C. P. Wadsworth. The relation between computational and denotational properties for Scott's D1-models of the lambda-calculus. SIAM J. Comput., 5(3):488-521, 1976. 34  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and only if, for every output value P , there is a program C P [M ] using effectively M as subprogram, such that C P [M ] evaluates to P . The fact that C P [M ] uses effectively M can be formalized as: not for all Q, C P [Q] evaluates to P . In the case of classical calculus, it has been proved [15] that, for all term M , if such a context C[ exists, then there is also a head context, i.e. a context of the shape: x 1 : x n : M 1 : Mm (for some m;n) with the same behaviour, where fx 1 ; x n g is the set of free variables of M ( FV (M) This means that the terms have a ....

.... M = fi x 1 : x n :x i M 1 : Mm (1 i n) for some n) From an operational point of view, solvable terms are the terminating programs, in the head reduction machine [12] From a semantic point of view, all unsolvable terms (i.e. the non terminating programs) can be all consistently equated [15]. From a logical point of view, a term M is solvable if and only if it can be typed in the intersection type assignment system defined by Coppo and Dezani [3] Let recall also the notion of solvability in the lazy calculus, introduced by Abramsky and Ong [1] for modelling the call by name lazy ....

C.P.Wadsworth. The relation between computational and denotational properties for scott D1-models of the lambda calculus. SIAM Journal of computing, vol.5, n.3, 1976, pp. 488-522. 36

....nite term xz 0 :x( z 1 :z 0 ( z 2 :z 1 ( Clearly, J is the normal form of the nite term Y ( fxy:x(fy) where Y is the x point combinator f: f(xx) f(xx) Both I and J are normal form wrt reduction. So I and J have di erent B ohm trees. Wadsworth showed that D1 j= I = J [Wad76] and from this one can infer that the 1 B ohm trees of I and J are the same [Bar84] We propose an alternative de nition of the 1 construction. We want to obtain the 1 B ohm trees as unique normal forms of reduction and the following strong form of reduction: where 1 denotes ....

C. P. Wadsworth. The relation between computational and denotational properties for Scott's D1-models of the lambda-calculus. SIAM Journal on Computing, 5(3):488-521, 1976. 10

....No first order calculus implementing Moggi s meta language is known. We seek here to understand the notion of call by value reduction that SKInT offers by finding and studying a satisfactory notion of labeled reduction, such as those introduced in the calculus by Hyland, Wadsworth and L evy [9, 15, 11]. There are a number of ways we can justify our choice of labels and our label calculus. We would have loved to explain our labels as abstract representations of paths, as in [2] However there are at least two difficulties here. The first is that paths are a way of connecting principal ports of ....

C. P. Wadsworth. The relation between computational and denotation properties for Scott's D1-models of the lambda-calculus. SIAM Journal of Computing, 5:488--521, 1976. 15

....languages, the execution of procedure bodies takes place in the current environment in which the name of the procedure is bound to its body, so that executing a recursive call results in reentering the body in the current environment. Techniques developed by Strachey s student Chris Wadsworth [9] turned out to be suitable for showing the equivalence of a semantics of recursive Lisp procedures using knots to a semantics using xed points. A corollory was the equivalence of the Lisp EVAL algorithm to a Scott Strachey style mathematical semantics [3] Although my study of the relation ....

C. P. Wadsworth. The relation between computational and denotational properties for Scott's D1 models of the lambda-calculus. SIAM Journal of Computing, 5:488-521, 1976.

.... approximation introduced by Honsell and Ronchi della Rocca [14] in the framework of continuous semantics, which can be naturally extended to a very large class of models, in particular to non sensible models (like Park s model) for which the standard approximation theorem of Wadsworth and Hyland [23, 24, 15] is not available. From now on D denotes either a continuous, stable or strongly stable model. Let us add a constant c 0 to the calculus, and consider the calculus induced by the fi reduction on (c 0 ) We call approximants the fi normal terms of (c 0 ) we denote by A the set of ....

...., for every n ; ffl W p n =D x x ; ffl ( p n 1 )a)b = p n ) a) p n )b. In order to prove that every stratified model satisfies the approximation property, we use an extended calculus, the labelled calculus. This calculus is very similar to the indexed calculus used by Hyland and Wadsworth [23, 24, 15]. Here we use the syntax introduced by Parigot in [19] Let C = fc n g n2 be a set of constants that we call labels. The set e of labelled terms is the subset of (C ) inductively defined by: ffl x 2 e , for every variable x, 25 ffl if u; v 2 e then (c n )u; c n )x u and (u)v belong to e ....

