L. Maranget. Optimal Derivations in Weak Lambda-calculi and in Orthogonal Term Rewriting Systems. In Conference Record of the 18th Annual ACM Symposium on Principles of Programming Languages, POPL'90, pages 255-269, ACM Press, 1990.  Home/Search   Document Details and Download   Summary   Related Articles   Check

.... ) the value of any expression of the sort Nat without entering into infinite computations. For instance, it is possible to evaluate sel(s(0) from(0) to s(0) i.e. s(0) 2 eval (sel(s(0) from(0) Similar kinds of annotations have been utilized in term (graph) rewriting [FKW00, KW95, Mar90, Ngu01, Pol01] They have mainly been used to define restrictions of rewriting that permit the implementation of lazy reductions via eager rewritings in a transformed TRS [FKW00, KW95, Ngu01] In [Luc98a, Luc01b, Luc02a] we have analyzed how these proposals relate to CSR. 11.3 Context sensitive ....

L. Maranget. Optimal Derivations in Weak Lambda-calculi and in Orthogonal Term Rewriting Systems. In Conference Record of the 18th Annual ACM Symposium on Principles of Programming Languages, POPL'90, pages 255-269, ACM Press, 1990.

....making it possible to express the inherent complexity as the number of fixed point unfoldings needed however, some operational reasoning nevertheless (as to be expected) sneaks into the theory. In section 4.10 we will sketch how to extend our model to encompass inherent complexity. Mar91] investigates a class of term rewriting systems where terms are labeled thereby implicitly defining DAGS but not cyclic graphs in general. By focusing upon the derivations where all redices with same label are reduced simultaneously, graph reduction is mimiced. Now, loosely speaking, a ....

Luc Maranget. Optimal derivations in weak lambda-calculi and in orthogonal terms rewriting systems. In ACM Symposium on Principles of Programming Languages, pages 255-- 269, 1991.

....f calculus which is a variant of calculus with emphasis on sharing. One important feature of this formalism is to perform fi reduction with shared environments instead of instantiations of terms. Several other formalisms which explicitly manipulate environments have been proposed in [1] 3][8]. They all use de Bruijn notation and copy environments. Our construction is based on weak calculus (which does not evaluate under s) and executes reductions without copying applications and environments in a clean and tractable formulation. The framework of computing is quite different from ....

....environments in body. 5 Conclusion We presented a new formal system called f calculus which stresses sharing as the most essential element and studied the basic properties as a functional calculus. The optimal reduction defined by L evy[7] is closely related to sharing, and two formalisms[3][8] using environments have been studied from the viewpoint of optimality in L evy s sense. However 3CCL[3] which executes full rules based on categorical combinators is insufficient for optimality. T l RS[8] whose reduction system is weak as well as ours, regards a parallel reduction as one step ....

[Article contains additional citation context not shown here]

L. Maranget. Optimal derivations in weak lambdacalculi and in orthogonal terms rewriting systems. POPL, 1991. 4

....sense were responsible for its creation Trying to capture how intermediate and final terms originate from the initial term is formalized in a notion called origin tracking [4, 5, 10] Origin tracking is based on so called residuals . Residuals have been used successfully in more theoretical work [15, 21, 23], for reasoning about optimal reduction strategies in TRSs. Figure 1: Example of a generated environment using origin tracking. 1.1 Applications Our motivation for working on origin tracking is its applicability to the automatic generation of tools from algebraic specifications of programming ....

....Secondary origins are represented by marking functions . This work was done in the framework of the Centaur system [6] In particular, the specification language Typol [16] has been extended with subject tracking [8] Closely related to origins are residual maps , descendants , or labelings [20, 15, 21, 13], which are used to study reduction strategies. Residuals indicate which redexes survive if a particular redex is contracted. One can think of this as giving interesting parts in the initial term a particular color, and then looking how this color survives during reduction. An interesting ....

L. Maranget. Optimal derivations in weak lambda-calculi and in orthogonal term rewriting systems. In Proceedings of the Eighteenth conference on Principles of Programming Languages POPL '91, pages 225--269, 1991.

....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] systems of recursion equations realize finite and infinite terms with sharing. As to the complexity of collapsing, arbitrary term graphs can be made fully collapsed in time O(n log n) where n is the size of term graphs. This bound reduces to O(n) for term graphs over finite ....

