## A Computational Model for Metric Spaces (1995)

Venue: | Theoretical Computer Science |

Citations: | 42 - 8 self |

### BibTeX

@ARTICLE{Edalat95acomputational,

author = {Abbas Edalat and Reinhold Heckmann},

title = {A Computational Model for Metric Spaces},

journal = {Theoretical Computer Science},

year = {1995},

volume = {193},

pages = {53--73}

}

### OpenURL

### Abstract

For every metric space X , we define a continuous poset BX such that X is homeomorphic to the set of maximal elements of BX with the relative Scott topology. The poset BX is a dcpo iff X is complete, and !-continuous iff X is separable. The computational model BX is used to give domain-theoretic proofs of Banach's fixed point theorem and of two classical results of Hutchinson: on a complete metric space, every hyperbolic iterated function system has a unique non-empty compact attractor, and every iterated function system with probabilities has a unique invariant measure with bounded support. We also show that the probabilistic power domain of BX provides an !-continuous computational model for measure theory on a separable complete metric space X . 1 Introduction In this paper, we establish new connections between the theory of metric spaces and domain theory, the two basic mathematical structures in computer science. For every metric space X, we define a continuous poset (not necessar...

### Citations

765 |
Geometric Measure Theory
- Federer
- 1969
(Show Context)
Citation Context ... for all opens O containing x. We call a measureswell-supported if S 0 = 0 holds for the complement S 0 of the support of . Obviously, every continuous measure is well-supported. By Theorem 2.2.16 in =-=[8]-=-, all measures on metric spaces have separable support. Hence, well-supported measures on metric spaces are continuous. Therefore, the notions `well-supported' and `continuous' coincide on metric spac... |

456 | Domain Theory
- Abramsky, Jung
- 1994
(Show Context)
Citation Context ...BX , which is a continuous dcpo by Theorem 6. On the other hand, we may construct BX, which is a continuous poset, consider (BX;) as an abstract basis and construct its rounded ideal completion I(BX) =-=[1]-=-, which is a continuous dcpo. We claim that the two continuous dcpo's B(X) and I(BX) are isomorphic. We may consider X as a subset of X such that d restricted to X is d. Since X is dense in X , the se... |

325 |
Fractals and self-similarity
- Hutchinson
(Show Context)
Citation Context ...ic space X from the dcpo fixed point theorem applied to a suitable pointed subdcpo of BX. The technique of this proof is then used to give a domain-theoretic proof of a classical result of Hutchinson =-=[10]-=-: on a complete metric space, every hyperbolic iterated function system (IFS) has a unique non-empty compact attractor (Section 4). In our proof, we work in the Plotkin power domain of BX, and complet... |

136 |
Probabilistic Non-determinism
- Jones
- 1990
(Show Context)
Citation Context ...be defined in terms of valuations on BX. A valuation on a topological space Y is a function from the opens of Y to R+ which is strict, modular, and Scott continuous. The probabilistic power domain PY =-=[12,11]-=- of Y consists of all valuationsson Y with (Y )s1, ordered by their values on Scott open sets. 18 For a in Y , the point valuation ffi(a) is defined by ffi(a)(O) = 1 if a in O, and = 0 otherwise. The ... |

125 | A probabilistic powerdomain of evaluations - Jones, Plotkin - 1989 |

68 | Dynamical systems, measures and fractals via domain theory
- Edalat
- 1995
(Show Context)
Citation Context ...ble (has a countable dense subset). All these results offer pleasing connections between classical notions of metric space theory and analogous notions of domain theory. These results extend those in =-=[5]-=- which were for a locally compact Hausdorff space. They present a considerably simpler framework than that of an !-algebraic cpo with distance and weight used by Weihrauch and Schreiber in [18] to emb... |

57 | Domain theory and integration
- Edalat
- 1995
(Show Context)
Citation Context ...plete metric spaces X. In this case, BX is an !-continuous dcpo, and PBX is !-continuous as well [12,11]. Thus it admits an effective treatment along the lines of [3]. This generalises the results of =-=[5,4]-=-, which were restricted to locally compact spaces. By Theorem 27, the set M 1 X of normalised measures of X corresponds to the subset SX of PBX. By [7], this correspondence is a homeomorphism if M 1 X... |

37 |
Logic of Domains
- Zhang
- 1991
(Show Context)
Citation Context ... be generalised to arbitrary posets; one only has to replace quantifications such as `for every directed set' by `for every directed set which has a least upper bound'. This is done in Section 1.4 of =-=[19]-=-, where continuous posets are defined and it is shown that their elementary properties are in complete analogy with those of continuous dcpo's. Thus for instance, the way7 below relation in a continuo... |

