machine recompute non linear values, so they should preserve the strong single pointer property. However, their abstract machines are formulated slightly too abstractly to capture memory ....

Yves Lafont. The linear abstract machine. Theoretical Computer Science, 59, 1988.

....analysis of sharing and garbage collection. More concretely, some of the overhead involved in the environment based approach could be avoided by detecting linear usage of variables and directly performing substitutions for such variables. These ideas have led to several abstract machines [16, 17, 18], but all run into diculties with correctly exploiting linearity information. In particular, Wadler [22] and Chirimar et al. 7] tried to give an operational semantics where terms of linear type have exactly one pointer to them. However, their attempts failed mainly because the linearity inherent ....

Yves Lafont. The linear abstract machine. Theoretical Computer Science, 59:157-180, 1988.

.... be valid to swap the contents of the two variables (preserving one reference to each object) but not to assign the value of one variable to the other (constructing two references to one object and leaving the other unreferenced) Linear objects are inspired by linear logic, introduced by Girard [Girard87] in which a proof must use each assumption exactly once. Scedrov presents a survey of that topic [Scedrov95] Linear objects can be used to naturally model some forms of resource: maintaining exactly one reference to the object reflects the fact that ownership of the resource can be passed ....

Jean-Yves Girard. Linear Logic. Theoretical Computer Science, 50:1--102, 1987. (p 133)

....Graph reduction captures this exactly, and does so in a typed framework. Linear logic provides type information for this new perspective into what a procedure does, with new insights as well into how to write programs, and does so in a way that is very different from continuation passing style [Gir87, Gir95] This insight gives us a different view of control threads and into naming, where information about functions is considerably refined. In linear logic, a procedure of type A Gammaffi B is one that accesses a datum of type A exactly once, and returns a datum of type B. This is ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50, 1987.

....timed Petri net reachability is equivalent to the provability in the subsystem of temporal linear logic for the corresponding sequent. Our final target is to analyze the dynamic behavior of timed Petri nets by means of the logic. 1 Introduction Linear logic which was introduced by Girard in 1987 [3] has been called a resource conscious logic. The expressive power is evidenced by some very natural encodings of computational models such as Petri nets (PN) 7, 15, 8, 10] Timed Petri nets (TPN) 2] are place transition nets enhanced with a definite notion of time. In this paper, we focus only ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987.

....computation, such as [24] on distributed computation, is yet to be clari ed. Another interesting piece of work at the intersection of logic (here, linear logic) and simplicial geometry is [3, 4] which provides sophisticated models for the multiplicative exponential fragment of linear logic [17] based on ane simplicial spaces with an extra homological constraint. Note that there are strong links between S4 and linear logic, see e.g. 34] 3. Logics, the Curry Howard Correspondence, and S4 3.1. Logics Consider any logic, speci ed as a set of deduction rules. So we have got a notion of ....

Jean-Yves Girard, Linear logic, Theoretical Computer Science 50 (1987), 1{ 102.

....cartesian product of the webs of X 0 and Y . There is a natural isomorphism between the space of stable functions from the cliques of X to those of Y and the cliques of s X ( Y . These two operations have logical counterparts which are made explicit as logical connectives in linear logic ([4, 8, 6] describe the coherence space semantics of linear logic) Van de Wiele observed that alternative definitions of the exponential operation on coherence spaces are available. More specifically, from a categorical viewpoint, the exponential is an endofunctor on the category of coherence spaces and ....

.... order and has the following functional counterpart: let X and Y be coherence spaces; let f and g be stable maps from X to Y ; then f 6B g iff 8x; y 2 C(X) x [ y 2 C(X) f(x y) f(x) g(y) For more information on the coherence spaces denotational semantics of linear logic, we refer to [4, 8, 6]. Let us now go back to the multiset exponentials. Let X and Y be coherence spaces. By use of the evaluation map of coK( m ) we associate to each clique t of m X ( Y a function from C(X) to C(Y ) sending a clique x of X to the following clique of Y : fb = 9 0 2 j m Xj ( 0 ; b) 2 t j ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987.