....orthogonal term rewriting systems. 53 9 Further Topics We briefly mention some topics in term graph rewriting that have not been discussed in the preceding sections. Optimality of reduction strategies in the sense of finding a normal form in a minimal number of steps is investigated in [95, 96, 97, 76]. Essential for these considerations is the subcommutativity of plain term graph rewriting over orthogonal systems. There are a few papers describing implementations of term graph rewriting. Socalled concurrent term rewriting is addressed in [40, 65] while [60, 39, 13] deal with the term graph ....

Luc Maranget. Optimal derivations in weak lambda calculi and in orthogonal term rewriting systems. In Proc. Annual ACM Symposium on Principles of Programming Languages, pages 255--269. ACM Press, 1991.

....(1) the concept of need as the foundation of laziness, 2) strategies for using narrowing in programming, and (3) implementations of narrowing in Prolog. 6. 1 Narrowing and need Seminal studies on the concept of need in rewriting appear in [24, 39] Subsequent variations and extensions, e.g. [7, 21, 27, 30, 33, 40, 41, 45, 48], do not address narrowing, but limit the discussion to rewriting. We have introduced a concept of need for narrowing that extends a similar concept for rewriting. We have shown that the concept of need for narrowing is inherently more complicated than that for rewriting. In orthogonal systems, a ....

L. Maranget. Optimal derivation in weak lambdacalculi and in orthogonal terms rewriting systems. In 17th Annual Symp. on Principles of Prog. Languages, pages 255--269. ACM, 1990.

....with non well founded relations which prevent proofs by induction. Observe that graphs di er from trees in that the latter naturally support de nition and proof by structural induction) In this paper we will annotate terms (trees) with global addresses [FF89, Ros96] Levy [L ev80] and Maranget [Mar92] propose using local addresses, but from the point of view of the operational semantics, global addresses describe better what is going in a computer or an abstract machine. With explicit global addresses we can keep track of the sharing that could be used in the implementation of a particular ....

L. Maranget. Optimal Derivations in Weak Lambda Calculi and in Orthogonal Rewriting Systems. In Principles of Programming Languages, pages 255-268, 1992.

....(1) the concept of need as the foundation of laziness, 2) strategies for using narrowing in programming, and (3) implementations of narrowing in Prolog. 6. 1 Narrowing and need Seminal studies on the concept of need in rewriting appear in [24, 39] Subsequent variations and extensions, e.g. [7, 21, 27, 30, 33, 40, 41, 45, 48], do not address narrowing, but limit the discussion to rewriting. We have introduced a concept of need for narrowing that extends a similar concept for rewriting. We have shown that the concept of need for narrowing is inherently more complicated than that for rewriting. In orthogonal systems, a ....

L. Maranget. Optimal derivation in weak lambdacalculi and in orthogonal terms rewriting systems. In 17th Annual Symp. on Principles of Prog. Languages, pages 255-269. ACM, 1990.

....(Weak) M;N : M fi fi MN fi fi n (Pure) s : W Delta s fi fi id (Substitution) Beta reduction. M ) s] W M [W Delta s] Bw ) Substitution elimination. MN ) s] M [s] N [s] App) 0 [W Delta s] W (FVar) n 1 [W Delta s] n [s] RVar) n [id] n (VarId) Figure 1: oe w . 10 Maranget (1991), 2 however, our notion of address is more abstract and allows us a better comprehension of implementations of machines for functional programming languages and their optimizations. 3.1 Explicit substitution and sharing Figure 3 presents the syntax and reduction rules of oe a w (it will be ....

Maranget, L. (1991), Optimal derivations in weak lambda calculi and in orthogonal rewriting systems, in `18th Principles of Programming Languages ', pp. 255--268.

....that must be eventually reduced to compute the normal form of t. Furthermore, if each needed redex is eventually reduced, the computation of a term with a normal form terminates. The strategy implicitly implemented by our approach reduces only needed redexes. Under certain additional conditions [34, 53], this strategy yields a reduction sequence of minimum length. Our approach to lazy rewriting improves several aspects of the indirect techniques references above. In particular, we remove certain common restrictions of the indirect approaches, such as the flattening of functional nestings, which ....

....reduction sequence may be obtained from a reduction by sharing the common subterms of a term. An early proposal in this direction is found in [53] A comprehensive discussion within the general framework of labelled term rewriting system and several references on the subject are found in [34]. Implementation techniques suitable for logic programming are found in [38, 39, 40, 41] and can be applied to our approach as well. Basically, they avoid the repeated reduction of two equal redexes by binding them together by means of a logic variable. Although in some cases this feature improves ....

