M. Hyland and L. Ong. On full abstraction for PCF. To appear in Information and Computation.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....motivation is purely mathematical. Extending games semantics to the probabilistic case is a natural step to take, just as probabilistic strategies are extremely natural in VonNeumann matrix games. Indeed, our category smoothly extends the deterministic games world charted out by Hyland Ong [13] and Nickau [20] and further developed by Abramsky McCusker [3, 4, 5, 19] Abramsky, Honda McCusker [2] and Laird [15] The category also fits in very nicely with the basic concepts of probability theory. The interaction of a strategy and a counter strategy on a game gives rise to a Borelian ....

J. M. E. Hyland and C.-H. L. Ong. On full abstraction for PCF: I, II and III. To appear in Information and Computation, 200?

....terms of rigidity, and not in terms of universality. Definition 3 Let A be an object of a category C. X is the set of all finite cliques (sets of points of the web of X) of X, and not of all finite multi cliques (multisets of points of the web of X) In the games considered e.g. in the papers [AJM94, HO94, Nic94], repeated moves are allowed in the interpretation of . 7 These sequential algorithms are not really standard: they are sequential algorithms on sequential data structures (see [Cur94] equipped with a notion of complete plays. This notion can be defined inductively on the construction of ....

Martin Hyland and Luke Ong. On full abstraction for PCF. To appear in Information and Computation, 1994.

.... (because upon receiving 1 for x the term launches a new computation instead of returning immediately) Bracketedness also has some considerable e#ect: namely throws are to the just above, and one can dispose of #s and [#]s altogether, which brings back the Hyland Ong finite canonical forms of [HO94]. Combining both constraints gets us even further down to the ground language of simply typed # calculus with unary partial functions on Integers, which can be described as rbt without #s. 4 Factorization We can rework the decomposition, but from within the model, in a more intrinsic way. We re ....

J. M. E. Hyland and C.-H. L. Ong. On full abstraction for PCF: I, II and III. To appear in Information and Computation, 200?, 1994.

....linear logic was given in [Lam94] while the ones in [LafStr91] or [Mey94] the latter for predicate logic without contractions) are not intensional. Subsequently this lead in 1993 to the development of intensional game theoretical models in the semantics of programming languages independently by [AbrJagMal94, HylOng1, Ni96]. These models proved to be very useful and provided e.g. a solution for the probably best known open problem in the semantics of programming languages, the Full Abstractness Problem for the programming language PCF [Pl77] by giving the first syntaxindependent fully abstract model. PCF is a ....

....of internal properties of the strategies involved (this will be made more precise below) 110 JAN J URJENS 3. DEFINITIONS To put the above intuitive examples on a more solid foundation, we will now provide the underlying definitions. They appeared in [McC98] and are essentially an adaption of [HylOng1], taking account of ideas in [AbrJagMal1] 3.1. Games and Strategies. Definition 1. An arena is a structure A = MA ; A ; A ) consisting of ffl a set of moves MA , ffl the labelling function A : MA fS; Eg Theta fQ; Ag (call moves labelled (S; l) resp. E; l) for l 2 fQ; Ag) S moves ....

J. M. E. Hyland, C.-H. L. Ong, On Full Abstraction for PCF: I, II and III, to appear in: Information and Computation

....: for both call by name and call by value evaluations. Introduction Game semantics has been used to interpret both logical systems and programming languages and to give completeness results. Most of the game models for typed languages have been restricted to intuitionistic logic [1, 12], corresponding to the technical hypothesis that one considers games where only Opponent may play the rst move. However this restriction prevents us from taking advantadge of the natural duality of game semantics, namely the symmetry Opponent Player. It has proved dicult to extend the game ....

....and call by value evaluations giving a uniform setting for interpreting them. Negative games are the usual intuitionistic ones, but we add their dual: positive games where the rst move belongs to Player. It is therefore not surprising that our model is a conservative extension of the HO one [12] which corresponds to the negative part of ours. In particular HO s tensor product is a negative operation (applied to and yielding a negative game) This seems to contradict the polarity rule of the linear tensor (which is a positive operation) and suggests that is a new construction. ....

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Martin Hyland and Luke Ong. On full abstraction for PCF. To appear in Information and Computation, 2000.

....games semantics to the probabilistic case is a natural step to take, just as probabilistic strategies are extremely natural in Von Neumann matrix games. Indeed, the category of probabilistic strategies that we construct smoothly extends the deterministic games world as charted out by Hyland Ong [13] and Nickau [20] and further developed by Abramsky McCusker [3, 4, 5, 19] Abramsky, Honda McCusker [2] and Laird [15] The category also fits in very nicely with the basic concepts of probability theory. The interaction of a strategy address: Equipe Preuves Programmes Syst emes, UFR de ....

J. M. E. Hyland and C.-H. L. Ong. On full abstraction for PCF: I, II and III. To appear in Information and Computation, 200?, 1994.

....Oxford University Computing Laboratory, Parks Road, Oxford. OX1 3QD November 1998, revised September 1999 Abstract We present a new denotational model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [HO94] and Nickau [Nic96] but the traditional distinction between question and answer moves is removed. We rst construct models D and DREC as global sections of a re exive object in the categories A and A REC of arenas and innocent and recursive innocent strategies respectively. We show that ....

