## Nonclassical Techniques for Models of Computation (1999)

Venue: | Topology Proceedings |

Citations: | 9 - 4 self |

### BibTeX

@INPROCEEDINGS{Martin99nonclassicaltechniques,

author = {Keye Martin},

title = {Nonclassical Techniques for Models of Computation},

booktitle = {Topology Proceedings},

year = {1999},

pages = {375--405}

}

### OpenURL

### Abstract

After surveying recent work and new techniques in domain theoretic models of spaces, we introduce a new topological concept called recurrence, and consider some of its applications to the model problem.

### Citations

457 | Domain theory
- Abramsky, Jung
- 1994
(Show Context)
Citation Context ...is section we review some basic ideas and results that will be needed throughout the course of this paper. 2.1 Domain Theory 2.1.1 Order The reader unfamiliar with the basics of domain theory willsnd =-=[-=-1] valuable. We touch on certain basic aspects which are not quoted very often. Denition 2.1 A partially ordered set (P; v) is a set P together with a binary relation v P 2 which is (i) re exive: ( 8... |

326 |
Classical Descriptive Set Theory
- Kechris
- 1994
(Show Context)
Citation Context ...ogical completeness of locally compact sober spaces. 6 2.2.2 The Choquet Phenomenon One of Choquet's original motivations for introducing the spaces we call Choquet complete (called strong Choquet in =-=[15-=-]) was to provide an elegant and unied approach to the classical Baire category arguments of analysis. Despite the success of the idea, it remains largely unknown to many. The Choquet phenomenon is th... |

321 |
A Compendium of Continuous Lattices
- Gierz, Hofmann, et al.
- 1980
(Show Context)
Citation Context ...in theory calls for the characterization of precisely those spaces which possess a model. 3.2 Countably Based Models The Scott topology on a continuous dcpo D is second countable i D is !-continuous [=-=-=-8]. Denition 3.8 A countably based model of a space X is a model (D; : X ' maxD) in which the continuous dcpo D is !-continuous. Lawson proved that a certain subset of countably based models capture ... |

191 | Outline of a mathematical theory of computation - Scott - 1970 |

46 |
Local compactness and continuous lattices
- Hofmann, Mislove
(Show Context)
Citation Context ...ree. 2.2 Topology We review topological ideas that have proven to be indispensable in the study of models. 2.2.1 Locally Compact Sober Spaces The standard reference on locally compact sober spaces is =-=[1-=-3]. Denition 2.20 A subset of a space is compact if each of its open covers has asnite subcover. Denition 2.21 A topological space X is locally compact if it has a basis of compact neighborhoods, that... |

37 | A Foundation for Computation
- Martin
- 2000
(Show Context)
Citation Context ...ons made using are reliable. For instance, Denition 3.14 says that if we observesthat a sequence (x n ) of approximations calculate x, then they actually do calculate x. For much more on this, see [19] and [20]. 14 Example 3.16 Domains and their standard measurements. (i) (IR; ) the interval domain with the length measurement [a; b] = b a. (ii) ([N * N]; ) the partial functions on the naturals w... |

37 |
Logic of Domains
- Zhang
- 1991
(Show Context)
Citation Context ... poset is algebraic if its compact elements form a basis. A poset is !-continuous if it has a countable basis. Continuity provides a denite notion of approximation for posets. Proposition 2.9 (Zhang [24]) Continuous posets have the interpolation property: x y ) (9z) x z y: A useful form of completeness is oered by a dcpo. 3 Denition 2.10 A poset is a dcpo if every directed subset has a suprem... |

32 |
Spaces of maximal points
- Lawson
- 1997
(Show Context)
Citation Context ...ace X is a model (D; : X ' maxD) in which the continuous dcpo D is !-continuous. Lawson proved that a certain subset of countably based models capture exactly the Polish spaces. Theorem 3.9 (Lawson [=-=17]-=-) For a topological space X, the following are equivalent: (i) The space X is Polish. (ii) There is an !-continuous dcpo D whose relative Scott and Lawson topologies on maxD agree such that X ' maxD. ... |

26 | Cartesian Closed Categories of Domains
- Jung
- 1998
(Show Context)
Citation Context ..., G f(S) exists & f ( G S ) = G f (S ): Denition 2.15 The Lawson topology on a continuous poset P has as a basis all sets of the form " "x n"F where x 2 P and F P issnite. 4 Propositio=-=n 2.16 (Jung [1-=-4]) The Lawson topology on a continuous dcpo is compact i it is Scott compact and the intersection of any two Scott compact upper sets is Scott compact. Denition 2.17 A Scott domain is a continuous dc... |

26 | The measurement process in domain theory
- Martin
- 2000
(Show Context)
Citation Context ...using are reliable. For instance, Denition 3.14 says that if we observesthat a sequence (x n ) of approximations calculate x, then they actually do calculate x. For much more on this, see [19] and [20]. 14 Example 3.16 Domains and their standard measurements. (i) (IR; ) the interval domain with the length measurement [a; b] = b a. (ii) ([N * N]; ) the partial functions on the naturals with f = ... |

22 |
A computational model for metric spaces, Theoretical Computer Science 193
- Edalat, Heckmann
- 1998
(Show Context)
Citation Context ...n relation is A B , B int(A). The upper space is a model of X because maxUX = ffxg : x 2 Xg ' X. Example 3.7 A model for complete metric spaces. Given a metric space (X; d), the formal ball model [5] BX = X [0; 1) is a poset when ordered via (x; r) v (y; s) , d(x; y) r s: 12 The approximation relation is characterized by (x; r) (y; s) , d(x; y)sr s: The poset BX is continuous. However, BX i... |