Luc Maranget. Optimal derivation in weak lambda-calculi and in orthogonal terms rewriting systems. In 17th Annual Symp. on Principles of Prog. Languages, pages 255--269. ACM, 1990.

....we do not lose the possibility of reducing root needed redexes. Theorem 4.4 Let R be an orthogonal TRS, be such that com R v and t be a non root stable term. Then, minimal(O R (t) is a root necessary set of redexes. This result improves the use of replacement restrictions by Maranget [11] to achieve optimal derivations by using parallel rewriting in a graph reduction framework. Due to the need to work in a graph reduction framework, Maranget restricts himself to a smaller class of orthogonal TRSs, for which V ar (r) V ar (l) and g V ar (r) g V ar (l) for all rules l r in ....

L. Maranget. Optimal Derivations in Weak Lambda-calculi and in Orthogonal Term Rewriting Systems In Conference Record of the 18th ACM Symposium on Principles of Programming Languages, pages 255-269, ACM Press, 1990.

....the concept of a redex family. The labelling is a modification of one used by Klop [29] for CRSs. This labelling system applies to all orthogonal ERSs (with one very minor restriction) Readers interested only in OTRSs and # calculusmay skip this section, as labelling systems due to Maranget [34] and Levy [30, 31] su#ce for these systems respectively. The concepts from this section that we use later are those of the label and the index of a redex, lab(u) and Ind(u) Fix some OERS R. For technical reasons, we assume that R does not contain any rules whose left hand side consists of just ....

....in C replace addresses in right hand sides of rules. In the case of Klop s labelling system [29] all constructors are empty . Redex labels occurring as the arguments in right hand sides of rewrite rules correspond to (constrained) labelled patterns in Maranget s labelling system for TRSs [34]. And in his labelling system for HRSs, van Oostrom [45] uses numbers where we use constructors in C, and he uses rules where we use label constructors in C # . However, our definitions are su#ciently close that Klop s results for labelled systems carry over to the present setting. The ....

L. Maranget. Optimal derivations in weak lambda-calculi and in orthogonal term rewriting systems. In 17 th ACM Symposium on Principles of Programming Languages, POPL'91, pages 255--269. ACM, 1991.

....two very different directions. On the formal side, it is important to automatize the manipulations we have described in this paper and to give a precise meaning to the computed origin functions. We can also try to address more complicated labelling schemes, such as the ones used in [Optimal] or [Weak] Such labelling schemes also have practical applications, as they permit to describe the history of parts of the execution state. On the practical side, the full range of applications for occurrence computations has yet to be explored, and real size debuggers, tracers, or profilers have to be ....

L. Maranget, "Optimal Derivations in Weak Lambda-Calculi and in Orthogonal Terms Rewriting Systems.", Proceeding of the eightteenth ACM Symposium on Principles of Programming Languages, Orlando, Florida, January 1991, pp. 255--269.

....while [85,94,60,15,37,64, 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 [77,75,83,84]. In [36,4,2,68] systems of recursion equations realize finite and infinite terms with sharing. As to the complexity of collapsing, arbitrary term graphs can be made fully collapsed in time O(n log n) where n is the size of term graphs. This bound reduces to O(n) for term graphs over finite ....

....orthogonal term rewriting systems. 1.9 Further Topics We briefly mention some topics in term graph rewriting that have not been discussed in the preceding sections. Optimality of reduction strategies in the sense of finding a normal form in a minimal number of steps is investigated in [97,98,99,77]. Essential for these considerations is the subcommutativity of plain term graph rewriting over orthogonal systems. There are a few papers describing implementations of term graph rewriting. So called concurrent term rewriting is addressed in [40,66] while [61,39,13] deal with the term graph ....

Luc Maranget. Optimal derivations in weak lambda calculi and in orthogonal term rewriting systems. In Proc. Annual ACM Symposium on Principles of Programming Languages, pages 255--269. ACM Press, 1991.

....NF terms as pure terms. 3 Calculi for Weak Reduction with Sharing In this section we generalize the weak explicit substitution calculus oe w defined by Curien et al. 1992) to include addresses in order to explicit pointer manipulations. Our starting point is reminiscent of the labeling used by Maranget (1991), 3 however, our notion of address is more abstract and allows us a better comprehension of implementations of machines for functional programming languages and their optimizations. Figure 1 defines syntax and reduction rules of oe a w . Like oe w it forbids substitution in abstractions by ....

