## On Natural Non-dcpo Domains

### BibTeX

@MISC{Sazonov_onnatural,

author = {Vladimir Sazonov and Boris Abramovich Trakhtenbrot},

title = {On Natural Non-dcpo Domains},

year = {}

}

### OpenURL

### Abstract

in his 87th year whose influence on me and help cannot be overstated. model for PCF of hereditarily sequential functionals is not ω-complete and therefore not continuous in the traditional terminology (in contrast to the old fully abstract continuous dcpo model of Milner). This is also applicable to a wider class of models such as the recently constructed by the author fully abstract (universal) model for PCF + = PCF + parallel if. Here we will present an outline of a general approach to this kind of “natural ” domains which, although being non-dcpos, allow considering “naturally ” continuous functions (with respect to existing directed “pointwise”, or “natural ” least upper bounds) and also have appropriate version of “naturally ” algebraic and “naturally ” bounded complete “natural ” domains. This is the non-dcpo analogue of the wellknown concept of Scott domains, or equivalently, the complete f-spaces of Ershov. In fact, the latter version of natural domains, if considered under “natural ” Scott topology, exactly corresponds to the class of f-spaces, not necessarily complete. 1

### Citations

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(Show Context)
Citation Context ...ntioned negative result in [6] and further positive results in [10] the domain theoretical structure of such models was essentially 1 As to the language PCF for sequential finite type functionals see =-=[4,7,9,11]-=-. Note also that the technical part of [10] — the source of considerations of the present paper — is heavily based on [8,9]. A. Avron et al. (Eds.): Trakhtenbrot/Festschrift, LNCS 4800, pp. 620–634, 2... |

214 |
On full abstraction for PCF
- Hyland, Ong
- 2000
(Show Context)
Citation Context ...g, where domains are not necessary directed complete partial orders (dcpos). As Dag Normann has recently shown [6], the fully abstract model of hereditarily-sequential finite type functionals for PCF =-=[1,3,5,10]-=- 1 is not ω-complete (hence non-dcpo) and therefore not continuous in the traditional terminology. This is also applicable to a potentially wider class of models such as the fully abstract model of (h... |

202 | Full abstraction for PCF
- Abramsky, Jagadeesan, et al.
- 2000
(Show Context)
Citation Context ...g, where domains are not necessary directed complete partial orders (dcpos). As Dag Normann has recently shown [6], the fully abstract model of hereditarily-sequential finite type functionals for PCF =-=[1,3,5,10]-=- 1 is not ω-complete (hence non-dcpo) and therefore not continuous in the traditional terminology. This is also applicable to a potentially wider class of models such as the fully abstract model of (h... |

48 | A type-theoretical alternative to ISWIM - Scott - 1993 |

30 |
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- 1977
(Show Context)
Citation Context ...ntioned negative result in [6] and further positive results in [10] the domain theoretical structure of such models was essentially 1 As to the language PCF for sequential finite type functionals see =-=[4,7,9,11]-=-. Note also that the technical part of [10] — the source of considerations of the present paper — is heavily based on [8,9]. A. Avron et al. (Eds.): Trakhtenbrot/Festschrift, LNCS 4800, pp. 620–634, 2... |

11 |
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(Show Context)
Citation Context ...s means existence of a lub for any bounded set, not necessarily finite. Algebraic and bounded complete dcpos with least element ⊥ are also known as Scott domains or as the complete f0spaces of Ershov =-=[2]-=- (or just Scott-Ershov domains). For the ‘natural’, non-dcpo version of these domains we need Definition 8. A natural pre-domain D is called naturally bounded complete if any two naturally finite elem... |

6 | On sequential functionals of type 3
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(Show Context)
Citation Context ...t brief outline of the so-called “natural” version of domain theory in the general setting, where domains are not necessary directed complete partial orders (dcpos). As Dag Normann has recently shown =-=[6]-=-, the fully abstract model of hereditarily-sequential finite type functionals for PCF [1,3,5,10] 1 is not ω-complete (hence non-dcpo) and therefore not continuous in the traditional terminology. This ... |

4 |
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- 1976
(Show Context)
Citation Context ...t have a (not necessarily natural) lub d ⊔ e in D which is also finitely restricted. Indeed, it can be obtained as the greatest lower bound in D of a finite nonempty set: d ⊔ e = ⊓{x [k] | x ⊒ d, e}. =-=(9)-=- By induction, given any (not necessary ‘basic’) naturally ω-algebraic and naturally bounded complete domains D and E with such projections, we should conclude that the composition Ψ [k] E ◦ f ◦ Ψ[k] ... |

3 |
Hereditarily-Sequential Functionals: A Game-Theoretic Approach to Sequentiality
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(Show Context)
Citation Context ...g, where domains are not necessary directed complete partial orders (dcpos). As Dag Normann has recently shown [6], the fully abstract model of hereditarily-sequential finite type functionals for PCF =-=[1,3,5,10]-=- 1 is not ω-complete (hence non-dcpo) and therefore not continuous in the traditional terminology. This is also applicable to a potentially wider class of models such as the fully abstract model of (h... |

3 |
Functionals computable in series and in parallel
- Sazonov
- 1976
(Show Context)
Citation Context ...lly 1 As to the language PCF for sequential finite type functionals see [4,7,9,11]. Note also that the technical part of [10] — the source of considerations of the present paper — is heavily based on =-=[8,9]-=-. A. Avron et al. (Eds.): Trakhtenbrot/Festschrift, LNCS 4800, pp. 620–634, 2008. c○ Springer-Verlag Berlin Heidelberg 2008sOn Natural Non-dcpo Domains 621 unknown. The point of using the term “natura... |

1 | Inductive Definition and Domain Theoretic Properties of Fully Abstract Models for
- Sazonov
(Show Context)
Citation Context ...oved to be quite simple, it was unclear at that moment whether anything reasonable could be obtained. What is new here is a general, abstract presentation of natural domains that does not rely, as in =-=[10]-=-, on a type structure like that of {Qα}. Unfortunately, it would take too much space to consider here the construction of the λ-model {Qα} — the source of general considerations of this paper. (See al... |