## Axioms for Definability and Full Completeness (2000)

Venue: | in Proof, Language and Interaction: Essays in Honour of Robin |

Citations: | 4 - 1 self |

### BibTeX

@INPROCEEDINGS{Abramsky00axiomsfor,

author = {Samson Abramsky},

title = {Axioms for Definability and Full Completeness},

booktitle = {in Proof, Language and Interaction: Essays in Honour of Robin},

year = {2000},

pages = {55--75},

publisher = {MIT Press}

}

### OpenURL

### Abstract

ion problem for PCF (see [BCL86, Cur93, Ong95] for surveys). The importance of full abstraction for the semantics of programming languages is that it is one of the few quality filters we have. Specifically, it provides a clear criterion for assessing how definitive a semantic analysis of some language is. It must be admitted that to date the quest for fully abstract models has not yielded many obvious applications; but it has generated much of the deepest work in semantics. Perhaps it is early days yet. Recently, game semantics has been used to give the first syntax-independent constructions of fully abstract models for a number of programming languages, including PCF [AJM96, HO96, Nic94], richer functional languages [AM95, McC96b, McC96a, HY97], and languages with non-functional features such as reference types and non-local control constructs [AM97c, AM97b, AM97a, Lai97]. A noteworthy feature is that the key definability results for the richer languages are proved by a reduction to...

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Citation Context ...tion. The motivation for this terminology comes from lattice theory (Joyal is generalizing Whitman's theorem on free lattices). A -atom in a lattice (also often called a meet-irreducible element, cf. =-=[DP90]-=-) is an element a such that n i=1 a isa =) 9i: a isa: Generalizing from posets to (enriched) categories, we get the definition given above. An object B is discrete if for each A the canonical map C s ... |

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Citation Context ... with specific reference to PCF. We present axioms on models of PCF from which the key results on definability and full abstraction can be proved. It should be emphasized that not only the results of =-=[AJM96]-=-, but also the top-level structure of the actual proofs, are captured by our axiomatic account. In particular, our main axioms are abstracted from key lemmas in [AJM96]. The axioms mostly take the for... |

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Citation Context ... (2) der y ; f = f: (3) There are moreover natural isomorphisms (the "exponential laws" ) e A;B : !(A \Theta B) = \Gamma! !A\Omega !B e 1 : !1s= \Gamma! 1 satisfying the coherence conditions=-= given in [Bie95]-=-. Every object !A has a cocommutative comonoid structure given by conA = !A !\Delta \Gamma! !(A \Theta A) e A;A \Gamma! !A\Omega !A weakA = !A !t A \Gamma! !1 e 1 \Gamma! 1: The co-Kleisli category K ... |

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Citation Context ...comonad, g y ; derB = g: The content of (A3) is that (f ; derB ) y = f i.e. that the two passages are mutually inverse. This property was proved for categories of games in [AJM96] and subsequently in =-=[McC96a], under the name of -=-the "Bang Lemma". The idea is that morphisms f : !A ! !B must display uniform behaviour in all "threads", i.e. in each copy of B together with its associated copies of A. This prop... |

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Citation Context ...xample [Fio96, Hyl91]. There has also been considerable progress in axiomatizing sufficient conditions for computational adequacy [FP94, Bra97, McC96a]. In another vein, the work on action structures =-=[MMP95]-=- can be seen as an axiomatics for process calculi and other computational formalisms. In the present paper we make the first contribution towards an axiomatic account of full abstraction, with specifi... |

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Citation Context ...eveloped in the present paper can be extended to yield this stronger result. We briefly sketch the necessary extensions. Firstly, we take our sequential categories to be enriched over enumerated sets =-=[AL91]-=- rather than just pointed sets. All the isomorphisms required in the axioms have then to be given effectively. This leads to an effective version of the Decomposition Theorem as in [AJM96]. The develo... |

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

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11 |
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Citation Context ...e form of assertions that some canonical map is an isomorphism, which is quite standard in categorical axiomatizations, for example of Synthetic Differential Geometry [Koc81], distributive categories =-=[Wal92]-=-, or dualizing objects [MOM91]. It is also noteworthy that, although our results apply to intuitionistic types, the axioms make essential use of the linear decompositions of these types [Gir87]. The p... |

5 | Free lattices, communication and money games - Joyal - 1997 |

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Citation Context ...ntinuously observable then K ! (C) is a computationally adequate model of PCF. The original proof by Plotkin for the Scott continuous function model [Plo77] goes through in our axiomatic setting (cf. =-=[Bra97]-=-). In practice, it is often more convenient to verify somewhat stronger axioms. We say that a sequential category C is normed if: 10 ffl It is enriched over algebraic cpo's, with the ?-morphisms being... |

4 |
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Citation Context ...s PCF is prototypical for higher-order programming languages, so is the pure simply typed -calculus for logical systems. (Definability results for game semantics of the pure calculus are discussed in =-=[DHR96]-=-, and were already known to the authors of [AJM96, HO96].) We shall indicate how our axiomatic approach can be modified (in fact: simplified) to deal with the pure calculus. We define a pure sequentia... |

4 |
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Citation Context ...t domains are discrete (any continuous function into a flat domain is either strict or constant); Coh, the category of coherence spaces and linear maps, is soft in the sense of [Joy95a, Joy95b]---see =-=[HJ97]. 2.4 Stan-=-dard datatypes Let C be a category with ?-maps as in Section 2.2. We assume given a class of objects of C which we will call "well-opened", which forms an exponential ideal, i.e. if B is wel... |

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1 |
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Citation Context ... strategies has all its elements definable. This is the strongest possible definability result, and is closely related to the notion of "Logical Full Abstraction" introduced by Longley and P=-=lotkin in [LP96]-=-, as shown loc. cit. The axiomatic methods developed in the present paper can be extended to yield this stronger result. We briefly sketch the necessary extensions. Firstly, we take our sequential cat... |

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Citation Context ... canonical map is an isomorphism, which is quite standard in categorical axiomatizations, for example of Synthetic Differential Geometry [Koc81], distributive categories [Wal92], or dualizing objects =-=[MOM91]-=-. It is also noteworthy that, although our results apply to intuitionistic types, the axioms make essential use of the linear decompositions of these types [Gir87]. The present paper is only a first s... |