....functional programming languages like PCF [1] In the framework of dilators (functors acting on ordinals) Girard discovered independently stability as a condition allowing for a finitary representation of these functors. He applied the same idea to the denotational semantics of system F (see [3]) and this led him to the crucial observation that this semantics (which is an extension of Berry s semantics of PCF) can be described in the framework of qualitative domains, and even in the one of coherence spaces, which are particular qualitative domains. Berry actually developed his semantics ....

Jean-Yves Girard. The system F of variable types, fifteen years later. Theoretical Computer Science, 45:159--192, 1986.

....proofs are equivalent when they have the same normal form. tortora uniroma3.it 1 Now, a very natural question arises: do these two equivalence relations (sometimes always) coincide Proofs of linear logic (LL) are represented as proof nets , a graph theoretic presentation (introduced in [Gir87]) which gives a more geometric account of proofs. Like for natural deduction proofs and calculus terms, with several sequent calculus proofs is associated a unique proof net. A net is both a canonical representative of a set of sequent calculus proofs and a computational object in itself (with a ....

.... appendix B: About injectivity for relational semantics. In chapter 1, we address the question of injectivity in a precise way, and give a positive answer in the multiplicative case. We start in section 1. 1, by generalizing to multiplicative and exponential LL the notion of experiment of [Gir87]: an experiment associates with every edge a of a proof net a multiset of elements of the web of the coherent space interpreting the type of a (the formula associated with the edge a) The result of an experiment of a proof net R, is the set of labels associated by the experiment with the ....

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Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1-102, 1987.

.... by jX 1 Phi X 2 j = f1g Theta jX 1 j) f2g Theta jX 2 j) and (f1g Theta u 1 ) f2g Theta u 2 ) 2 Gamma(X 1 Phi X 2 ) if u 2 = and u 1 2 Gamma(X 1 ) or u 1 = and u 2 2 Gamma(X 2 ) 3 The poset so defined belongs to the class of qualitative domains introduced by Girard in [Gir86]. Qualitative domains can equivalently be considered as dI domains where all prime elements are atomic. 5 It is the De Morgan dual of with. Using the same notations as in the description above of the n ary version of the with, u is coherent in X 1 Phi Delta Delta Delta Phi X n iff u is ....

Jean-Yves Girard. The system F of variable types, fifteen years later. Theoretical Computer Science, 45:159--192, 1986.

....of temporal linear logic and applications to computer science. At the end of this chapter, the organization of the thesis is described. Background In order to express dynamic change in process environment, it is useful to consider a concept of resource such as data consumption. Linear logic (LL) [6] introduced by Girard in 1987 has been called a resource conscious logic. The expressive power is so rich that one can construct a counter machine within the propositional fragment of linear logic. Some computational models of concurrency are applications of LL [26, 2, 20] In particular, the ....

....Furthermore, linear logic is useful as a formal logical system. The cut elimination theorem holds in LL. The theorem plays an important part in logic programming, uniform proof and proof search, and so on. The full propositional fragment of LL has a complete semantics in terms of phase spaces [6]. We think that linear logic has been given various applications in computer science through its resourceconsciousness and usefulness as a formal system. The Motivation and Aim of this Study Linear logic can represent a dynamically changing environment. However, it is not enough to treat dynamic ....

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Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987.

....detailed summary of the theory. This is the object of the unusually long preliminary section 1. This paper requires from the reader a reasonable knowledge of the denotational semantics and of the phase space semantics of linear logic. Many good texts on these topics are available, see for instance [Gir87, Gir95, AC98]. 1 A summary of indexed linear logic In this section, we recall the main ideas, de nitions and properties of indexed linear logic and symmetric product phase spaces. The material summarized here is presented in full details in [BE99] 1.1 Notations We rst x some terminology and notations. If ....

....A of LL(I) such that A = R and hAi = and such that the sequent J A is provable in LL(I) For each formula A of LL(I) such that A = R and hAi = the sequent J A is provable in LL(I) 1. 4 Symmetric product phase spaces Remember that phase spaces have been introduced by Girard in [Gir87] for giving a truth value semantics to linear logic. As this is standard material, we just recall here the basic de nitions and a few easy properties that we use implicitly in the sequel. A phase space is a pair (Q; where Q is a commutative monoid and is a subset of Q on which no special ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1-102, 1987.