Maranget, L. (1991). Optimal derivations in weak lambda calculi and in orthogonal rewriting systems. 18th Principles of Programming Languages. pp. 255--268.

....the program by himself. In this case, however, there is no way to determine which kind of modification of the semantics or computational behavior is introduced by the annotations. Syntactic replacement restrictions have been utilized in term rewriting (referred to as syntactic annotations) KW95, Mar90] Such syntactic replacement restrictions have been used to define rewriting restrictions that permit the implementation of lazy reductions via eager rewritings in a transformed TRS [KW95] and also permit optimal reductions by using graph rewriting techniques [Mar90] All these proposals have ....

.... syntactic annotations) KW95, Mar90] Such syntactic replacement restrictions have been used to define rewriting restrictions that permit the implementation of lazy reductions via eager rewritings in a transformed TRS [KW95] and also permit optimal reductions by using graph rewriting techniques [Mar90] All these proposals have used syntactic annotations as an auxiliary tool for modifying preexistent computational mechanisms. Context sensitive rewriting takes the symmetric approach; it can be thought of as a mechanization of the syntactic annotations themselves. We do not assume any extra ....

L. Maranget. Optimal Derivations in Weak Lambda-calculi and in Orthogonal Term Rewriting Systems. In Conference Record of the 18th Annual ACM Symposium on Principles of Programming Languages, POPL'90, pages 255269, ACM Press, 1990.

....mutation, still through rewriting rules. It is natural to apply these techniques in the setting of object languages to the notion of destructive updates, or update of the value of a eld (as for instance the increment of a counter encapsulated in an object) 1 Levy [L ev80] and Maranget [Mar92] propose using local addresses, but from the point of view of the operational semantics, global addresses describe better what is going on a computer or on an abstract machine. 3 b b b a a Cyclic graph Corresponding addressed term Figure 2: Representation of cycles through addresses ....

L. Maranget. Optimal Derivations in Weak Lambda Calculi and in Orthogonal Rewriting Systems. In Principles of Programming Languages, pages 255-268, 1992.

....reasons, to set the cost of a multistep equal to the number of positions narrowed in the step. A justi cation of this choice will be given after the de nition of cost. The notions of family of redexes, cost of a derivation, and complete narrowing step de ned next extend those for rewriting [9, 41, 50]. For any set I and equivalence relation on I, jIj denotes the cardinality of I, and I= denotes the quotient of I modulo . De nition 17 Let = t 0 ; hp i 1 ; R i 1 ; i 1 i i2I1 t 1 ; hp i 2 ; R i 2 ; i 2 i i2I2 be a narrowing (multi)derivation. Let n be a ....

....right hand side of a rule) since the reducts of blood related subterms are all equal, too. We show that complete, outermost needed narrowing derivations have minimum cost and minimum length. The proof relies on the analogous result for orthogonal systems formulated for reduction sequences only [41, 50]. Formally, we must give a meaning to the notion of need when a non elementary step is computed. To achieve optimality, we require multisteps only as far as blood related terms are concerned. Thus, it suces to consider multisteps in which only one complete family is narrowed. These steps are quite ....

[Article contains additional citation context not shown here]

L. Maranget. Optimal derivation in weak lambda-calculi and in orthogonal terms rewriting systems. In 18th Annual Symp. on Principles of Prog. Languages, pages 255-269. ACM, 1991.

....calculus. 3 Calculi for Weak Reduction with Sharing In this section we generalize the weak explicit substitution calculus oe w defined by Curien, Hardin and L evy (to appear) to include addresses in order to explicit pointer manipulations. Our starting point is reminiscent of the labeling used by Maranget (1991), 2 however, our notion of address is more abstract and allows us a better comprehension of implementations of machines for functional programming languages and their optimizations. Figure 2 defines the reduction rules we will use. The calculus includes the 2 In particular the use of ....

Maranget, L. (1991). Optimal Derivations in Weak Lambda Calculi and in Orthogonal Rewriting Systems. 18th POPL. pp. 255268.