C.P. Wadsworth, The relation between computational and denotational properties for Scott's D1-models of the lambda calculus, SIAM J. of Computing 5(1976) 488-521.

....proof has the structure presented by diagram in Figure 2. The embedding maps untyped terms into terms in the simply typed lambda calculus using constants f and g that can be thought of as a retraction pair used in the interpretation of the simply typed lambda calculus (see Scott [14] Wadsworth [19], and Meyer [10] From the syntactical point of view, blocks all redexes, replacing ( x:M)N by f(g( x:M) N . The notion of o reduction ( o ) is introduced to play an analogous role to the lambda abstraction marking ( it replaces a blocked redex f(g( x:M) N by the unblocked redex ( x:M)N ....

Wadsworth, C.P.: The relation between computational and denotational properties for Scott's D1-models of the lambda calculus. SIAM Journal of Computing 5(3) (1976) 488-521.

....argument, which we do not reproduce in entirety. The full details of this fact for the model D1 can be found in [3, x19.2] and uses only properties of D1 which we have proved for D and D REC . The argument uses the technique of labelled reduction introduced by Hyland in [8] and Wadsworth in [19]. The properties of the approximation are used to show that labelled reduction is monotone in the model, and hence that labelled normal forms are maximal. This gives rise to an approximation theorem in the model any term is the union of its approximate normal forms (introduced by Wadsworth in ....

C. P. Wadsworth. The relation between computational and denotational properties for Scott's D1-models of the -calculus. SIAM Journal of Computing, 5:488-521, 1976. 55

....we give a semantics based on the notion of unfolding for our parallel and non deterministic extension of classical calculus. This is not achieved by means of trees, but by using the equivalent notion of approximant originated, in the case of calculus, from the works of L evy [28] and Wadsworth [45]. In the first section of the paper we introduce the syntax of the calculus and two reduction relations. The first one explicitly makes the into a choice operator, while the second one, instead, simulates the choice by a distribution law. Adapting to the present case the notion of head reduction ....

....the relation v O is a precongruence. The set SOL, when restricted to pure terms, is the set of terms having a head normal form, that is those terms which are solvable in the classical sense. Hence the restriction of O to pure terms is the theory of D1 by a well known result of Wadsworth [45]. Proposition15. The following (in) equations hold: i) x:M )N O M [N=x] vii) x: MkN ) O x:Mkx:N ; ii) M N )L O ML NL; viii) M N v O M;N ; iii) L(M N ) v O LM LN ; ix) L v O M;N ) L v O M N ; iv) MkN )L O MLkNL; x) M;N v O MkN ; v) LMkLN v O L(MkN ....

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C.P. Wadsworth, " The Relation between Computational and Denotational Properties for Scott's D1-models of the Lambda Calculus", SIAM J. of Comp. 5, 1976, 488-521.

....6.5 (Strong normalisation of derivation reduction) If D : B r E t:oe, then SN (D) Proof: If D : B r E t:oe, then, taking R such that x R = x, by Theorem 6.4, Comp (D : B r E t:oe) Then, by Lemma 6. 2 (i) SN (D) 7 Approximants Now we will develop, essentially following [22] (see also [6] a notion of approximant for combinator terms. This will be done by introducing a special symbol into the definition of terms. The general idea is that a term a directly approximates a term t if they are identical but for those places where a has an occurrence of . 20 ....

C.P. Wadsworth. The relation between computational and denotational properties for Scott's D1 -models of the lambda-calculus. SIAM J. Comput., 5:488--521, 1976. 32

....B E t : oe, then by Lemma 17:2, also B hti : oe. Since t 0 t, by the remark above also ht 0 i fi hti . Since is closed for fi expansion, we have B ht 0 i : oe. Then, by Theorem 17:1, we have B E t 0 : oe. 3 Approximants Now we will develop, essentially following [20] (see also [6] a notion of approximant for combinator terms. This will be done by introducing a special symbol into the definition of terms. Definition 22 (Combinator terms with ) 1. The set T(C; X; is defined by: t : j x j C j Ap (t 1 ; t 2 ) 2. The notion of rewriting of ....

C.P. Wadsworth. The relation between computational and denotational properties for Scott's D1-models of the lambda-calculus. SIAM J. Comput., 5:488--521, 1976.

....coincide with (maximal) superdevelopments. To capture properties of reduction, it is sometimes useful to decorate terms with labels. Several ways of labeling terms appear in the literature; in particular, labels are often used to either drive or record the history of reduction (see e.g. [38, 27, 62]) A typical example is the so called 0 reduction (see [6] Ch.11) in which labels act as markers for redexes which are allowed to be contracted. 0 reduction reveals itself as a powerful tool to give elegant proofs of fundamental theorems about reduction. 11 The labeling of terms which ....

