Robin Milner. Fully abstract models of typed lambda-calculus. Theoretical Computer Science, 4:1--22, 1977.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in [36] reveals that the fully abstract model for the base language VPCF need only be constructed out of continuous, extensional functions; other mathematically attractive properties of the model are simply not needed. In a companion paper [38] we show how to construct using techniques of Milner [21] fully abstract models of VPCF and FPC (a sequential language with recursive types [9] in which the semantics of VPCF S can be encoded) called V and S. While our proofs only require the model V to be fully abstract, the intimate connection between V and S requires that the model S also not ....

....xpoint model V of VPCF. One obvious way to construct a fully abstract model of V is to take the term model consisting of equivalence classes of all closed terms. Unfortunately the term model for VPCF, while trivially fully abstract, is not continuous. However, by a suitable context lemma (cf. [21]) the term model for a purely functional call by value language is always monotone. Thus, if all types in the term model are nite, then it is also continuous. We rely upon this insight to prove Theorem 12. Instead of VPCF, we consider a language FVPCF in which all types are nite and which will ....

R. Milner. Fully abstract models of the typed lambda calculus. Theoretical Computer Sci., 4:1-22, 1977.

....examples do not exist in other words, when denotational approximation coincides with operational approximation the denotational model is said to be fully abstract. Shortly after Plotkin s paper, Milner proved that there was exactly one fully abstract model of PCF meeting certain conditions [16]. Until recently, all descriptions of this fully abstract model have used operational semantics (see, for instance, 19, 34] New constructions using games semantics [2, 10, 20] and logical relations [22, 33] have yielded a more abstract understanding of PCF. The sequentiality problem is ....

....values are not definable, and yields insight into the structure of sequential computation. There are other ways to build fully abstract models for FPC. For instance, Riecke and Viswanathan [31, 32] give a dcpo based model for call by value FPC. The construction uses Milner s syntactic methods of [16]. This construction sheds little light into the structure of FPC, except that the model validates least fixpoint reasoning. Games semantics has also been applied to full abstraction questions for FPC. In [15] McCusker builds a model of call by name FPC. The sums in this model are separated: ....

R. Milner. Fully abstract models of the typed lambda calculus. Theoretical

....nontermination. This variant gives us contextual equivalence where contexts produce closed terms of unit type, and in which we allow ourselves to observe termination. The order on M(S) generates a pre order on terms. This is the pre order used to quotient the fully abstract model of PCF by Milner [9]. So, if we have an observable type which contains the values for both termination and non termination, we can handle the conventional form of contextual equivalence. 4. Obs = ob(L) ob(L) and L = C. When everything is observable, is just equality. 3. ABSTRACT TYPES An equationally speci ed ....

R. Milner. Fully abstract models of typed lambda-calculus. Theoretical Computer Science, 4:1-20, 1977.

....Denotational equivalence. ffl Unsoundness for operational equivalence. 13.4 Equational Theories ffl Unrolling equation for recursive functions. ffl Definedness value restrictions. ffl Sequentiality conditions. ffl Correctness of equational theory via p.e.r. interpretation. 13.5 References [56, 43, 40] 104 ....

Robin Milner. Fully abstract models of the typed lambda calculus. Theoretical Computer Science, 4:348--375, 1977.

....is ordered by the relation v oe , where [M ] v oe [N ] if for all contexts C[ Delta] in which C[M ] halts at a numeral n, then C[N ] halts at n. It is not hard to check that this is a well defined partial order. The fact that T is a frame follows from the context lemma for PCF: Lemma 20 (Milner [7]) For any closed PCF terms M;N of type (oe ) M j name PCF N iff for all closed terms P of type oe, M P ) j name PCF (N P ) In other words, two functional terms are observationally congruent iff they are congruent when applied. Thus, the frame satisfies the extensionality property. To ....

Robin Milner. Fully abstract models of the typed lambda calculus. Theoretical Computer Sci., 4:1-- 22, 1977.

....[36] reveals that the fully abstract model for the base language VPCF need only be constructed out of continuous, extensional functions; other mathematically attractive properties of the model are simply not needed. In a companion paper [38] we show how to construct using techniques of Milner [21] fully abstract models of VPCF and FPC (a sequential language with recursive types [9] in which the semantics of VPCF S can be encoded) called V and S. While our proofs only require the model V to be fully abstract, the intimate connection between V and S requires that the model S also not ....

