## A Simple Adequate Categorical Model for PCF (1997)

Venue: | In Proceedings of Third International Conference on Typed Lambda Calculi and Applications |

Citations: | 4 - 0 self |

### BibTeX

@INPROCEEDINGS{Braüner97asimple,

author = {Torben Braüner},

title = {A Simple Adequate Categorical Model for PCF},

booktitle = {In Proceedings of Third International Conference on Typed Lambda Calculi and Applications},

year = {1997},

pages = {82--98},

publisher = {Springer-Verlag}

}

### OpenURL

### Abstract

Usually types of PCF are interpreted as cpos and terms as continuous functions. It is then the case that non-termination of a closed term of ground type corresponds to the interpretation being bottom; we say that the semantics is adequate. We shall here present an axiomatic approach to adequacy for PCF in the sense that we will introduce categorical axioms enabling an adequate semantics to be given. We assume the presence of certain "bottom" maps with the role of being the interpretation of non-terminating terms, but the order-structure is left out. This is different from previous approaches where some kind of order-theoretic structure has been considered as part of an adequate categorical model for PCF. We take the point of view that partiality is the fundamental notion from which order-structure should be derived, which is corroborated by the observation that our categorical model induces an order-theoretic model for PCF in a canonical way.

### Citations

440 | Computational lambda-calculus and monads
- Moggi
- 1989
(Show Context)
Citation Context ...converse to adequacy at exponential types by introducing a monad with appropriate properties and then interpret types in a way that reflects the evaluation strategy following the ideas put forward in =-=[Mog89]. This is -=-the approach taken in [Gun92, Win93] where PCF-like languages are interpreted in the category cpo using the usual "lift" monad. The use of a monad to give an adequate interpretation with the... |

397 |
LCF considered as a programming language
- Plotkin
- 1977
(Show Context)
Citation Context ...as cpos and terms as continuous functions. The possibility of non-termination of programs (that is, closed terms) is then reflected in the presence of bottom elements in cpos. It was shown by Plotkin =-=[Plo77]-=- that non-termination of a program of ground type actually corresponds to the interpretation being the bottom element of the appropriate cpo. A semantics with that property is called adequate; the ade... |

267 | Semantics of Programming Languages: Structures and Techniques. Foundations of Computing - Gunter - 1992 |

263 |
Event structures
- Winskel
- 1986
(Show Context)
Citation Context ...s and continuous stable functions, dI. They are both well known from the literature; the first category is for example described in the textbooks[Win93, Gun92], and the latter category in the article =-=[Win87]-=-. Very recently, there has been considerable progress in axiomatising su#cient conditions for obtaining full abstraction for PCF, [Abr96]. The intuitions leading to this work stem from the theory of g... |

239 | The Formal Semantics of Programming Languages - Winskel - 1993 |

197 |
On full abstraction for PCF
- Hyland, Ong
(Show Context)
Citation Context ...n simply be shown that a pointed cartesian closed category where all maps are strict is equivalent to the one-object-one-map category. 2.2 Numerals The following definition is essentially as given in =-=[HO96]-=-. Note that booleans are represented by numerals. Definition 2. An object of numerals in a cartesian category is an object N equipped with maps zero : 1 # N and succ, pred : N # N such thats0; pred =s... |

192 | Full abstraction for pcf
- Abramsky, Jagadeesan, et al.
- 2000
(Show Context)
Citation Context ...e some kind of order-theoretic structure has been considered as part of an adequate categorical model for PCF. For example, in [BCL86] cpo-enrichment is taken as part of a categorical model, which in =-=[AJM96]-=- is replaced by rationality, that is, poset-enrichment such that only certain increasing chains are assumed to have least upper bounds. Our motivation for adhering to the point of view that partiality... |

75 | Operationally-based theories of program equivalence - Pitts - 1997 |

62 | Axiomatic Domain Theory in Categories of Partial Maps - Fiore - 1994 |

53 |
Full Abstraction for sequential languages: the state of the art
- Berry, Curien, et al.
- 1985
(Show Context)
Citation Context ...accordance with [Fio94b]). This is di#erent from previous approaches where some kind of order-theoretic structure has been considered as part of an adequate categorical model for PCF. For example, in =-=[BCL86]-=- cpo-enrichment is taken as part of a categorical model, which in [AJM96] is replaced by rationality, that is, poset-enrichment such that only certain increasing chains are assumed to have least upper... |