....space X has as web the set of all finite cliques of X, and the linear function space of two coherence spaces X 0 and Y has as web the cartesian product of the webs of X 0 and Y . These two operations have logical counterparts which are made explicit as logical connectives in linear logic ([Gir87, GLT89, Gir95] describe the coherence space semantics of linear logic) Van de Wiele observed that alternative definitions of the exponential operation on coherence spaces are available. More specifically, from a categorical viewpoint, the exponential is an endofunctor on the category of coherence spaces and ....

.... it is a cartesian closed category, there is a canonical notion of application of to a clique x of X, which yields the following clique of Y : x) fb = 9 0 2 j m Xj ( 0 ; b) 2 and j 0 j xg: For more information on the coherence spaces denotational semantics of linear logic, we refer to [Gir87, GLT89, Gir95]. We call N the discrete coherence space that has, as points of its web, the natural numbers. Observe that, since N is discrete, its cliques are the singletons and the empty set. 3 The full hierarchy of simples types is defined inductively by oe : j oe oe, where is the type of natural ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987.

....functional programming languages like PCF [Ber78] In the framework of dilators (functors acting on ordinals) Girard discovered independently stability as a condition allowing for a finitary representation of these functors. He applied the same idea to the denotational semantics of system F (see [Gir86]) and this led him to the crucial observation that this semantics (which is an extension of Berry s semantics of PCF) can be described in the very simple framework of coherence spaces. Berry actually developped his semantics in the framework of dI domains (Scott domains satisfying some further ....

Jean-Yves Girard. The system F of variable types, fifteen years later. Theoretical Computer Science, 45, 1986.

....an operational perspective of our result. We give a sequential execution model of Linear CHAM based on Abramsky s idea of a stack of coequations and a name queue, and then we consider a concurrent multi thread implementation of the model. 1 Introduction Girard s Classical Linear Logic (CLL) [6] is expected to give new theoretical foundations of parallel computation [7] Abramsky [1] gave a computational interpretation of CLL using the framework of Berry and Boudol s Chemical Abstract Machine (CHAM) 3] In this computational system which is called Linear Chemical Abstract Machine ....

.... Determinacy: If P Q and P R, then Q j R. Convergence: For every typable proof expression P , there is a proof expression Q such that P Q. 4 Translation 4. 1 Translation of 0 terms into proof expressions Formulas in IL are translated into formulas in CLL by Girard s translation [6] as follows: A ffi = A when A is atom, A oe B) ffi = A ffi P B ffi ) This translation is extended to the proof level, i.e. translation of a proof of Gamma A into a proof of Gamma ffi ; A ffi . Using this idea, we define a translation ( Gamma) ffi (x x) ffi = ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987.

....proof theoretic formulation of the logic of the TLLP language. We then present a series of resource management systems designed to implement not only interpreters but also compilers based on an extension of the standard WAM model. 1 Introduction Linear logic was introduced by J. Y. Girard in 1987 [4] as a resource conscious refinement of classical logic. Since then a number of logic programming languages 2 M. Banbara, K. Kang, T. Hirai, and N. Tamura based on linear logic have been proposed: LO[1] ACL[12] Lolli[3] 8] 9] Lygon[5] Forum[13] and LLP[2] 15] These languages suggest a ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987.

....of explicit memory management operations for individual objects. These type systems provide information about the last use of a data structure, and clearly, if we are guaranteed that a data structure has been used for the last time, we can safely deallocate it. The simplest linear type systems [19, 1] actually guarantee that linear data structures are used exactly once. After this one use, the data structure is a deallocated. More sophisticated type systems [30, 6, 18, 15] make it possible to use linear objects several times, but still provide support for detecting the last use of such ....

....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 [11] and Reynolds syntactic control of interference [24] Linear type systems were later studied by many researchers [19, 30, 1, 3, 6, 29, 34, 15]. Type and e ect systems were introduced by Gi ord and Lucassen [10] and they too have been explored by many others [17, 26, 28, 21] More recently, a number of new linear type systems, or more generally, substructural type theories, have been developed such as Kobayashi s quasi linear types ....

Yves Lafont. The linear abstract machine. Theoretical Computer Science, 59:157-180, 1988.

