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## Full Abstraction for PCF (1996)

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Venue: | INFORMATION AND COMPUTATION |

Citations: | 253 - 16 self |

### Citations

4101 | Communicating sequential processes
- Hoare
- 1978
(Show Context)
Citation Context ...ne σ‖τ = {s ∈ (MA +MB +MC)⋆ | s↾A,B ∈ σ, s↾B,C ∈ τ} σ; τ = {s↾A,C | s ∈ σ‖τ}even. This definition bears a close resemblance to that of “parallel composition plus hiding” in the trace semantics of CSP =-=[Hoa85]-=-; see [AJ94a] for an extended discussion of the analogies between game semantics and concurrency semantics, and [Abr94] for other aspects. We now describe composition in terms of the functions inducin... |

1417 |
Introduction to Lattices and Orders
- Davey, Priestley
- 1990
(Show Context)
Citation Context ...ting all occurrences of symbols not in Y from s. We write s ⊑ t if s is a prefix of t, i.e. for some u, su = t. We always consider sequences under this prefix ordering and use order-theoretic notions =-=[DP90]-=- without further comment. Given a family of sets {Xi}i∈I we write ∑ i∈I Xi for their disjoint union (coproduct); we fix ∑ i∈I Xi = {(i, x) | i ∈ I, x ∈ Xi} as a canonical concrete representation. In p... |

562 |
Recursively enumerable sets and degrees
- Soare
- 1987
(Show Context)
Citation Context ...Recursive Strategies We shall develop effective versions of G and E . Our treatment will be very sketchy, as the details are lengthy and tedious, but quite routine. We refer to standard texts such as =-=[Soa87]-=- for background. We say that a game A is effectively given if there is a surjective map eA : ω →MA with respect to which λA (with some coding of {P,O,Q,A}) and the characteristic functions of PA and ≈... |

536 | Domain theory
- Abramsky, Jung
- 1994
(Show Context)
Citation Context ...n trees. However, infinite strategies will not admit a well-founded inductive decomposition process. Instead, we must appeal to notions of continuity and approximation, in the spirit of Domain Theory =-=[AJ94b]-=-. We define a PCF type-in-context ([Cro94]) to be a type of the form (T1& . . .&Tp)⇒ U where T1, . . . , Tp, U are PCF types. Given such a type-in-context T , we will write Str(T ) for the set of stra... |

453 |
LCF considered as a programming language
- Plotkin
- 1977
(Show Context)
Citation Context ...shion. There have, to our knowledge, been just four models of PCF put forward as embodying some semantic analysis. Three are domain-theoretic: the “standard model” based on Scott-continuous functions =-=[Plo77]-=-; Berry’s bidomains model based on stable functions [Ber79]; and the Bucciarelli-Ehrhard model based on strongly stable functions [BE91]. The fourth is the Berry-Curien model based on sequential algor... |

240 | Games and full completeness for multiplicative linear logic
- Abramsky, Jagadeesan
- 1994
(Show Context)
Citation Context ...ts of the present paper are recast in an abstract, axiomatic form in [Abr00]. There have also been some significant applications of game semantics, notably [MH99, GM00]. 2 The Model We shall refer to =-=[AJ94a]-=- for general background and motivation on game semantics. We begin by fixing some notation. If X is a set, we write X⋆ for the set of finite sequences (words, strings) on X . We shall use s, t, u, v a... |

174 |
The Lambda Calculus: Its Syntax and Semantics. NorthHolland, revised edition
- Barendregt
- 1984
(Show Context)
Citation Context ...nguish between these observationally equivalent terms. However, note that our results in Section 4 apply directly to PCF. 3.2 Evaluation Trees We shall now describe a suitable analogue of Böhm trees =-=[Bar84]-=- for PCFc. These give an (infinitary) notion of normal forms for PCFc terms, and provide a bridge between syntax and semantics. We use Γ,∆ to range over type environments x1 : T1, . . . , xk : Tk. We ... |

122 |
Fully abstract models of typed lambda-calculus
- Milner
- 1977
(Show Context)
Citation Context ...terization” of the fully abstract model (by which we mean the inequationally fully abstract order-extensional model, which Milner proved to be uniquely specified up to isomorphism by these properties =-=[Mil77]-=-). The problem is to understand what should be meant by a “semantic characterization”. Our view is that the essential content of the problem, what makes it important, is that it calls for a semantic c... |