34 | An Axiomatisation of Computationally Adequate Domain-Theoretic Models of FPC - Fiore, Plotkin - 1994 |

31 |
A note on inconsistencies caused by fixpoints in a cartesian closed category
- Huwig, Poigné
- 1990
(Show Context)
Citation Context ... . This would be equivalent to using the natural isomorphism (A +B) �� # # = (A �� # ) + (B �� # ) given by the observation that the functor (-)��# has a right adjoint and thus preserv=-=es sums. But in [HP90]-=- a cartesian closed category with finite sums together with a fixpoint operator is shown to be inconsistent , that is, it is equivalent to the category consisting of one object and one map. So we have... |

26 | An Axiomatic Approach to Adequacy
- Bräuner
- 1996
(Show Context)
Citation Context ...stricted version of rational openness is not only su#cient, but also necessary for the interpretation to be adequate. A more detailed presentation of the results given in this article can be found in =-=[Bra96]. 2 PCF - -=-Semantic Issues 2.1 Undefinedness In this subsection we introduce a notion of undefinedness by assuming the existence of "undefined" maps #A : 1 # A which are supposed to be the interpretati... |

22 |
Lambda definability and logical relations
- Plotkin
(Show Context)
Citation Context ...tatement of the defining diagram for an internal fixpoint operator, Definition 6. ## 5 Adequacy In our adequacy proof for PCF we will use the now standard technique of logical relations introduced in =-=[Plo73]-=-. The original adequacy proof in [Plo77] made use of a unary logical relation CompA # TA called the computability predicate. Recall that TA is the set of programs of type A. The logical relation CompA... |

20 |
Diagonal Arguments and Cartesian Closed Categories
- Lawvere
- 1969
(Show Context)
Citation Context ...ny other element but bottom, and equipped with appropriate extensions of the maps from the standard object of numerals. 2.3 Fixpoints The main concern of this subsection is fixpoints as introduced in =-=[Law69]-=-. Definition 3. Let C be a category with a terminal object. A map h : 1 # B is a fixpoint of a map f : B # B i# h; f = f . A fixpoint operator for an object B is an operation on maps (-) # B : C(B, B)... |

17 |
A type-theoretic alternative to CUCH, ISWIM, OWHY, unpublished manuscript
- Scott
- 1969
(Show Context)
Citation Context ...in a canonical way. 1 Introduction The programming language PCF (an acronym for Programming language for Computable F unctions) was in 1969 introduced by Scott in the until recently unpublished paper =-=[Sco93]-=-. The formal system of PCF consists of the #-calculus augmented with booleans and numerals together with recursion where the usual reduction rules are replaced by an operational semantics. Scott's pio... |

16 |
Axioms for full abstraction and full completeness
- Abramsky
- 1996
(Show Context)
Citation Context ...extbooks[Win93, Gun92], and the latter category in the article [Win87]. Very recently, there has been considerable progress in axiomatising su#cient conditions for obtaining full abstraction for PCF, =-=[Abr96]-=-. The intuitions leading to this work stem from the theory of games rather than from traditional ordertheoretic domain theory, but rational openness is, however, recognised 2 as the essential notion f... |

16 | Continuity and E#ectivity in Topoi - Rosolini - 1986 |

10 | Categories of partial morphisms and the partial lambda-calculus - MOGGI - 1986 |

3 |
First steps on the representation of domains
- Fiore
- 1994
(Show Context)
Citation Context ...nating programs, but the order-structure is left out motivated by the point of view that partiality is the fundamental notion from which order-structure should be derived (which is in accordance with =-=[Fio94b]-=-). This is di#erent from previous approaches where some kind of order-theoretic structure has been considered as part of an adequate categorical model for PCF. For example, in [BCL86] cpo-enrichment i... |