....or to redistribute to lists, requires prior specific permission and or a fee. ICFP 01, September 3 5, 2001, Florence, Italy. Copyright 2001 ACM 1 58113 415 0 01 0009 . 5.00. this problem have repeatedly found success: Linear type systems, which have been derived from Girard s linear logic [11] and Reynolds syntactic control of interference [24] and The type, region and e ect discipline developed by Gifford and Lucassen [10] and re ned by Jouvelot, Talpin and Tofte [17, 26, 28] Despite the individual successes of these techniques, there has been little research that attempts to ....

....then a reference counting solution could be used: 9 : # rgn( 1 ( list at ) 1 list 6. RELATED AND 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 [11] and Reynolds syntactic control of interference [24] Linear type systems were later studied by many researchers [19, 30, 1, 3, 6, 29, 34, 15] Type and e ect systems were introduced by Gi ord and Lucassen [10] and they too have been explored by many others [17, 26, 28, 21] More recently, a ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1-102, 1987.

....to identify the unique path which survives reduction, and on the other hand, to calculate algebraically its weight, which is invariant throughout reduction, and equal, in fact, to the value of the term. The geometry of interaction has been developed as a semantics for linear logic proof nets [4]. Combined with a standard translation of the calculus into these nets, the results may then be lifted to the scope of functional programs. The nodes in the graph of each term are logical symbols with premises and Research done whilst staying at Laboratoire d Informatique (CNRS UMR 7650) ....

Jean-Yves Girard. Linear Logic. Theoretical Computer Science, 50(1):1-102, 1987.

....a modality for P . One particular example of this would be that of phase spaces: Recall that a phase space M consists of a commutative monoid and a subset of M . For 22 subsets X of M , negation is de ned via X : fm 2 M j 8n 2 X:mn 2 g: In that case let P : fX M j X = X g Girard [6] calls these sets facts . They form a complete lattice with respect to since facts are closed under arbitrary intersection. The tensor is given via X Y : fmn j m 2 X; n 2 Y g . We will not repeat here how the other connectives are de ned. It seems, however, interesting to point out how ....

....value 1 exactly on the diagonal. As some minor calculations show, the modalities we obtain for 3 sets if we take to be the nite powerset functor on Rel are similar to the ones described in [5] If we use the nite multiset functor instead we obtain linear exponentials similar to the ones in [6]. However, the construction we introduced in the last section has one major di erence: The underlying set for (A; is always A. In other words cannot be used to determine a subset of A as the underlying set instead, the way it is done in the usual version of the modalities for coherence ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1-102, 1987.

....we de ne a cut elimination procedure which associates with every proof net of multiplicative and additive linear logic a unique cut free one. MSC: 03F05; 03F07; 03F52 Keywords: Linear Logic, Cut Elimination, Proof Nets, Additives, Multiboxes. 1 Introduction Linear Logic (LL) introduced in [Gir87], is a re nement of classical and intuitionistic logic: the standard connectives ( and and or ) are split in two classes (additive and multiplicative) and the exponentials (the new connectives of LL) give a logical status to the structural rules of classical and intuitionistic sequent ....

.... of view, it is dicult to imagine a greater gap than the one separating the multiplicative from the additive world: we have on the one hand the perfect theory of multiplicative proof nets, developped from all angles, with (almost) half a dozen of correctness criterions (from the longtrip one of [Gir87] to the homological one of [M et94] and on the other hand the lack of a satisfying theory of additive proof nets, and a very sparse litterature on the subject. To deal with additive proof nets is so dicult, that the additives are usually simply ignored, and when this is impossible, they are ....

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Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1-102, 1987.

.... model of first order propositional linear logic, a linear category (see [Bie95] whose objects are the finiteness spaces, and we also extend this semantics to the second order, using new ideas presented in [Bac00] for extending to non uniform settings the constructions that Girard performed in [Gir86] for qualitative domains and coherence spaces. 1 Or successively; the multiset can be interpreted (e.g. in the forthcoming example) as giving the different values taken by the argument during the computation. In the non deterministic game model of [DH00] the same information is made available, ....