121 | Towards a geometry of interaction - Girard - 1989 |

93 |
Sequential algorithms on concrete data structures
- Berry, Curien
- 1982
(Show Context)
Citation Context ...y’s bidomains model based on stable functions [Ber79]; and the Bucciarelli-Ehrhard model based on strongly stable functions [BE91]. The fourth is the Berry-Curien model based on sequential algorithms =-=[BC82]-=-.1 Of these, we can say that the standard model gives a good account of functional computation at higher types, but fails to capture sequentiality, while the sequential algorithms model gives a good a... |

93 |
Categories for Types
- Crole
- 1993
(Show Context)
Citation Context ... not admit a well-founded inductive decomposition process. Instead, we must appeal to notions of continuity and approximation, in the spirit of Domain Theory [AJ94b]. We define a PCF type-in-context (=-=[Cro94]-=-) to be a type of the form (T1& . . .&Tp)⇒ U where T1, . . . , Tp, U are PCF types. Given such a type-in-context T , we will write Str(T ) for the set of strategies on the game S(T ). The Unicity of D... |

75 | Hereditarily sequential functionals
- Nickau
- 1994
(Show Context)
Citation Context ...a, Mal93]. Further work is needed to understand more fully the relationship between the two approaches. 3 Independently, Hanno Nickau obtained essentially the same model and results as Hyland and Ong =-=[Nic94]-=-. A very different description of the fully abstract model for PCF was obtained by Peter O’Hearn and Jon Riecke, using Kripke logical relations [OR95]. This construction is very interesting, and proba... |

73 | Full abstraction for pcf (extended abstract
- Abramsky, Jagadeesan, et al.
- 1994
(Show Context)
Citation Context ...ne and September 1993. An extended abstract of the present paper appeared in the Proceedings of the Second Symposium on Theoretical Aspects of Computer Science, which was held in Sendai in April 1994 =-=[AJM94]-=-. Independently, and essentially simultaneously, Martin Hyland and Luke Ong gave a different model construction, also based on games and strategies, which led to the same model of PCF, and essentially... |

70 |
The Formal Semantics of Programming Languages. Foundation of Computing Series
- Winskel
- 1993
(Show Context)
Citation Context ...therwise : Ω endcase else Ω These functions have been defined using some “syntactic sugar”. Standard techniques can be used to transform these definitions into PCF syntax. In particular Bekic̆’s rule =-=[Win93]-=- can be used to transform a finite system of simultaneous recursion equations into iterated applications of the Y combinator. The universal term UT can then be defined by UT = FT emptycontextT . It re... |

70 | Game semantics
- Abramsky, McCusker
- 1998
(Show Context)
Citation Context .... These results all build on the concepts, methods and results developed in the present paper, and that of Hyland and Ong. For an expository account of some of these results, and some references, see =-=[AM99]-=-; there is an overview in [Abr97]. The main results of the present paper are recast in an abstract, axiomatic form in [Abr00]. There have also been some significant applications of game semantics, not... |

63 | A Per model of polymorphism and recursive types - Abadi, Plotkin - 1990 |

57 |
Full abstraction for sequential languages: the state of the art
- Berry, Curien, et al.
- 1986
(Show Context)
Citation Context ....6(ii), using continuity, and hence the fact that only finitely many calls to D,H and B are made in evaluating FS⌈σ⌉ΓM̃ . (This can be made precise using Berry’s Syntactic Approximation Lemma for PCF =-=[BCL85]-=-). 46 Theorem 5.8 (Universality Theorem) For all PCF types T and recursive strategies σ ∈ Str(T ) with n = ⌈σ⌉, M(K!(G))JUTnK ≃T σ. Thus every functional in M(Erec) (equivalently, every functional i... |

54 |
Proofs as processes
- Abramsky
- 1994
(Show Context)
Citation Context ...ion is the copy-cat strategy as defined in [AJ94a]. As a set of positions, this strategy is defined by: idA = {s ∈ P even A⊸A | s↾1 = s↾2}. In process terms, this is a bi-directional one place buffer =-=[Abr94]-=-. These copy-cat strategies are the identity morphisms in G. 8 Composition The composition of (history-free) strategies can similarly be defined either in terms of the set representation, or via the u... |

49 |
Geometry of interaction 1: Interpretation of system F
- Girard
- 1989
(Show Context)
Citation Context .... For example, the twist map MPA +M O A ∼=MOA +M P A corresponds to the matrix ( 0 idMO A idMPA 0 ) where 0 is the everywhere-undefined partial function. (Compare the interpretation of axiom links in =-=[Gir89a]-=-.) The strategy induced by this function is the copy-cat strategy as defined in [AJ94a]. As a set of positions, this strategy is defined by: idA = {s ∈ P even A⊸A | s↾1 = s↾2}. In process terms, this ... |

