Wadsworth, Christopher P. Semantics and Pragmatics of The Lambda Calculus. PhD thesis, University of Oxford (1971).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is duplicated (since 2 has to be inserted on y s place in one of the copies but not in the other) Hence h 4 will be evaluated twice. However, one can do better: actually there is no need to copy (h 4) since (h 4) does not contain y free in [Jon87, p. 246] this observation is attributed to [Wad71] An implementation clever enough to avoid such unnecessary copying is termed fully lazy [Jon87, p. 210] Usually, one does not implement functional languages by means of # reductions of # expressions instead, one transforms into supercombinators by lambda lifting ( Jon87] By doing so, E ....

C.P. Wadsworth. Semantics and Pragmatics of the Lambda Calculus. PhD thesis, Oxford University, 1971.

....simply typed calculus via a Curry Howard correspondence: given a proof net representation of a term, the type of the term can be obtained as the judgement proved by the proof net. In fact, proof net representations of terms exhibit an isomorphism with a more conventional graph representation [Wad71] of the terms. In this paper we draw the proof nets as forest structures. We associate a positive or negative polarity to the nodes of the forest. The nodes with positive polarity are drawn as , and the nodes with negative polarity are drawn as . The axiom links of the proof net are represented ....

....nally applying abstraction for the variables f , x, and y gives the proof net (d) c) d) f f Lemma 1. Given a simply typed term, the translation of the term using T[ results in a proof net; moreover this proof net is isomorphic to a traditional graph representation of the term [Wad71]. Proof. By induction on the structure of the term. The isomorphism with traditional graph representation of terms maps sequences of abstraction nodes to nodes and sequence of application nodes to nodes. In the rest of the paper we will refer to terms and proof nets interchangeably in ....

C Wadsworth. Semantics and Pragmatics of the Lambda-Calculus. PhD thesis, Oxford University, 1971.

....of identical terms through pointers, and avoids repeated evaluation of identical terms as it is commonly done in normal order reduction. Graph reduction for the calculus was proposed by Wadsworth in order to bring together the advantages of both the applicative and the normal order evaluation [20]. Wadsworth also formally proved the correctness of his graph reduction technique. As an aside, Wadsworth also showed that his graph reduction did not capture enough sharing to lead to an optimal interpreter. More recently a new graph structure, which allows sharing of contexts , has been ....

....[18] In this paper, however, we are not concerned with optimality questions, and we restrict our attention to argument sharing in a language which is simpler than the calculus. Much of the past work on graph rewriting has been to prove its correctness with respect to either the calculus [20] or Term Rewriting Systems [7, 8, 9, 16] In contrast, this paper explores graph rewriting as a system in its own right, and makes no attempt to prove the correctness of a graph implementation with respect to a tree (or unshared) view of the computation. Motivated by what we have observed in ....

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C. Wadsworth. Semantics And Pragmatics Of The Lambda-Calculus. PhD thesis, University of Oxford, Oxford, UK, September 1971.

....mutation of objects. This makes our framework more abstract, as it involves no particular structure for computing. Moreover it provides a small step semantics of object mutation. This is in fact a generalization of Wadsworth s graph reduction technique of implementation of functional languages [23, 21], which, by essence, forbids destructive updates. Moreover, our graphs are represented as special terms, called addressed terms, exploiting the idea of simultaneous rewriting, already mentioned in [18, 4] and slightly generalized in this paper (see [13] The framework Obj a is much more ....

C. P. Wadsworth. Semantics and pragmatics of the lambda calculus. PhD thesis, Oxford, 1971.

....objects by considering the state of the object as what has been computed so far [ASS85] The semantics of sharing. Ecient implementations of lazy functional languages (or of computer algebras) require some sharing mechanism to avoid multiple computations of a single argument. Term graphs [Wad71, Tur79, BVEG 87, Plu99] have been studied as a representation of program expressions intermediate between abstract syntax trees and concrete representations in memory, and term graph rewriting provides a formal operational semantics of functional programming sensitive to sharing. However, ....

C. P. Wadsworth. Semantics and pragmatics of the lambda calculus. PhD thesis, Oxford, 1971.

....representations of intuitionistic propositional logic. Proof nets can be considered as representations of terms in the simplytyped calculus via a Curry Howard correspondence. In fact, proof net representations of terms exhibit an isomorphism with a more conventional graph representation [44] of the terms. Given a proof net representation of a term, the type of the term can be obtained as the judgement proved by the proof net. In this paper we draw the proof nets as forest structures. We associate a positive or negative polarity to the nodes of the forest. The nodes with positive ....

....following proof net where x is represented by the empty set. R . f g x P . Lemma 3. 2 Given a simply typed term, the translation of the term using T[ results in a proof net; moreover this proof net is isomorphic to a traditional graph representation of the term [44]. Proof: By induction on the structure of the term. The isomorphism with traditional graph representation of terms maps sequences of abstraction nodes to nodes and sequence of application nodes to nodes. In the rest of the paper we will refer to terms and proof nets interchangeably in ....