....here is a pure, self contained rewriting system. The work described here has been inspired by the substitution pushing used by the Call by Mix strategy of Grue [11] The current trend of optimal reduction also touches on sharing although only as a method to minimise the number of reductions [18, 17, 10, 19] In fact a lot of the work done within term graph rewriting [6] seems to be aiming at achieving the same results as we are: to provide a convenient yet formal specification method that makes it possible to reason about rather than abstract away from sharing and recursion; cf. 13, 12, 15, 16, 27, ....

L. Maranget, Optimal Derivations in Weak Lambda-Calculi and in Orthogonal Terms Rewriting Systems, in POPL '91---Eightteenth Annual ACM Symposium on Principles of Programming Languages (Orlando, Florida), January 1991, pp. 255-- 269.

.... can be performed in any context, one simply adds rules of the following form, for any operator f and any rank i: t i t 0 i f(t 1 ; t i ; t n ) f(t 1 ; t 0 i ; t n ) Conditional rewriting systems can be obtained by removing some of these rules as in [15], or by adding a predicate P that limits the applicability of the rule, yielding rules of the form: t i t 0 i P f(t 1 ; t i ; t n ) f(t 1 ; t 0 i ; t n ) As shown by the Centaur experiment [4, 10] these rules can be directly transformed into executable ....

....may be useful both for generalizing proofs (removal of unused hypotheses) or debugging proving tactics. The methods used in this paper permit to describe the simple labelings that are related to descendance. More elaborate labelings have also been developed to study other aspects of reductions [6, 11, 13, 15]. The labels are not simple letters in an alphabet, but rather structured terms, based on extra operators (like underlining in [6, 11, 13] It is interesting to see how these operators interact with those of our language Orig, in order to provide an implementation of these elaborated labelings. ....

L. Maranget, "Optimal Derivations in Weak Lambda-Calculi and in Orthogonal Term Rewriting Systems", Proceedings of the Eightteenth ACM Symposium on Principles of Programming Languages, Orlando, Florida, 1991, pp. 255--269.

....to define it for orthogonal Combinatory Reduction Systems. In this case the approach based on labels seems problematic since labels should keep much more structure (they should be labeled trees) In particular, we would eventually face the same problems of orthogonal term rewriting systems in (Maranget 1991). On the other hand, the extraction process seems much more robust: so much to support the following Conjecture: The extraction process still works in the case of orthogonal Combinatory Reduction Systems, providing the correct family relation. There is another problem concerning optimal ....

L. Maranget (1991) Optimal Derivations in Weak lambda-calculi and in Orthogonal Terms Rewriting Systems. In Proceedings 17 th ACM Symposium on Principles of Programmining Languages, pages 255 -- 269.

....respect to sharing in the evaluators 2 . 1 In this paper we write call by need rather than lazy to avoid a name clash with the work of Abramsky [2] which describes call by name reduction to values. 2 Ironically, this problem immediately showed up in the dif A number of researchers [1, 13, 20, 31, 28] have studied reductions that preserve sharing in calculi with explicit substitutions, especially in relation to optimal reduction strategies. Having different aims, the resulting calculi are considerably more complex than those presented here. Closest to our treatment is Yoshida s weak lambda ....

....that we avoid duplication only of argument evaluations. In the program (f:fI(fI) w: II)w) the redex II in the argument will be reduced twice. Put differently, our calculus captures neither full laziness as described by Wadsworth [30] nor the sharing required by optimal calculus interpreters [13, 15, 16, 17, 20]. However, as observed by Arvind, Kathail and Pingali [7] full laziness can always be obtained by extracting the maximal free expressions of a function at compiletime [25] 6 Although we rejected the liberalized let V rule which produced this particular expression, it is perfectly reasonable to ....

L. Maranget. Optimal derivations in weak lambdacalculi and in orthogonal term rewriting systems. In Proc. ACM Conference on Principles of Programming Languages, Orlando, Florida, January 1991.

.... before) When considering a function call f(t 1 ; t k ) only the arguments whose indices are present in the list are evaluated (in the ordering which the list has specified) Syntactic replacement restrictions have been utilized in term rewriting, referred to as syntactic annotations) [KW95, Mar90]. Such syntactic replacement restrictions have been used to define rewriting restrictions which permit the implementation of lazy reductions via eager rewritings in a transformed TRS [KW95] and also permit optimal reductions by using graph rewriting techniques [Mar90] In [OF97] the authors gave ....