....n 1 ) F (X 1 ; X n ) and similarly for morphisms) admits a right adjoint 8 : Fin[n 1] Fin[n] Since an adjoint is unique when it exists, there is no choice in the definition of this operation 8. Let F 2 Fin[1] be a stable functor. We associate to this functor a trace TrF , like in [Gir86]. For this purpose, we define first a set kFk: its elements are all the pairs (n; a) where n 2 N and a 2 jF (n)j, such that, if V n and a 2 jF (V )j then V = n (where n = f0; n Gamma 1g) Remember that a finite set has only one possible structure of finiteness space, so that this ....

Jean-Yves Girard. The system F of variable types, fifteen years later. Theoretical Computer Science, 45:159--192, 1986.

....other hand, denotational semantics identi es too many proofs. The seek of an object sticking as much as possible to the computational nature of proofs led to the introduction of a new syntax for logic: proof nets, a graph theoretic presentation which gives a more geometric account of proofs (see [5]) This discovery was achieved by a sharp (syntactical and semantical) analysis of the cut elimination procedure. Any person with a little knowledge of the multiplicative framework of LL, has no doubt that proofnets are the canonical representation of proofs. But as soon as one moves from such a ....

.... less pure . A reasonable solution for the multiplicative and exponential fragment of LL (with quanti ers) does exist (combining [2] and [7] like in [16] but trying to add the additive connectives means entering a true jungle. Of course, it is possible to survive in this jungle: it is shown in [5] how to compute with the additives, and in [16] an even (apparently much) more savage way of computing is de ned and shown to yield a unique cut free object in some important cases. So what The problem is that the objects (the proof nets) used are de nitely not canonical. Some better ( more ....

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Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1-102, 1987.

....in di erent nodes of its graph, and then allowing the di erent processing threads to communicate so that the weight of a nished segment can be used to calculate the weight of another (longer) segment. The geometry of interaction has been developed as a semantics for linear logic proof nets [4]. Combined with a standard translation of the calculus into these nets, the results may then be lifted to the scope of functional programs. The nodes in the graph are logical symbols with premises and conclusions, and each orientated edge links a conclusion of a node to a premise of another ....

Jean-Yves Girard. Linear Logic. Theoretical Computer Science, 50(1):1-102, 1987.

....from [AFV96] is to avoid the use of trees fl such that: 9p 2 D fl #fl(p) fl(p Delta 1) and p Delta 2 62 D fl It means there is no tree that have an X labeled node whose unique leaf is also an X labeled node. 1. 2 Lexicalized Proof Nets Proof nets in linear logic have become familiar [Gir87, Ret93, Abr95]. In this paper, we refer to [Ret96] s notations of proof nets, extended to the ordered calculus [Ret97] It defines proof nets as bicolored (Red and Blue, or Regular and Bold) graphs with the five links corresponding to the axiom, the tensor ( Omega ) the before ( the par (P) and Table 1. ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987.

....to bids rather than bidders. We note that other purely logical means for distinguishing sharable and nonsharable resources may be possible, rather than relying on whether a multiple good occurrences lie above the bid level or below it. For instance, resourceoriented logics (e.g. linear logic [Girard, 1987]) are designed primarily to deal with the issue of resource consumption and sharing. The connections to this work seem worthy of deeper exploration. 2 When a bidder offers multiple GLBs, we must enforce substitutability constraints by using dummy goods. This is not necessary when the bid is ....

....deserves further exploration. Clearly an important concept for CAs, L GB s ability to make this distinction implicitly is very desirable for the natural, concise expression of preferences. Finally, we are currently pursuing the connection to work in resource oriented logics (e.g. linear logic [Girard, 1987] ) though existing logics do not seem to able to handle complementarities. ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987.

....multisets since there could be di erent clauses C i and C j which result in the same axiom T (C i ) T (C j ) To keep 2 the presentation simple we do not do this. However, there are well known formal frameworks dealing with multisets, for instance substructural logics or linear logic, SHD93, Gir87] The concepts de ned in this paper can be easily worked out for these frameworks, too. Let (P ) be a property which is provable in A[T (P ) When we change P into P 0 by replacing the clause C i by the clause C 0 i , we can ask whether (P 0 ) still holds in A [ T (P 0 ) But, if T (C ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1-102, 1987.

