## Logical Full Abstraction and PCF (1996)

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Venue: | Tbilisi Symposium on Language, Logic and Computation. SiLLI/CSLI |

Citations: | 13 - 5 self |

### BibTeX

@INPROCEEDINGS{Longley96logicalfull,

author = {John Longley and Gordon Plotkin},

title = {Logical Full Abstraction and PCF},

booktitle = {Tbilisi Symposium on Language, Logic and Computation. SiLLI/CSLI},

year = {1996},

pages = {333--352},

publisher = {SiLLI/CSLI}

}

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### Abstract

ion and PCF John Longley Gordon Plotkin March 15, 1996 Abstract We introduce the concept of logical full abstraction, generalising the usual equational notion. We consider the language PCF and two extensions with "parallel" operations. The main result is that, for standard interpretations, logical full abstraction is equivalent to equational full abstraction together with universality; the proof involves constructing enumeration operators. We also consider restrictions on logical complexity and on the level of types. 1 Introduction The study of denotational semantics seeks to provide mathematical descriptions of programming languages by giving denotations of programs in terms of previously understood mathematical structures. For example, if P is a program that takes an input and produces an output, we might take its denotation to be a function from a set of input-values to a set of output-values. The most widely-known approach to denotational semantics is that of traditiona...

### Citations

398 |
LCF considered as a programming language
- Plotkin
- 1977
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Citation Context ...ght take its denotation to be a function from a set of input-values to a set of output-values. The most widely-known approach to denotational semantics is that of traditional domain theory (see e.g., =-=[14]-=-), where the mathematical structures involved are certain kinds of complete partial order (cpo). Other kinds of mathematical structure have also been used successfully---for a selection of different a... |

264 | Foundations of Constructive Mathematics - Beeson - 1985 |

192 |
Lectures on a mathematical theory of Computation
- Scott
- 1981
(Show Context)
Citation Context ... here of logical full abstraction results rather in intensional concerns, such as the study of definability and so of computability. These distinctions harken back to Scott's original explicit choice =-=[16, 17]-=- to investigate first extensional theories and only then to consider questions of computability and of the relation with symbolic computation. They also bring to mind the much more recent programme of... |

144 |
Semantics of Programming Languages
- Gunter
- 1992
(Show Context)
Citation Context ... paper should apply not just to the languages we have considered but to a wider class. As regards functional languages, one would certainly wish to consider the lazy and call-by-value variants of PCF =-=[6, 11]-=-. A further useful extension would be to recursively typed languages, such as FPC [6, 4]. It would then be natural to consider polymorphic extensions of PCF; this seems not to be a straightforward mat... |

140 |
P.J.: Introduction to Higher-Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...interpretation is cartesian-closed (by which we mean that the underlying category is cartesian closed, and that the higher-order types, - abstraction and application are interpreted accordingly---see =-=[10]-=-; this property implies that the interpretation models fij-conversion). It follows from operational soundness that any EFA interpretation is sound. Note that the usual interpretation in cpos is sound,... |

132 |
Towards a mathematical semantics for computer languages
- Scott, Strachey
- 1971
(Show Context)
Citation Context ... here of logical full abstraction results rather in intensional concerns, such as the study of definability and so of computability. These distinctions harken back to Scott's original explicit choice =-=[16, 17]-=- to investigate first extensional theories and only then to consider questions of computability and of the relation with symbolic computation. They also bring to mind the much more recent programme of... |

66 | Malacaria P., “Full Abstraction for PCF (extended abstract
- Abramsky, Jagadeesan
- 1994
(Show Context)
Citation Context ...athematical structures involved are certain kinds of complete partial order (cpo). Other kinds of mathematical structure have also been used successfully---for a selection of different approaches see =-=[1, 11, 13]-=-. One of the principal aims of denotational semantics is to deepen our understanding of the logic of programming languages, and to provide conceptual and mathematical tools for reasoning about program... |

60 |
Realizability Toposes and Language Semantics
- Longley
- 1994
(Show Context)
Citation Context ...athematical structures involved are certain kinds of complete partial order (cpo). Other kinds of mathematical structure have also been used successfully---for a selection of different approaches see =-=[1, 11, 13]-=-. One of the principal aims of denotational semantics is to deepen our understanding of the logic of programming languages, and to provide conceptual and mathematical tools for reasoning about program... |