.... syntactic annotations) KW95, Mar90] Such syntactic replacement restrictions have been used to define rewriting restrictions which permit the implementation of lazy reductions via eager rewritings in a transformed TRS [KW95] and also permit optimal reductions by using graph rewriting techniques [Mar90]. In [OF97] the authors gave a rewritingbased implementation of the evaluation strategy used in the algebraic languages OBJ and CafeOBJ. All these proposals have used syntactic annotations as an auxiliary tool for modifying preexistent computational mechanisms. To our knowledge, few (or no) ....

L. Maranget. Optimal Derivations in Weak Lambda-calculi and in Orthogonal Term Rewriting Systems. In Conference Record of the 18th Annual ACM Symposium on Principles of Programming Languages, POPL'90, pages 255-269, ACM Press, 1990.

....which we can take as the starting point for the derivation and verification of lower level implementations. In short, the calculus is a good calculus for a call by need language in Plotkin s sense. Unlike many others, we are not interested in capturing the optimality of reduction strategies (Maranget, 1991; Yoshida, 1993; Field, 1990; Kathail, 1990; Gonthier y Abramsky and Ong (1990; 1988) explored the model theoretic properties of this calculus. They called it the lazy lambda calculus , even though this calculus does not at all address the laziness of implementations. The Call By Need Lambda ....

Maranget, L. (1991). Optimal derivations in weak lambda-calculi and in orthogonal term rewriting systems. Pages 255--269 of: Proc. ACM Conference on Principles of Programming Languages, Orlando, Florida.

....In particular, the new cons operator does not evaluate its arguments. This is similar to impose ( Since Lisp functions are evaluated by processing lists built from these operations, the restrictions over the basic operators, in some sense, are raised to arbitrary Lisp functions. In [KW95, Mar90], the specification of replacement restrictions is similar to ours: in a first stage, the authors also 4 The Journal of Functional and Logic Programming 1998 1 Lucas Context Sensitive Computations 1.1 limit the replacements that are allowed on the arguments of function symbols. Next, the ....

....In fact, few and weak results on computational properties for lazy rewriting (whose definition is more complex than ours) are given. Thus, a practical use of the restricted reduction relation itself is not feasible. A more detailed comparison is given after our technical presentation. In [Mar90] the implementation of lazy reductions by means of graph reduction techniques based on a restricted class of TRSs is proposed. In fact, the reduction relation defined by Maranget (called conditional reduction, although it is not related to conditional TRSs) is more similar to csr than is the lazy ....

[Article contains additional citation context not shown here]

L. Maranget. Optimal derivations in weak lambda-calculi and in orthogonal term rewriting systems. In<F5.483e+05> Conference Record of the 18th Annual ACM Symposium on Principles of Programming Languages,<F5.359e+05> POPL'90, pages 255--269, New York, 1990. ACM Press.

.... several calculi for explicit substitution have recently been developed by Curien and L evy [ACCL90, Cur86, Cur91, HL89, Hin77] An attempt to formalize weak reduction , i.e. the kind of reduction that is actually done by most functional language implementations, is described by Maranget [Mar91] Barendregt et al. have put forth a calculus to capture sharing in graph reduction implementation of Term Rewriting Systems (TRS) BBvE 87, BvEG 87, Ken90, BvEvLP87] In the same vein, we want to develop a calculus to capture the sharing of subexpressions in a more general class of ....

L. Maranget. Optimal derivations in weak lambda-calculi and in orthogonal term rewriting systems. In Proc. ACM Conference on Principles of Programming Languages, Orlando, Florida, January 1991.

....reduction systems are related to variable and substitution as found in calculus, and in particular where the latter needs to be extended. One particularly interesting direction is what can be modeled by combining explicit substitution with various forms of labeling of terms in the style of L evy [13, 14]. It seems to be possible to use this to refine the definition of unfolding used to be that of graph traversal : the example in the introduction should intuitively be represented directly as fib(n) let fibwalk ( n 0 ; n 1 ; n ) if ffl = 0 then ffl else ffl (ffl ; ffl ffl ; ffl Gamma1) ....

L. Maranget, Optimal Derivations in Weak Lambda-Calculi and in Orthogonal Terms Rewriting Systems, in "POPL '91---Eightteenth Annual ACM Symposium on Principles of Programming Languages" (Orlando, Florida), January 1991, pp. 255--269.