47 | Abramsky and Radha Jagadeesan. Games and full completeness for multiplicative linear logic - Samson - 1994 |

42 |
Modèles complètement adéquats et stables des lambda-calculs typés. Thèse de Doctorat d’Etat, Université de Paris VII
- Berry
- 1979
(Show Context)
Citation Context ...of PCF put forward as embodying some semantic analysis. Three are domain-theoretic: the “standard model” based on Scott-continuous functions [Plo77]; Berry’s bidomains model based on stable functions =-=[Ber79]-=-; and the Bucciarelli-Ehrhard model based on strongly stable functions [BE91]. The fourth is the Berry-Curien model based on sequential algorithms [BC82].1 Of these, we can say that the standard model... |

42 | Observable sequentiality and full abstraction - Cartwright, Felleisen - 1992 |

37 | Reasoning about Idealized Algol using regular languages - Ghica, McCusker - 2000 |

34 | Kripke logical relations and PCF
- O’Hearn, Riecke
- 1995
(Show Context)
Citation Context ...lly the same model and results as Hyland and Ong [Nic94]. A very different description of the fully abstract model for PCF was obtained by Peter O’Hearn and Jon Riecke, using Kripke logical relations =-=[OR95]-=-. This construction is very interesting, and probably of quite general applicability, but does not appear to us to embody a specific semantic analysis of sequentiality. Since the results described in ... |

33 |
Ein dialogisches Konstruktivitätskriterium. In: Infinitistic Methods
- Lorenzen
- 1961
(Show Context)
Citation Context ...ve been obtained by somewhat different routes. Hyland and Ong’s approach is based on dialogue games and innocent strategies, in the tradition of Lorentzen’s dialogue interpretations of logical proofs =-=[Lor60, Lor61]-=-, and the work by Kleene and Gandy on the semantics of higher-type recursion theory [Gan93], while our approach is closer to process semantics and the Geometry of Interaction [AJ94a, Mal93]. Further w... |

30 |
Logik und agon
- Lorenzen
- 1960
(Show Context)
Citation Context ...ve been obtained by somewhat different routes. Hyland and Ong’s approach is based on dialogue games and innocent strategies, in the tradition of Lorentzen’s dialogue interpretations of logical proofs =-=[Lor60, Lor61]-=-, and the work by Kleene and Gandy on the semantics of higher-type recursion theory [Gan93], while our approach is closer to process semantics and the Geometry of Interaction [AJ94a, Mal93]. Further w... |

29 |
Fully Abstract Models of Programming Languages
- Stoughton
- 1988
(Show Context)
Citation Context ...reorder at all types. In particular, it is reflexive at all types. This says that all elements of the model have extensional (functional) behaviour—there is no junk. We can now apply Theorem 7.2.2 of =-=[Sto88]-=- to conclude that M can be collapsed by a continuous homomorphism to the fully abstract model. In short, the fully abstract model is the extensional collapse of any intensionally fully abstract model.... |

28 | Full abstraction in the lazy -calculus - Abramsky, Ong - 1989 |

26 |
Algebraic Theories. Volume 26 of Graduate Texts in Mathematics
- Manes
- 1976
(Show Context)
Citation Context ...o the pairing function. Proposition 2.14 For all σ : !A→ B, τ : !B → C: (m1) σ†; τ† ≈ (σ†; τ)† (m2) der†A;σ ≈ σ (m3) σ†; derB ≈ σ. As an immediate consequence of this Proposition and standard results =-=[Man76]-=-: Proposition 2.15 (!, der, (·)†) is a comonad in “Kleisli form”. If we define, for σ : A → B, !σ = (derA;σ)† : !A→ !B, and δA : !A → !!A by δA = id † !A , then (!, der, δ) is a comonad in the standar... |

26 |
Full Abstraction and Semantics Equivalence
- Mulmuley
- 1987
(Show Context)
Citation Context ...collapsing map is a homomorphism, and in particular preserves application. This contrasts sharply with “collapses” of the standard model to obtain the fully abstract model, as in the work of Mulmuley =-=[Mul87]-=- and Stoughton and Jung [JS93], which are only homomorphic on the “inductively reachable” subalgebra. Thus we propose that a reasonable factorization of the full abstraction problem is to look for a s... |

25 | Projecting sequential algorithms on strongly stable functions
- Ehrhard
- 1996
(Show Context)
Citation Context ... to non-functional aspects of the behaviour of their arguments. The bidomains model also contains non-sequential functions; while the strongly stable model, in the light of a recent result by Ehrhard =-=[Ehr]-=-, can be seen as the “extensional collapse” of the sequential algorithms model. In short, all these models are unsatisfactory because they contain “junk”. On the other side of the coin, we have Milner... |