....##) in letlin k2 = #k. z : 2; k ##) in if x then letlin k = ###. k1 k0) in k2 k else letlin k = ###. k2 k0) in k1 k Fig. 1. Examples of Information Flow in CPS k is not used in one of the branches, then information about x can be learned by observing z. Linear type systems [2, 13, 33, 34] can express exactly the constraint that k is used in both branches. By making k s linearity explicit, the type system can use the additional information to recover the precision of the source program analysis. Fragment D illustrates our simple approach; in addition to a normal let construct, we ....

....of the source term. This result also shows that the CPS language is at least as precise as the source. Lemma 6 (Type Translation) # # # e : s # # # y : s [#] # [ # # # e : s] y. 6 Related Work The constraints imposed by linearity can be seen as a form of resource management [13], in this case limiting the set of possible future computations. Linear continuations have been studied in terms of their category theoretic semantics [11] and also as a computational interpretation of classical logic [5] Polakow and Pfenning have investigated the connections between ordered ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987.

....a theorem prover into our language in order to decide the validity of the logical formulae. 6. 2 Related Work Our type system builds upon foundational work by other groups on syntactic control of interference [31] linear logic [13] and linear type systems in functional programming languages [20, 42, 1, 15, 3, 8, 40]. Our research also has much in common with efforts to define program logics for reasoning about aliasing [6, 9, 26, 32, 17] In particular, if we view propositions as types, there are striking similarities with recent work by Reynolds [32] who builds on earlier research by Burstall [6] ....

Yves Lafont. The linear abstract machine. Theoretical Computer Science, 59:157--180, 1988.

....The disadvantage of this approach is that we would have to integrate a theorem prover into our language in order to decide the validity of the logical formulae. 6. 2 Related Work Our type system builds upon foundational work by other groups on syntactic control of interference [31] linear logic [13] and linear type systems in functional programming languages [20, 42, 1, 15, 3, 8, 40] Our research also has much in common with efforts to define program logics for reasoning about aliasing [6, 9, 26, 32, 17] In particular, if we view propositions as types, there are striking similarities with ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987.

....to stateful programming. These sections are primarily interested in the static semantics of LTT and treat its operational semantics only informally. We then present the framework for modular development of operational semantics in Section 6. This paper assumes familiarity with linear logic [6, 18] and with the propositions as types correspondence [9] Additional familiarity with LF and logical frameworks in general will be helpful, but is not required. 2 Intuitionistic LTT The LTT type theory consists of two parts: a proof sub language, and a computational programming language built ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1-102, 1987.

....[AJP94] are the originators of the specific form of games described in this report. Other types of games have also been proposed by Blass [Bla92] and Hyland and Ong [HO94, HO95] Games naturally incorporate a notion of linearity and there is a strong correspondence with Girard s linear logic [Gir87]. The syntax chosen for the calculus associated with linear logic used in this report has been particularly influenced by the work of Wadler [Wad93] and Bierman [BBdPH92] The standard categorical semantics for linear logic was developed originally by Seely [See89] however, here we only make use ....

....identity functor on the category C, and curry is a currying morphism, and eval is an evaluation morphism. Given a morphism, oe : A Omega B Gammaffi C, then using the currying morphism, we have that curry(oe) A (B Gammaffi C) 6 Linear Logic Linear logic was originally developed by Girard [Gir87] and attempts to be sensitive to resources. The structural rules of contraction and weakening are discarded in linear logic, requiring that assumptions are used exactly once in proofs. Formulas thus correspond to finite resources leading to the view of linear logic as a logic of resources. ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987.

....we refer to [75, 43, 51, 61, 23, 89] In the literature there exists a variety of definitions of term graphs. Besides hypergraphs, directed graphs, terms with labels, and recursion equations have been used as underlying structures. Acyclic graphs have been dealt with in [34, 95, 96, 97] while [83, 92, 59, 15, 37, 63, 32] also consider cyclic graphs. By equipping function symbols with additional labels, sharing of different occurrences of a subterm in a term can be expressed through identical labels. Such labelled terms correspond to acyclic term graphs and have been studied in [76, 74, 82] In [36, 4, 2, 67] ....

Jean-Claude Raoult. On graph rewritings. Theoretical Computer Science, 32:1--24, 1984.