C. Wadsworth. The relation between computational and denotational properties for Scott's D1 models of the lambda-calculus. SIAM J. Comput 5:488-521, 1976.

....can be rephrased as an approximation property such that the semantics of a type is completely determined by the semantics of its finite syntactic approximations. In fact, this is a very desirable property in the semantics of programming languages (see, for example, the approximation theorem in [29]) 3.3 Tree Expansions As we have seen, simple unfolding does not induce a sufficiently strong notion of type equivalence. A stronger condition of approximation seems required to deal with infinite expansions. Let us first explain how to associate a finitely branching, labeled, regular tree with ....

Wadsworth, C. The relation between computational and denotational properties for Scott's D models of the lambda-calculus, SIAM J. of Computing, 5, pp 488-521, 1976.

....and top normal forms. In this paper we will consider the case that the set of values is the set of the top normal forms, that is we will consider Berarducci trees. 1. 1 Berarducci Trees Berarducci trees arise in a natural way when we look at the parsing trees of lambda terms (see for example [25]) In this representation the abstraction with respect to a given variable is an unary operator and the application (explicitly denoted by ) is a binary operator. For example the term ( x:x) y:y) is represented by x y x y : As a matter of fact, x:x) y:y) reduces to y:y, so we can ....

....original proof of B ohm s theorem . B ohm s theorem states that given two distinct normal forms there is a context C[ such that C[M ] x and C[N ] y, where x; y are arbitrary distinct variables. The method used to nd such a context is called the B ohm out technique [4] Section 10.3) In [25] Wadsworth, generalizing B ohm s theorem, shows that two lambda terms M;N have the same B ohm tree modulo in nite expansions i for all contexts C[ the following holds: C[M ] has a head normal form ( C[N ] has a head normal form. The proof technique used to obtain this result is the B ohm ....

[Article contains additional citation context not shown here]

C. P. Wadsworth. The relation between computational and denotational properties for Scott's D1-models of the lambda-calculus. SIAM J. Comput., 5(3):488-521, 1976. 24

....to a term in (head )normal form. A rewrite system is strongly normalizing (or terminating) if all the rewrite sequences are finite; it is (head )normalizing if every term is (head )normalizable. A term is called unsolvable if it has no head normal form. Now we will develop, essentially following [22] (see also [6] a notion of approximant for combinator terms. This will be done by introducing a special symbol into the definition of terms. The general idea is that a term a directly approximates a term t if they are identical but for those places where a has an occurence of . Definition 5 ....

C.P. Wadsworth. The relation between computational and denotational properties for Scott's D1-models of the lambda-calculus. SIAM J. Comput., 5:488--521, 1976.

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C. P. Wadsworth. The relation between computational and denotational properties for Scott's D1-models of the lambda-calculus. SIAM J. Comput., 5(3):488-521, 1976. 34

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C.P. Wadsworth. The relation between computational and denotational properties for Scott's D#-models of the lambda-calculus. SIAM J. Comput., 5:488--521, 1976. 18

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C. P. Wadsworth. The relation between computational and denotational properties for Scott's D1-models of the lambda-calculus. SIAM J. Comput., 5(3):488-- 521, 1976.

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Wadsworth C.P., "The Relation between Computational and Denotational Properties for Scott 's D1 Models of the -calculus", Siam J. Comput. 5, 1976, 488-521.

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C. P. Wadsworth. The relation between computational and denotational properties for Scott's D1-models of the lambda-calculus. SIAM J. Comput., 5(3):488--521, 1976. 24

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C. P. Wadsworth. The relation between computational and denotational properties for Scott's D-models of the lambda-calculus. SIAM Journal on Computing, 5(3):488--521, Sept. 1976.

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C.P. Wadsworth, " The relation between computational and denotational properties for Scott's D1-models of the lambda-calculus", SIAM J. of Comp. 5, 1976. 109

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C.P. Wadsworth, " The relation between computational and denotational properties for Scott's D1-models of the lambda-calculus", SIAM J. of Comp. 5, 1976.

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C. P. Wadsworth. The relation between computational and denotational properties for Scott's D# -models of the lambda-calculus. SIAM J. Comput., 5(3):488--521, 1976.

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C.P. Wadsworth, " The relation between computational and denotational properties for Scott's D1-models of the lambda-calculus", SIAM J. of Computing 5, 1976. 66

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C.P. Wadsworth, "The Relation between Computational and Denotational Properties for Scott's D1 Models of the Lambda-calculus", Siam J. Comput. 5, 1976, 488-521. 23

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