49 |
Tω as universal domain
- Plotkin
(Show Context)
Citation Context ...closed L-terms L oes: oe !sand R oe :s! oe such that x oe :R oe (L oes(x))sx oe :x holds in L. Lemma 13 Every type is a PCF ++ -retract of ' ! '. Proof We use the "effective universality" re=-=marked in [15]-=-, that every effectively given coherent !-continuous cpo is a computable retract of T ! . In the interpretation C of PCF ++ provided by the classical category of cpos, every C[[ oe ]] is such a cpo; f... |

46 |
Fully abstract models of typed -calculi
- Milner
- 1977
(Show Context)
Citation Context ... ; N h : oe h and numerals n, MN 1 : : : N h ! n iff M 0 N 1 : : : N h ! n (a more general version for open terms is easily derived). An operational proof of the Context Lemma for PCF can be found in =-=[12, 3]-=- and similar proofs can be obtained for PCF + and PCF ++ ; for these latter two languages it is also a consequence of the facts that the usual cpo model is adequate and that all finite elements are de... |

34 | An Axiomatisation of Computationally Adequate Domain-Theoretic Models of FPC
- Fiore, Plotkin
- 1994
(Show Context)
Citation Context ... regards functional languages, one would certainly wish to consider the lazy and call-by-value variants of PCF [6, 11]. A further useful extension would be to recursively typed languages, such as FPC =-=[6, 4]-=-. It would then be natural to consider polymorphic extensions of PCF; this seems not to be a straightforward matter. It would be also interesting to formulate an appropriate notion that would allow ou... |

27 |
Fully abstract models of typed λ-calculi
- Milner
- 1977
(Show Context)
Citation Context .... . , Nh : σh and numerals n, MN1 . . . Nh → ∗ n iff M ′ N1 . . . Nh → ∗ n (a more general version for open terms is easily derived). An operational proof of the Context Lemma for PCF can be found in =-=[12, 3]-=- and similar proofs can be obtained for PCF + and PCF ++ ; for these latter two languages it is also a consequence of the facts that the usual cpo model is adequate and that all finite elements are de... |

2 |
Full Abstraction and the Context Lemma (Preliminary Report
- Jim, Meyer
- 1991
(Show Context)
Citation Context ...ting to formulate an appropriate notion that would allow our results to be presented at their natural level of generality. This should at least include suitable extensions of PCF (in which regard see =-=[9]-=-), and perhaps a greater degree of abstraction is obtainable. Finally, we have said very little about axioms and inference rules for J L . It would be useful to work out the details of an axiomatizati... |

2 |
Interdefinability of Parallel Operations in PCF
- Stoughton
- 1991
(Show Context)
Citation Context ...14]; the formulations here differ in two inessential respects: one is the absence of a Boolean type; the other is the use of a "parallel-or" constant rather than a parallel conditional (for =-=which see [18]). The typ-=-es of PCF are built up from a single ground type ' (the natural numbers) using the right-associative binary type constructor !; we write M : oe to mean "M is a term of type oe". For each typ... |

2 |
On representation of sequential and parallel functions
- Trakhtenbrot
- 1975
(Show Context)
Citation Context ...erpretation of Milner's EFA interpretation for PCF [12] does not provide an LFA interpretation, as there exist first-order functions that are effective and sequential but not PCF-definable (see e.g., =-=[19]-=-). We do not know of any natural LFA interpretations for PCF + . Although in this paper we have concentrated mainly on LFA interpretations for the whole of J L , it is also natural to consider logical... |

1 |
see also On Full Abstraction for PCF: I, II and III, by the same authors
- Hyland, Ong
- 1995
(Show Context)
Citation Context ...working in, for example, the effective topos [7]. One should also 2 remark that intensional aspects may nonetheless play a role in the study of equational full abstraction---see the study of games in =-=[1, 8]-=-. The rest of the paper is structured as follows. In Section 2 we review the definitions of the three versions of PCF that we will consider. We also define the syntax of a program logic for these lang... |