.... [29] and Prolog [28] These encodings have been successfully used in the specification of a wide range of computations, including the evaluation of functional programming languages [4, 15, 27] abstract machines [16] and process calculi [26] Various recent papers suggest that linear logic [11] can be used as well to make this style of specification more expressive. For example, specifications of imperative and concurrent programming language features [5, 6, 9, 23, 24] and the sequential and concurrent (pipe line) semantics of a RISC processor [6] have been modeled using linear logic. ....

....be introduced when needed. We shall generally treat types implicitly, but shall include types in examples to help make the specifications more readable. We shall assume that the reader is familiar with the usual two sided sequent calculus presentation of intuitionistic logic [10] and linear logic [11]. In this paper, we consider sequents of the form Delta Gamma B, where B is a formula and Delta is either a multiset or a set of formulas, depending on whether we are working in linear or intuitionistic logic. We shall also assume that the reader is familiar with substitutions and their basic ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987.

....by various translation from linear logic to interaction nets. It can be seen as an intermediate translation giving a theoretical justification for those coding. Another interesting aspect of second order exponentials is their similarities to combinatorial exponentials. 1 Introduction Linear logic [Gir87], which may be seen as a logic that takes resources into account, has a sequent calculus presentation. Another presentation consists in using proof structures and proof nets composed of connectives and boxes linked together to form nets. The aim of this notion is to identify certain not ....

....two parts, one managing the context and the other introducing the connective. This leads to formula transformations and proof transformations that preserve the elimination of main cuts. However, like CPS transformations, this mechanism eliminates non essential cuts known as commutative conversions [Gir87, GLT88]. A benefit of this strategy is to force cut elimination to be ChurchRosser. In fact there are two motivations for those transformations. The first one is that this method inhibits commutative conversion. As a consequence, this strategy can be compared to codings of linear logic into interaction ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987.

....imperative operations such as update and deallocation of memory. Over the past 15 years, three techniques for solving this problem have repeatedly found success, particularly for the domain of functional programming languages: 1. Girard s linear logic [13] and related work on linear type systems [19, 1, 37] and syntactic control of interference [29] control sharing and or the number of uses of important computer resources such as memory. These systems make it possible to deallocate and reuse storage safely. 2. Moggi s computational lambda calculus [23] separates pure values from e ectfull ....

....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 ....

Yves Lafont. The linear abstract machine. Theoretical Computer Science, 59:157-180, 1988.

....the behavior of their code in the presence of imperative operations such as update and deallocation of memory. Over the past 15 years, three techniques for solving this problem have repeatedly found success, particularly for the domain of functional programming languages: 1. Girard s linear logic [13] and related work on linear type systems [19, 1, 37] and syntactic control of interference [29] control sharing and or the number of uses of important computer resources such as memory. These systems make it possible to deallocate and reuse storage safely. 2. Moggi s computational lambda calculus ....

....type systems that, to a rst approximation, might be called linear. Although the di erences may appear small they can result in signi cantly di erent memory management properties. True linear type systems, those type systems pulled along the Curry Howard isomorphism from Girard s linear logic [13], such as Abramsky s intuitionistic linear type system [1] contain a collection of multiplicatives, including 1 ( 2 , a function type that requires its argument to be used exactly once, and 1 2 , a pair in which each component is used exactly once. In order to retain the expressiveness of ....

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Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1-102, 1987.

.... and Bar Hillel, we obtain a kind of propositional calculus with a directionally sensitive implication, where the structural rules (permutation, contraction, thinning) are suspended [49] B;A] permute [A;B] thin ; A [A;A] contract [A] And in linear logic [30, 31], substructural logics lacking the structural rules have received systematic study. In these logics, the constituents of an expression act like resources: they may be consumed and produced in the course of a proof. This is easy to explain by analogy with automata or Petri nets, as explained in ....

....them with modalities and used techniques from modal logic [93, 95, 62] and also [64] with a slightly different approach. Nevertheless 4 the relation to other logics, in particular with intuitionistic logic and classical logic has only been clarified with the invention of linear logic by Girard [30]: the full power of intuitionistic logic is recovered by modalities which restore (and control) the so called structural rules of weakening and contraction. The embedding into intuitionistic logic is especially important since that is the very reason for the simple interface with Montague ....

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Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50(1):1--102, 1987.