....model V of VPCF. One obvious way to construct a fully abstract model of V is to take the term model consisting of equivalence classes of all closed terms. Unfortunately the term model for VPCF, while trivially fully abstract, is not continuous. However, by a suitable context lemma (cf. [21]) the term model for a purely functional call by value language is always monotone. Thus, if all types in the term model are finite, then it is also continuous. We rely upon this insight to prove Theorem 12. Instead of VPCF, we consider a language FVPCF in which all types are finite and which ....

R. Milner. Fully abstract models of the typed lambda calculus. Theoretical Computer Sci., 4:1--22, 1977.

....However, as in the lazy calculus, the j rule holds if M is convergent: Theorem8 conditional extensionality. H[ M ] hpi p implies H[ x:Mx] H[ M ] for x 62 fv(M ) Proof: Use Proposition 4 and the definition of the encoding. 6 Full abstraction Full abstraction, first studied by Milner [Mil77] and Plotkin [Plo77] is the problem of finding a denotational interpretation for a programming language such that the resulting semantic equality coincides with a notion of operational indistinguishability. Inspired by the work of Milner and Park in concurrency [Par81, Mil89] Abramsky [Abr87] ....

.... semantic domain (restrictive approach) ffl to enrich the language (expansive approach) These two approaches are exemplified by the solutions to the full abstraction problem for PCF (a typed calculus extended with fixed points, boolean and arithmetic features) proposed by Milner [Mil77] and Plotkin [Plo77] in the latter, PCF is augmented with a parallel or operator. We shall see that in our case both directions lead to interesting constructions. 6.1 The restrictive approach The first approach exploits the possibility of quantifying barbed bisimulation on a particular class ....

R. Milner. Fully abstract models of typed lambda calculus. Theoretical Computer Science, 4:1--22, 1977.

....language, observational congruences can be used to construct a model in the sense described here. More precisely, a model can be built for most call by name, functional languages in which observational congruence satisfies the extensionality property (also called Milner s Context Lemma [12]) The reader should keep in mind that the theorems are generally applicable to program optimizations, even though the presentation will be based on the more abstract characterization of models. 2 Algebra and the Simply Typed calculus We now briefly review the syntax and Henkin semantics of the ....

R. Milner. Fully abstract models of the typed lambda calculus. Theoretical Computer Sci., 4:1--22, 1977.

....such model to be fully abstract for pcf. The condition is that all compact elements of the model be pcf definable (see Theorem 3.4.2) Plotkin s result leaves open the question of whether there is a denotational model which is fully abstract for pcf proper. This was quickly answered by Milner [ Milner, 1977 ] Theorem 1.1.2 (Milner) There is a unique (up to isomorphism) orderextensional, continuous, inequationally fully abstract model for pcf. 1.2 Full abstraction problem for pcf What then is the full abstraction problem for pcf While there is widespread agreement that the problem is a ....

....sense) for pcf. As Plotkin already intimated in [ Plotkin, 1977 ] the key to the solution is an abstract characterization of sequential computation. To do that, one needs a proper understanding of sequentiality. The matter is straightforward in the case of first order computation. Milner [ Milner, 1977 ] and others have already obtained a number of satisfactory abstract descriptions of first order sequential functions. Intuitively, the meaning of sequential computation is clear enough: it is to do one thing at a time at any intermediate stage of the computation, and possibly in a specific ....

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R. Milner. Fully abstract models of typed lambda-calculus. Theoretical Computer Science, 4:1--22, 1977.

....VPCF K also work for VPCF Gammapif0 K: Theorem 8 Suppose M and N are closed VPCF Gammapif0 terms of type oe. Then the operational semantics of Table 6 satisfies the three properties of Theorem 7. The proof techniques used in proving Theorem 4 work in a Milner style model of VPCF Gammapif0 (cf. [8]) We are currently investigating whether such theorems can be extended to more general situation, e.g. VPCF Gammapif0 S. There is also a large amount of practical work left to be done. For instance, the optimizations described in Section 4 should have similar counterparts in VPCF K and other ....

R. Milner. Fully abstract models of the typed lambda calculus. Theor. Comp. Sci., 4:1--22, 1977.

....C[X] and for any program value n, if C[s] and C[t] are both programs and if C[s] n then C[t] n. The Full Abstraction Problem for pcf is concerned with the search for an abstract, syntaxindependent order fully abstract model of pcf. This problem has been studied intensively for over 15 years [Plo77, Mil77, Cur93, Ong95]. The next two sections set the scene: we give a racy overview of the category of arenas and innocent strategies in section 2, and sketch the fully abstract game interpretation of pcf in section 3. The reader is directed to [HO94] for a full account in which proofs of all results cited in the ....