10 |
A Louveau (1990), Glimm-E¤ros dichotomy for Borel equivalence relations
- Harrington, Kechris, et al.
- 2000
(Show Context)
Citation Context ...sdor space is Choquet complete. (iii) A metric space is Choquet complete i it is completely metrizable. (iv) A Gssubset of a Choquet complete space is Choquet complete. 7 A proof of (iv) appears in [9=-=-=-], while the others are all due to Choquet [2]. Interestingly, Choquet's completeness includes the most well-known form of completeness in topology. Corollary 2.31 Every Cech-complete space is Choque... |

9 |
Domain theory and integration. Theoretical Computer Science, 151:163–193
- Edalat
- 1995
(Show Context)
Citation Context ...m UX via X ' maxUX = ffxg : x 2 Xg UX where the topology on maxUX UX is the relative Scott topology inherited from UX . Because of this homeomorphism, we say that UX is a model of the space X. In [4=-=-=-], Abbas Edalat used only this link to the classical world to dene a generalization of the Riemann integral for bounded functions f : X ! R on a compact metric space X. Once it was observed that X cou... |

9 |
Computational models for ultrametric spaces
- Flagg, Kopperman
- 1997
(Show Context)
Citation Context ...cott and Lawson topologies on maxD agree such that X ' maxD. In [18], the author asked whether or not every Polish space had a model by a Scott domain. The reason was that earlier Flagg and Kopperman =-=[7]-=- had proven it for zero-dimensional Polish spaces. Theorem 3.10 (Ciesielski, Flagg & Kopperman [3]) Every Polish space is homeomorphic to the maximal elements of an !-continuous Scott domain. 13 The c... |

8 |
Characterizing topologies with bounded complete computational models
- Ciesielski, Flagg, et al.
- 1999
(Show Context)
Citation Context ...not every Polish space had a model by a Scott domain. The reason was that earlier Flagg and Kopperman [7] had proven it for zero-dimensional Polish spaces. Theorem 3.10 (Ciesielski, Flagg & Kopperman =-=[3]-=-) Every Polish space is homeomorphic to the maximal elements of an !-continuous Scott domain. 13 The converse, in view of Prop. 2.19, follows from Lawson's theorem. Finally, we confront the real issue... |

8 | Domain theoretic models of topological spaces
- Martin
- 1998
(Show Context)
Citation Context ...or a topological space X, the following are equivalent: (i) The space X is Polish. (ii) There is an !-continuous dcpo D whose relative Scott and Lawson topologies on maxD agree such that X ' maxD. In =-=[18]-=-, the author asked whether or not every Polish space had a model by a Scott domain. The reason was that earlier Flagg and Kopperman [7] had proven it for zero-dimensional Polish spaces. Theorem 3.10 (... |

7 |
Computation on metric spaces via domain theory. Topology and its applications, 85:247–263
- Lawson
- 1998
(Show Context)
Citation Context ...oset whose Scott topology is not even sober, much less Hausdor. Recurrence is a rare form of denseness indeed. 6 Applications We now apply the techniques discussed previously. 20 Theorem 6.1 (Lawson [=-=16-=-]) The Lawson topology on an !-continuous dcpo is Polish. Proposition 6.2 Let D be an !-continuous dcpo. If X maxD is a Gssubset of D, then (i) X is a second countable, Choquet complete T 1 space. (i... |

5 |
General topology. Polish Scienti
- Engelking
- 1977
(Show Context)
Citation Context ...s a Gssubset of its metric completion ^ X. (v) X is a Gssubset of a compact Hausdor space. 16 proof We need only establish the equivalence of (i), (ii) and (iii). See the masterful text by Engelking [=-=-=-6] for the others. (i) ) (iii): Example 3.16(v) gives a model (BX;) of a complete metric space X with X ' ker = maxBX: As the kernel of a measurement, X ' maxBX is a Gssubset of the continuous dcpo B... |

4 | The regular spaces with countably based models
- Martin
(Show Context)
Citation Context ...Tychono , Choquet completeness allows for the possibility of no separation whatsoever. The importance of this is made certain by the next result, a straightforward generalization of onesrst given in [21]. Theorem 2.32 Every locally compact sober space is Choquet complete. proof Let (X; ) be locally compact sober. Dene the approximation scheme a : f(U; x) : x 2 U 2 g ! as follows: Given an open s... |

3 |
Baire spaces
- Haworth, McCoy
(Show Context)
Citation Context ...intersection of countably many open dense sets is dense. Denition 2.35 A space is completely Baire if all of its closed subsets are Baire. The terminology \completely Baire" is from Kechris [15].=-= In [10]-=-, they are called Baire spaces in the strong sense, where a proof of the following may be found. Proposition 2.36 If X is a completely Baire space, then every Gssubset of X is completely Baire. The re... |

2 | Domain Environments - Heckmann |

2 |
A note on Baire spaces and continuous lattices
- Hofmann
- 1980
(Show Context)
Citation Context ... completeness of a second countable space is equivalent to its being completely Baire, provided one carries a weak additional assumption (like local compactness, for example). 9 Theorem 2.38 (Hofmann =-=[12]-=-) For a second countable, locally compact space X, the following are equivalent: (i) The space X is sober. (ii) Every closed subset of X is Baire. Here is another example from descriptive set theory t... |

1 |
Topology and measurements on domains
- Martin, Reed
(Show Context)
Citation Context ... is called a development provided that fSt(x; U n ) : n 0g is a basis at x where St(x; U n ) = [ fA : x 2 A 2 U n g: A space with a development is termed developable. 15 Theorem 3.19 (Martin & Reed [=-=22-=-]) A space is developable and T 1 i it is the kernel of a measurement on a continuous poset. For the case of a continuous dcpo, see [22]. In addition, one may capture metric spaces and complete metric... |