....a new denotational model for untyped calculus, using the techniques of game semantics. A game model of the lazy calculus was given by Abramsky and McCusker in [AM95] using the history free strategies of [AJM94] The particular variety of game used here is a fairly simple modi cation of that in [HO94] for a fully abstract model of PCF the distinctions of question and answer are not needed. In Section 2 we present some standard de nitions, rephrased to take this into account. To begin with, we de ne an arena, which details what moves may be made in the game, and the notion of well formed ....

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J. M. E. Hyland and C.-H. L. Ong. On full abstraction for PCF: I, II and III. To appear in Information and Computation, ftp://ftp.comlab.ox.ac.uk/pub/ Documents/techpapers/Luke.Ong/, 1994.

....it has allowed various full abstraction results to be obtained, showing that games models capture the behaviour of programs in a precise way. Furthermore, the paradigm is flexible enough to furnish fully abstract models of a wide variety of languages, ranging from purely functional languages [1, 4, 13, 16, 17, 19], to languages with control operators [15] mutable variables [3, 5] and even general references [2] However, despite being grounded on mathematical structures usually associated with concurrency, all of this work has focussed on sequential, deterministic languages. Contribution In this paper we ....

J. M. E. Hyland and C.-H. L. Ong. On full abstraction for PCF: I, II and III. To appear in Information and Computation, 199?

.... We aim to construct a universal model (i.e. every element of the model is the denotation of some term) of the pure untyped calculus which induces the B ohm tree theory B, by building on the game models presented in [KNO01] Although the general approach is innocent in the sense of [HO00] and [Nic96] the two player games we use are simpler and can be considered a special case where moves are neither questions nor answers but simply declarations . A notable feature of game semantics is that the de nable strategies are e ective methods for copying moves uniformly from one ....

....Review of Previous Work We begin with a brief summary of the results of [KNO01] on which we build in the body of this paper. The de nitions in this section are only reviewed in a cursory way; the reader is referred to [KNO01] for a more detailed exposition and proofs of the later results, and to [HO00] and [McC98] for proofs of the initial results regarding categories of games. 2.1 Arenas and Innocent Strategies We begin with a word on trees. Usually, when we talk about a tree we mean a countablybranching labelled tree, and presented in a concrete way. Often, the labels will be sequences in ....

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J. M. E. Hyland and C.-H. L. Ong. On full abstraction for PCF: I, II and III. To appear in Information and Computation, 2000.

....Andrew D. Ker, Hanno Nickau and C. H. Luke Ong Oxford University Computing Laboratory, Parks Road, Oxford. OX1 3QD Abstract We present a new denotational model for the untyped calculus, using the techniques of game semantics. The strategies used are innocent in the sense of Hyland and Ong [9] and Nickau [17] but the traditional distinction between question and answer moves is removed. We rst construct models D and DREC as global sections of a re exive object in the categories A and A REC of arenas and innocent and recursive innocent strategies respectively. We show that these ....

....a new denotational model for untyped calculus, using the techniques of game semantics. A game model of the lazy calculus was given by Abramsky and McCusker in [2] using the history free strategies of [1] The particular variety of game used here is a fairly simple modi cation 2 of that in [9] for a fully abstract model of PCF the distinctions of question and answer are not needed. In Section 2 we present some standard de nitions, rephrased to take this into account. To begin with, we de ne an arena, which details what moves may be made in the game, and the notion of well formed ....

[Article contains additional citation context not shown here]

J. M. E. Hyland and C.-H. L. Ong. On full abstraction for PCF: I, II and III. To appear in Information and Computation, ftp://ftp.comlab.ox.ac.uk/pub/ Documents/techpapers/Luke.Ong/, 1994.

.... 1 Introduction We aim to construct a universal model (i.e. every element of the model is the denotation of some term) of the pure untyped calculus which induces the Bohm tree theory B, by building on the game models presented in [KNO98] Although the general approach is innocent in the sense of [HO94] and [Nic96] the two player games we use are simpler and can be considered a special case where moves are neither questions nor answers but simply declarations . A notable feature of game semantics is that the definable strategies are effective methods for copying moves uniformly from one ....

....asked for a game model that induces the Bohm tree theory. This paper answers that question. 2 Arenas and Innocent Strategies This Section and the next give a quick introduction to the basic ideas underpinning the main result of the paper. We refer the reader to [KNO98] for further details and to [HO94] and [McC98] for proofs of all results quoted. We define an arena to be a finite tuple of nonempty trees of moves. The root of each tree is called an initial move. Our trees are considered upside down with the root at the top, rather like family trees. We can also refer to the child of a node, ....

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J. M. E. Hyland and C.-H. L. Ong. On full abstraction for PCF: I, II and III. To appear in Information and Computation, ftp-able, 133 pp., 1994.

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M. Hyland and L. Ong. On full abstraction for PCF. To appear in Information and Computation.

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J.M.E.Hyland, C.-H. L. Ong, On Full Abstraction for PCF: I, II and III, to appear in: Information and Computation

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