R. Milner. Fully abstract models of typed lambda-calculus. Theoretical Computer Science, 4:1--22, 1977.

....operational approximation the denotational model is said to be fully abstract. Shortly after Plotkin s paper, Milner proved that there was exactly Basic Research in Computer Science, Centre of the Danish National Research Foundation. one fully abstract model of PCF meeting certain conditions [16]. Until recently, all descriptions of this fully abstract model have used operational semantics (see, for instance, 19, 34] New constructions using games semantics [2, 10, 20] and logical relations [22, 33] have yielded a more abstract understanding of PCF. The sequentiality problem is enduring ....

....values are not definable, and yields insight into the structure of sequential computation. There are other ways to build fully abstract models for FPC. For instance, Riecke and Viswanathan [31, 32] give a dcpo based model for call by value FPC. The construction uses Milner s syntactic methods of [16]. This construction sheds little light into the structure of FPC, except that the model validates least fixpoint reasoning. Games semantics has also been applied to full abstraction questions for FPC. In [15] McCusker builds a model of call by name FPC. The sums in this model are separated: ....

R. Milner. Fully abstract models of the typed lambda calculus. Theoretical Computer Sci., 4:1--22, 1977.

.... Delta Cm [t] and so, since s app t, we have tC 1 [t] Delta Delta Delta Cm [t] j C[t] v. The Context Lemma tells us that applicative contexts alone are enough to determine the observational preorder (between closed terms) The proof we have presented is due to Berry [2] Milner [15] has a similar result for a family of languages expressed in the formalism of Combinatory Logic. 4.3 Another proof of the Context Lemma We present a proof due to Martin Hyland. Fix hole variables X and Y . We define by simultaneous recursion: C Y : Y j c j x j (x A :C Y ) j (C Y C Y ) j Y ....

....] ae d 1 Delta Delta Delta d n : Proposition 5.1(ii) By algebraicity again, we have [ s ] ae v [ t ] ae and we are done. 8. 5 A restrictive approach to full abstraction 39 Milner has shown that there is an order extensional fully abstract model for PCF, and it is unique up to isomorphism [15]. Theorem 8.5 (Milner) There is a unique (up to isomorphism) inequationally fully abstract, order extensional, continuous model of PCF. Milner s model is essentially a term model. The construction uses a suitable completion of terms quotiented by an appropriate equivalence relation in a way ....

R. Milner. Fully abstract models of typed lambda-calculus. Theoretical Computer Science, 4:1--22, 1977.

....category of Scott domains all of whose compact elements are definable in pcf. Using Stoughton s Theorem [Sto88] the model can be extensionally collapsed by means of a continuous homomorphism to the least fixpoint, order extensional, fully abstract model which is shown to be unique by Milner [Mil77]. It is unclear at this stage how our model relates to that of Abramsky et al. Our model of computation is based on a kind of game in which each play consists of a dialogue of questions and answers between two players. This approach is very concrete and in nature goes back to Kleene [Kle78] and ....

R. Milner. Fully abstract models of typed lambda-calculus. Theoretical Computer Science, 4:1--22, 1977.

....7 Gamma1 = Gamma1 s Gamma1 s Delta j Figure 2: A semi sub lattice of preorders improvement relation if and only if they are contained in a certain kind of simulation relation. This is a form of context lemma e.g. (Milner 1977; Abramsky 1990; Howe 1989) and the proof of the characterisation uses previous technical results concerning a more general class of improvement relations (Sands 1991) In (Sands 1991) we abstracted over the way improvement is defined in terms of the operational semantics, and following Howe ....

Milner, R. (1977). Fully abstract models of the typed lambda-calculus. Theoretical Computer Science 4(1), 1--22.

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Robin Milner. Fully abstract models of typed lambda-calculus. Theoretical Computer Science, 4:1--22, 1977.

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R. Milner. Fully abstract models of typed lambda calculus. Theoretical Computer Science, 4:1--22, 1977.

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R.Milner. Fully abstract models of typed lambda-calculus. Theoretical Computer Science 4, 1-22, 1977.

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