33 |
Domain representability of metric spaces
- Blanck
- 1997
(Show Context)
Citation Context ... by Weihrauch and Schreiber in [18] to embed a complete separable metric space into a domain, or that of the equivalence classes of converging ideals of closed neighbourhood systems used by Blanck in =-=[2]-=- to give a domain representation of a complete metric space. Our construction is also much simpler than Lawson's recent construction of an MP hull for a Polish space [16]. Furthermore, our results lea... |

32 |
Spaces of maximal points
- Lawson
- 1997
(Show Context)
Citation Context ...hood systems used by Blanck in [2] to give a domain representation of a complete metric space. Our construction is also much simpler than Lawson's recent construction of an MP hull for a Polish space =-=[16]-=-. Furthermore, our results lead to a simple computational model for Hilbert and Banach spaces. In fact, for a normed vector space X, BX is isomorphic to the poset of closed balls ordered by reverse in... |

30 | Power domains and iterated function systems
- Edalat
- 1996
(Show Context)
Citation Context ...c: We use the space of formal balls to show that an iterated function system with probabilities on a complete metric space has a unique invariant measure with bounded support (Section 6). As shown in =-=[6]-=- for locally compact Hausdorff spaces, our domain-theoretic framework can be used to derive algorithms for generating invariant measures and for computing expected 2 values of functions with respect t... |

22 |
Valuations on continuous lattices
- Lawson
- 1982
(Show Context)
Citation Context ...ns can be identified on metric spaces. A corresponding result for continuous dcpo's is not known. For !-continuous dcpo's however, it is known that every valuation extends uniquely to a Borel measure =-=[14,17]-=-. 5.2 Valuations on BX For a complete metric space X, let M 1 X be the set of normalised wellsupported measures on X. We embed M 1 X into the probabilistic power domainsPBX of BX, which forms a contin... |

14 |
Embedding metric spaces into cpo’s
- Weihrauch, Schreiber
- 1981
(Show Context)
Citation Context ...hose in [5] which were for a locally compact Hausdorff space. They present a considerably simpler framework than that of an !-algebraic cpo with distance and weight used by Weihrauch and Schreiber in =-=[18]-=- to embed a complete separable metric space into a domain, or that of the equivalence classes of converging ideals of closed neighbourhood systems used by Blanck in [2] to give a domain representation... |

13 |
Existence theorems for measures on continuous posets, with applications to random set theory. Mathematica Scandinavica
- Norberg
- 1989
(Show Context)
Citation Context ...ns can be identified on metric spaces. A corresponding result for continuous dcpo's is not known. For !-continuous dcpo's however, it is known that every valuation extends uniquely to a Borel measure =-=[14,17]-=-. 5.2 Valuations on BX For a complete metric space X, let M 1 X be the set of normalised wellsupported measures on X. We embed M 1 X into the probabilistic power domainsPBX of BX, which forms a contin... |

9 |
The Scott topology induces the weak topology
- Edalat
- 1996
(Show Context)
Citation Context ... lines of [3]. This generalises the results of [5,4], which were restricted to locally compact spaces. By Theorem 27, the set M 1 X of normalised measures of X corresponds to the subset SX of PBX. By =-=[7]-=-, this correspondence is a homeomorphism if M 1 X is equipped with the weak topology and SX with the relative Scott topology. By !-continuity, every valuationson BX extends uniquely to a Borel measure... |

7 |
Self-similar sets as Tarski's fixed points
- Hayashi
- 1985
(Show Context)
Citation Context ...d in the proof of Theorem 18, a domain-theoretic proof of this result can be given which completely avoids the Hausdorff metric. This extends the corresponding result for a compact metric space as in =-=[9,6]-=-. The proof is performed in the Plotkin power domain [1] of the domain BX. We first recall some basic definitions (see [1]). Let D be a continuous dcpo. For subsets A and B of D, we define -- A OE L B... |

6 |
Total objects of domains
- Kamimura, Tang
- 1984
(Show Context)
Citation Context ... for a Polish space by choosing a separable complete metric for it. Some authors have studied the subspace of maximal points of a domain with the relative Scott topology (MP space). Kamimura and Tang =-=[13]-=- considered MP spaces of bounded complete !-continuous domains; in the algebraic case, these spaces were shown to be Polish spaces. This result has now been elegantly generalised by Lawson [16]. He sh... |

3 | Continuous Information Categories
- Edalat
- 1991
(Show Context)
Citation Context ... X + with the relative Scott or Lawson topology. If X is a separable complete metric space, then BX is an !-continuous dcpo by Cor. 10. Thus, BX can be given an effective structure along the lines of =-=[3]-=-. This structure can be used to derive an effective structure for X via the homeomorphism between X and X + . This provides a computational framework for X. Recall that a Polish space is a topological... |