....] If e is of the form if e 0 then Nil else Cons 1 t , for example, then the computation of e 0 will be repeated for every element of the infinite list. Sharing has been lost. Much work has been done on making substitution explicit, the most relevant for our purposes being that by Maranget [Mar91], where he develops a framework of Labelled Terms Rewriting Systems. Using these he studies the weak calculi and shows the lazy strategy to be optimal. The resulting semantics for laziness is significantly more complex than that presented here (having been developed with different goals in ....

L.Maranget, Optimal Derivations in Weak Lambda-calculi and in Orthogonal Term Rewriting Systems, in Proc SIGPLAN POPL 91, Orlando, pp 255-269, 1991.

....actually the only important one, because it prevents any critical pairs. The following result also holds when reduction under is allowed. Proposition 2.1 oe cw is confluent. Proof: There are no critical pairs. Thus the system is orthogonal in an extended sense, which is defined and studied in [35]. The reader may want to work out a direct proof by defining a notion of parallel reduction in the most obvious way, and by showing that the diamond property holds for this (Beta ) x:a) s] b a[ b=x) Delta s] App) ab) s] a[s] b[s] VarId) x[id ] x (VarCons) x[ a=x) Delta s] ....

L. Maranget, Optimal Derivations in Weak Lambda-calculi and in Orthogonal Term Rewriting Systems, POPL 91.

....particular, the patterns used to specify the animation behavior for Tip s animator become much simpler, and the adaptation of the abstract syntax proposed by Dinesh to improve his origins becomes unnecessary. On the theoretical side, origins are related to so called residuals or descendants [HL91, Mar91] which are used in the search for optimal reduction strategies. Currently, Field and Tip are extending residuals to creation residuation tracking using ideas from incremental rewriting as described by [Fie93] The notion of a program scheme [Cou90] is a general device to understand control ....

L. Maranget. Optimal derivations in weak lambda-calculi and in orthogonal term rewriting systems. In Proceedings of the Eighteenth conference on Principles of Programming Languages POPL '91, pages 225--269, 1991.

....NF terms as pure terms. 3 Calculi for Weak Reduction with Sharing In this section we generalize the weak explicit substitution calculus oe w defined by Curien et al. 1992) to include addresses in order to explicit pointer manipulations. Our starting point is reminiscent of the labeling used by Maranget (1991), 3 however, our notion of address is more abstract and allows us a better comprehension of implementations of machines for functional programming languages and their optimizations. Figure 1 defines syntax and reduction rules of oe a w . Like oe w it forbids substitution in abstractions by ....

Maranget, L. (1991). Optimal derivations in weak lambda calculi and in orthogonal rewriting systems. 18th Principles of Programming Languages. pp. 255--268.

....to let constructs, and gives an optimal reduction strategy. Her calculus subsumes several of our reduction rules as structural equivalences. However, due to a different notion of observation, reduction in this calculus is not equivalent to reduction to WHNF. A number of researchers [Fie90, ACCL90, Mar91] have studied reductions that preserve sharing in calculi with explicit substitutions, especially in relation to optimal reduction strategies. Having different aims, the resulting calculi are considerably more complex than those presented here. The rest of this paper is organized as follows. ....

Luc Maranget. Optimal derivations in weak lambda-calculi and in orthogonal term rewriting systems. In Proc. 19th ACM Symposium on Principles of Programming Languages, Orlando, Florida, pages 255--269. ACM Press, January 1991.

.... the creation Trying to capture how intermediate and final terms originate from the initial term is formalized in a notion called origin tracking [Ber91, Ber92, DKT93] Origin tracking is based on so called residuals, which have been used successfully in more theoretically oriented papers [HL91, Mar91] for reasoning about optimal reduction strategies in TRSs. 1.1 Applications Our motivation to work on origin tracking was that we needed it for the automatic generation of tools from algebraic specifications of programming languages. As an example, Supported by the European Communities, ....

....by marking functions. Part of his work has been implemented in the Centaur system [BCD 89] In particular, the specification language Typol [Kah87] has been extended with subject tracking [Des88] Closely related to origins are residual maps, descendants, or labelings [L ev75, HL91, Mar91, Fie91] which are used to study reduction strategies. Residuals indicate which redexes survive if a particular redex is contracted. One can think of this as giving interesting parts in the initial term a particular color, and then looking how this color survives during reduction. An interesting ....

L. Maranget. Optimal derivations in weak lambda-calculi and in orthogonal term rewriting systems. In Proceedings of the Eighteenth conference on Principles of Programming Languages POPL '91, pages 225--269, 1991.

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