C. Wadsworth. Semantics and Pragmatics of the LambdaCalculus. PhD thesis, Oxford University, 1971. 10

....languages, such as LISP (MCCARTHY et al. 1962) used a strict reduction scheme rather than the lazy reduction scheme. Copyright c 1992 Alan Jeffrey. This work was supported by SERC project GR H 16537. Miranda is a trademark of Research Software Limited. Graph reduction was introduced by WADSWORTH (1971) as a means of efficiently implementing the lazy reduction strategy. Rather than reducing syntax trees, we reduce syntax graphs which allows a more efficient representation of sharing. For example, we can represent the reduction of E i 1 as: lx E i # . x x ## E i 2i ....

WADSWORTH, C. P. (1971). Semantics and Pragmatics of the Lambda Calculus. D.Phil thesis, Oxford University.

....implicit in the data structures. Later abstract machines include some variation in the used reduction strategy but remain dedicated to one (some early, seminal examples are Plotkin 1977, Henderson 1980, Cardelli 1983) The first issue, sharing to avoid duplication, was addressed already by Wadsworth (1971) who proposed graph reduction which is the simple idea that duplication should be delayed as long as possible by representing subterms of common origin by identical subgraphs. This way all duplicates profit from any reductions happening to that particular subterm (or graph, as it were) This ....

....for weak calculus (reflecting that functional languages share the restriction that reduction never happens under a ) into a calculus, oe a w , with explicit substitution, naming, and addresses. Moreover, it naturally permits two update principles that are readily identifiable as graph reduction (Wadsworth 1971) and environment based evaluation (Curien 1991) In Section 4 we show how oe a w adequately describes sharing with any (weak) reduction strategy; the proof is particularly simple because it can be tied directly to addresses; to illustrate this we prove that oe a w includes the let calculus ....

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Wadsworth, C. (1971), Semantics and pragmatics of the lambda calculus, PhD thesis, Oxford.

....Although it has important termination properties, it is highly inefficient due to the multiple evaluation of the same subexpression. Graph reduction is an optimised form of reduction, proposed by Wadsworth, which overcame this inefficiency and turned reduction into a practical technique [134]. It consists in introducing sharing of subexpressions and thus of their results, so that multiple evaluation of subexpressions is avoided. This is equivalent to representing expressions as graphs, where subexpressions consist the nodes of the graph, linked together with pointers. In real ....

C. P. Wadsworth. Semantics and pragmatics of the lambda calculus. PhD thesis, Oxford University, 1971.

....preserve this sharing; otherwise our proof terms will blow up in size. Any logic programming system is likely to implement sharing of terms obtained by copying multiple pointers to the same subterm. In Terzo, this can be seen as the implementation of a reduction algorithm described by Wadsworth [25]. But we require even more sharing. The similar terms obtained by applying a term to different arguments should retain as much sharing as possible. Therefore some intelligent implementation of higherorder terms within the metalanguage such as Teyjus s use of explicit substitutions [16, ....

C. P. Wadsworth. Semantics and Pragmatics of the Lambda Calculus. PhD thesis, Oxford University, 1971.

.... rewriting systems, such as Term Graph Rewriting [KKSV93] Jungle Rewriting [HoPl91] DAG (Directed Acyclic Graph) Rewriting [Mar91] and many others (in particular, covering cyclic graph reduction as well) inspired by Wadsworth s original work on graph based implementation of the calculus [Wad71], created the need to develop an abstract notion of family general enough to cover all the existing notions, and refined enough to enable proof of normalization and optimality results. Such structures were indeed introduced by the 2 authors in [GlKh96] as Deterministic Family Structures (DFSs) ....

Wadsworth C.P. Semantics and pragmatics of the Lambda-calculus. PhD. thesis, University of Oxford, 1971.

....will be reduced to hM j ff = Ai, that is, A is put in the environment, as in [HM76] or, following the terminology of [AKP84] A is flagged so that it will not be copied in case the redex is shared. This suggests that in order to avoid the extra complication of detecting mfe s at run time, as in [Wad71], a term can be first pre processed by well known techniques [Hug82, Joh85] Then doing sharing of arguments is enough to capture the amount of sharing offered by Wadsworth s interpreter. We can now extend OE 2 with term rewriting rules. Theorem 9.25 Let R be an orthogonal term rewriting system. ....

C. Wadsworth. Semantics and Pragmatics of the Lambda-Calculus. 1971. PhD thesis, University of Oxford.

....substitutions invariably happen in a more controlled way. This is due to practical considerations, relevant in the implementation of both logics and programming languages. The term afb=xg may contain many copies of b (for instance, if a = xxxx) without sophisticated structure sharing mechanisms [18], performing substitutions immediately causes a size explosion. Therefore, in practice, substitutions are delayed and explicitly recorded; the application of substitutions is independent, and not coupled with the fi rule. The correspondence between the theory and its implementations becomes ....

