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.

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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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