24 | Definability and full abstraction - Curien |

24 | Non-deterministic games and program analysis: an application to security - Malacaria, Hankin - 1999 |

21 | Geometry of interaction 2: deadlock-free algorithms - Girard - 1988 |

17 |
Studying the fully abstract model of PCF within its continuous function model
- Jung, Stoughton
- 1993
(Show Context)
Citation Context ...m, and in particular preserves application. This contrasts sharply with “collapses” of the standard model to obtain the fully abstract model, as in the work of Mulmuley [Mul87] and Stoughton and Jung =-=[JS93]-=-, which are only homomorphic on the “inductively reachable” subalgebra. Thus we propose that a reasonable factorization of the full abstraction problem is to look for a semantic presentation of an int... |

13 |
Finitary PCF is undecidable
- Loader
- 2000
(Show Context)
Citation Context ...t each functional type of Finitary PCF. The same point occurs in one form or another with all the currently known descriptions of the fully abstract model for PCF. A remarkable result by Ralph Loader =-=[Loa96]-=- shows that this is in fact inevitable. Loader proved that observation equivalence for Finitary PCF is undecidable. This shows that an intensional description of the fully abstract model is the best t... |

12 | Games and full abstraction for the lazy λ-calculus - Abramsky, McCusker - 1995 |

11 | Games in the Semantics of Programming Languages
- Abramsky
(Show Context)
Citation Context ... concepts, methods and results developed in the present paper, and that of Hyland and Ong. For an expository account of some of these results, and some references, see [AM99]; there is an overview in =-=[Abr97]-=-. The main results of the present paper are recast in an abstract, axiomatic form in [Abr00]. There have also been some significant applications of game semantics, notably [MH99, GM00]. 2 The Model We... |

8 | Axioms for Definability and Full Completeness
- Abramsky
- 2000
(Show Context)
Citation Context ...or an expository account of some of these results, and some references, see [AM99]; there is an overview in [Abr97]. The main results of the present paper are recast in an abstract, axiomatic form in =-=[Abr00]-=-. There have also been some significant applications of game semantics, notably [MH99, GM00]. 2 The Model We shall refer to [AJ94a] for general background and motivation on game semantics. We begin by... |

7 |
Extensional embedding of a strongly stable model of PCF
- Bucciarelli, Ehrhard
- 1991
(Show Context)
Citation Context ...etic: the “standard model” based on Scott-continuous functions [Plo77]; Berry’s bidomains model based on stable functions [Ber79]; and the Bucciarelli-Ehrhard model based on strongly stable functions =-=[BE91]-=-. The fourth is the Berry-Curien model based on sequential algorithms [BC82].1 Of these, we can say that the standard model gives a good account of functional computation at higher types, but fails to... |

3 |
Game semantics for exponentials. Announcement on the types mailing list
- Abramsky, Jagadeesan
- 1993
(Show Context)
Citation Context ...ned in general on history-free strategies in G; however, it can be defined on the co-Kleisli category for the comonad !, as we will see. 2.7 Exponentials Our treatment of the exponentials is based on =-=[AJ93]-=-. The game !A is defined as the “infinite symmetric tensor power” of A. The symmetry is built in via the equivalence relation on positions. • M!A = ω ×MA = ∑ i∈ωMA, the disjoint union of countably man... |

3 |
Dialogues, Blass games and sequentiality for objects of finite type. Unpublished manuscript
- Gandy
- 1993
(Show Context)
Citation Context ...s and innocent strategies, in the tradition of Lorentzen’s dialogue interpretations of logical proofs [Lor60, Lor61], and the work by Kleene and Gandy on the semantics of higher-type recursion theory =-=[Gan93]-=-, while our approach is closer to process semantics and the Geometry of Interaction [AJ94a, Mal93]. Further work is needed to understand more fully the relationship between the two approaches. 3 Indep... |

3 | Dalle macchine a ambienti alla geometria dell’interazione. Unpublished manuscript - Malacaria - 1993 |

3 | Crole. Categories for Types - Roy - 1993 |

1 | Abramsky. Proofs as processes - Samson - 1994 |

1 | write \titlerunninghead{<(Shortened) Article Title>} in file ! 59 AJ93 - Please - 1993 |

1 | Abramsky and Guy McCusker. Game Semantics - Samson - 1999 |

1 | Blass games and sequentiality for objects of finite type. Unpublished manuscript - Dialogues - 1993 |