C.P. Wadsworth, Semantics and Pragmatics of the Lambda Calculus, Dissertation, Oxford University, 1971. 56

....our simulation environment in Section 5. Simulation results are presented and analysed in Section 6. Further issues and related work are reviewed in Section 7, and we present our conclusions in Section 8. 2 Parallel graph reduction The graph reduction model was first proposed by Wadsworth [45] and forms the basis of most modern compiled implementations of functional languages. Essentially the idea is that a parameter expression can be represented in unevaluated form by a closure : a heap cell containing a code pointer and pointers to the variables on which the expression depends. In ....

C. P. Wadsworth. Semantics and Pragmatics of the lambda calculus. PhD thesis, Oxford University, Oxford, U.K., 1971.

....store in explicating the sharing interpretation; this was a natural choice, because the concept of store is so basic and immediately meaningful. But it could be especially valuable to have a resource model that worked without appeal to imeperative ideas, based perhaps on call graphs of terms [52]. We would also like to have realizability models. Second, we have made numerous comparisons between linear and bunched typing, but we wonder whether it is possible to have the best of both. Ideally, we would like a way to combine linear and bunched typing in a way that simultaneously accounts ....

C. P. Wadsworth. Semantics and pragmatics of the lambda calculus. Ph.D. thesis, University of Oxford, 1971.

....languages [Lan66, Sto77] We are mainly interested in the role of calculus in implementations. The semantics of sharing. Ecient implementations of lazy functional languages (or of computer algebra) require some sharing mechanism to avoid multiple computations of a single argument. Term graphs [Wad71, Tur79, BVEG 87, Plu99] have been studied as a representation of program expressions 2 Acyclic graph a b b a b Corresponding addressed term Figure 1: Representation of sharing through addresses (schematic) intermediate between abstract syntax trees and concrete representations in ....

C. P. Wadsworth. Semantics and pragmatics of the lambda calculus. PhD thesis, Oxford, 1971.

.... in the input clause, like in ( Moreover, this characterisation of is useful to understand better its intrinsic meaning: For instance, it shows that the instantiation of the parameter of an input should happen only when needed , in a way which resembles the call by need semantics of calculus [Wad71]. An attractive property of is the simple axiomatisation (for finite terms) The treatment of matching, the calculus conditional construct, is of technical interest. In the equational theories in [MPW92, Hen91, BD92] a conditional construct is discharged upon evaluation of its boolean ....

C. P. Wadsworth. Semantics and pragmatics of the lambda calculus. PhD thesis, University of Oxford, 1971. 32

....the metalanguage must preserve this sharing; otherwise our proof terms will blow up in size. Any Prolog system implements sharing of terms obtained by copying multiple pointers to the same subterm. In lProlog, this can be seen as the implementation of a reduction algorithm described by Wadsworth [Wad71]. But we require even more sharing. The similar terms obtained by applying a l term to different arguments should retain as much sharing as possible. Therefore some intelligent implementation of higher order terms within the meta language such as Nadathur s use of explicit substitutions ....

C. P. Wadsworth. Semantics and Pragmatics of the Lambda Calculus. PhD thesis, Oxford University, 1971.

....forms it is sometimes better to reduce sub terms in head normal forms. For example, in lazy graph reduction, the implementation of b reduction (lx.E)F b E[F x] implies making a copy of the body E before the substitution. It is well known that this may lose sharing and work may be duplicated [18]. Program transformations, such as fully lazy lambdalifting [10] aim at maximizing sharing but duplication of work can still occur. Another approach used to avoid recomputation is to consider alternative evaluation strategies. If the expression to reduce is (lx.E)F we know that the whnf of the ....

C.P. Wadsworth. Semantics and Pragmatics of the Lambda-Calculus. PhD thesis, Oxford, 1971.

....choices that I discovered to be beneficial. One such detail is the implementation of substitution. For example, the straightforward implementation of the substitution [ M x ]N would introduce as many copies of M as there are free occurrences of x in N , leading to excessive memory usage [Wad71]. A better strategy, also employed in the Twelf implementation of LF [PSC] is to use explicit substitutions [ACCL91] so that the substitution is performed lazily when the resulting term is later examined. Thus, each LF type or object is represented as a pair of a regular LF type or object with ....

C. P. Wadsworth. Semantics and Pragmatics of the Lambda Calculus. PhD thesis, Oxford University, 1971.

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Wadsworth, Christopher P. Semantics and Pragmatics of The Lambda Calculus. PhD thesis, University of Oxford (1971).

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C.P. Wadsworth. Semantics and Pragmatics of the Lambda Calculus. PhD thesis, Oxford Universtity, 1971.

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C.P. Wadsworth. Semantics and Pragmatics of the Lambda Calculus. PhD thesis, Oxford University, 1971.

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Wadsworth, C.P.: Semantics and Pragmatics of the Lambda Calculus. D.Phil. thesis, Oxford University, September 1971. 435

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C. P. Wadsworth, `Semantics and pragmatics of the lambda calculus, Ph.D thesis, Oxford University, England, 1971.

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