....Nancy, France, September 1997. Springer Verlag LNAI 1582 Grammars For Unbounded Dependencies Joshua S. Hodas Computer Science Department Harvey Mudd College Claremont, CA, 91711 hodas cs.hmc.edu Abstract. A number of researchers have proposed applications of Girard s Linear Logic [7] to computational linguistics. Most have focused primarily on the connection between linear logic and categorial grammars. In this work we show how linear logic can be used to provide an attractive encoding of phrase structure grammars for parsing structures involving unbounded dependencies. ....

....Phrase Structure Grammars [4, 5] As part of the presentation we show how a variety of issues, such as island and coordination constraints can be dealt with in this model. 1 Introduction Over the last several years a number of researchers have proposed applications of Girard s Linear Logic [7] to computational linguistics. On the semantics side, Dalrymple, Lamping, Pereira, and Saraswat have shown how deduction in linear logic can be used to enforce various constraints during the construction of semantic terms [1, 2, 3] On the syntax side, Hepple, Johnson, Moortgat, Morrill, Oehrle, ....

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Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987.

....called as t protocol[5] CR property is recovered by adding some restrictions on logical rules to LK. Despite of these restrictions, soundness and completeness w.r.t. classical provability is still retained. More importantly one can map a derivation of LKT to a derivation of Girard s Linear Logic[9]. In fact, this mapping called linear decoration is the key to prove SN and CR property of LKT. Linear decoration establishes the mapping between LKT and MELL, not only for the formulas, sequents and derivations but also the cut elimination, hence the isomorphism. This is also called as stringo ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1--102, 1987.

....informal reasoning 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 ....

Yves Lafont. The linear abstract machine. Theoretical Computer Science, 59:157--180, 1988. 31

....ne a modality for P . One particular example of this would be that of phase spaces: Recall that a phase space M consists of a commutative monoid and a subset of M . For 21 subsets X of M , negation is de ned via X : fm 2 M j 8n 2 X:mn 2 g: In that case, let P : fX M j X = X g in[5], Girard calls these sets facts . They form a complete lattice with respect to since facts are closed under arbitrary intersection. The tensor is given via X Y : fmn j m 2 X; n 2 Y g . We will not repeat here how the other connectives are de ned. It seems, however, interesting to point ....

....of the embedding G consists of all 3 sets whose structure map takes the value 1 exactly on the diagonal. As some minor calculations show, the modalities we obtain for 3 sets if we take to be the nite powerset functor on Rel correspond to the ones described in [4] To obtain the ones from [5] we have to adjust our de nition of , for the construction we introduced in the last section has one major fault: The underlying set for (A; is always A, ie cannot be used to determine a subset of A to use instead, the way it is done in the usual version of the modalities for coherence ....

Jean-Yves Girard. Linear logic. Theoretical Computer Science, 50:1-102, 1987.

....Birmingham, School of Computer Science. URL: http: www.cs.bham.ac.uk exr .Research supported under the EPSRC project no. GR L28296, x SLAM: The Explicit Substitutions Linear Abstract Machine. 1 particular, linearity has been suggested as a possible solution since the beginning of linear logic [Laf88]: the idea was that if a variable is linear, it is used once, hence it has only one pointer to it, hence update in place is possible. Unfortunately this approach does not work, as the array demonstrates: several read operations are possible if the update operation is executed after the last ....

Yves Lafont. The linear abstract machine. Theoretical Computer Science, 59:157-180, 1988.

....(such as JACK [1] or dMARS) whilst others are an abstract specification of necessary constructs such as Agent0 [4] In order to develop and reason about such programs, an ability to take into account concurrency and dynamic state is required. Linear logic, introduced by Girard in 1987 [2] has such properties, and has been successfully applied to modelling updates, reasoning about the environment, and implementing concurrent behaviour. Linear logic is often described as resource sensitive, and has been the basis for a number of programming languages including Lygon [3] Lolli, ....

Jean-Yves Girard. Linear Logic. Theoretical Computer Science, 50:1--102, 1987.

No context found.

Yves Lafont. The Linear Abstract Machine. Theoretical Computer Science, 59:157--180, 1988.

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Yves Lafont. The linear abstract machine. Theoretical Computer Science, 59(1,2):157 { 180, 1988.

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Jean-Claude Raoult. On graph rewritings. Theoretical Computer Science, 32:1-- 24, 1984.

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