## A Duality Theory for Quantitative Semantics (1998)

Venue: | Computer Science Logic. 11th International Workshop, volume 1414 of Lecture Notes in Computer Science |

Citations: | 4 - 3 self |

### BibTeX

@INPROCEEDINGS{Heckmann98aduality,

author = {Reinhold Heckmann and Michael Huth},

title = {A Duality Theory for Quantitative Semantics},

booktitle = {Computer Science Logic. 11th International Workshop, volume 1414 of Lecture Notes in Computer Science},

year = {1998},

pages = {255--274},

publisher = {Springer Verlag}

}

### OpenURL

### Abstract

. A continuous predicate on a domain, or more generally a topological space, can be concretely described as an open or closed set, or less obviously, as the set of all predicates consistent with it. Generalizing this scenario to quantitative predicates, we obtain under certain wellunderstood hypotheses an isomorphism between continuous functions on points and supremum preserving functions on open sets, both with values in a fixed lattice. The functions on open sets provide a topological foundation for possibility theories in Artificial Intelligence, revealing formal analogies of quantitative predicates with continuous valuations. Three applications of this duality demonstrate its usefulness: we prove a universal property for the space of quantitative predicates, we characterize its inf-irreducible elements, and we show that bicontinuous lattices and Scott-continuous maps form a cartesian closed category. 1 Introduction It is well-known that a predicate p on a set X, i.e. a function p ...

### Citations

456 | Domain Theory
- Abramsky, Jung
- 1994
(Show Context)
Citation Context ...ile p(x) can be simply obtained as P (fxg), or in a more elaborated way as V S3x P (S). This simple scenario can be generalized from sets (discrete topological spaces) to arbitrary topological spaces =-=[13, 15, 1]-=-. According to the spirit of topology, only continuous predicates should be considered. Of course, continuity of a function from a topological space X to f0; 1g depends on the topology defined for f0;... |

321 |
A compendium on continuous lattices
- GIERZ
- 1980
(Show Context)
Citation Context ... a continuous lattice. Proof. By Theorem 17, the lattice P L op (X) is continuous. By Theorem 3, the latter is isomorphic to the order-dual of [X ! L op op ] = [X ! L]. ut A lattice L is bicontinuous =-=[2]-=- if both L and L op are continuous. Let BCONT be the category of bicontinuous lattices with Scott-continuous maps as morphisms. Since continuous lattices and Scott-continuous maps form a cartesian clo... |

136 |
Probabilistic Non-determinism
- Jones
- 1990
(Show Context)
Citation Context ...etween topological spaces X and Y , P L (f): P L (X) ! P L (Y ) is defined as P L (f) O = (f \Gamma1 (O)) for alls2 P L (X) and all O 2 O(Y ). This definition corresponds to those used for valuations =-=[7]-=- and in conventional measure theory [3]. Theorem 7. The map P L (\Delta) is well-defined and results in a monotonic functor from TOP to SUP, the category of complete sup-lattices and sup-maps. When re... |

125 |
A probabilistic powerdomain of evaluations
- Jones, Plotkin
- 1989
(Show Context)
Citation Context ...Intelligence. Note that the complete lattice P L (X) sits inside the complete lattice [O(X) ! L] such that the inclusion is Scott-continuous. The same is true for the dcpo VX of continuous valuations =-=[6] whic-=-h consists of all strict mapss2 [O(X) ! L] such thatssatisfies the classical modular law (U [ V ) + (U " V ) = (U ) + (V ) for all opens U and V . If we interpret addition in this equation in a f... |

107 | Probabilistic predicate transformers - Morgan, McIver, et al. - 1996 |

69 |
Powerdomains and predicate transformers: a topological view
- Smyth
- 1983
(Show Context)
Citation Context ...ile p(x) can be simply obtained as P (fxg), or in a more elaborated way as V S3x P (S). This simple scenario can be generalized from sets (discrete topological spaces) to arbitrary topological spaces =-=[13, 15, 1]-=-. According to the spirit of topology, only continuous predicates should be considered. Of course, continuity of a function from a topological space X to f0; 1g depends on the topology defined for f0;... |

61 | Quantitative analysis and model checking - Huth, Kwiatkowska - 1997 |

16 |
Completely distributive complete lattices
- Raney
- 1952
(Show Context)
Citation Context ...on. The difference is that one refers to arbitrary suprema instead of directed suprema. A complete lattice is completely distributive if every element is the supremum of all elements way-way-below it =-=[11, 12]-=-. If L is completely distributive then we obtain similarly that P L (X) is a supprojection of [O(X) ! L] and the way-way-below relation on P L (X) is induced by the way-way-below relation on [O(X) ! L... |

13 | Power domains and second order predicates
- Heckmann
- 1993
(Show Context)
Citation Context ...l of the ordering, fsg holds in F L (X) iff for all x in X, fxsgx holds in L. It is folklore knowledge that the lattice of closed sets, or F 2 (X), is isomorphic to O(X) \Gamma ffi 2 (for a proof see =-=[4]-=-). This suggests to define P L (X) = O(X)\Gamma ffi L for arbitrary spaces X and complete lattices L, and to compare F L (X) and P L (X). We call the elements of P L (X), i.e. sup-maps from the lattic... |

8 |
Tight Galois connections and complete distributivity
- Raney
- 1960
(Show Context)
Citation Context ...on. The difference is that one refers to arbitrary suprema instead of directed suprema. A complete lattice is completely distributive if every element is the supremum of all elements way-way-below it =-=[11, 12]-=-. If L is completely distributive then we obtain similarly that P L (X) is a supprojection of [O(X) ! L] and the way-way-below relation on P L (X) is induced by the way-way-below relation on [O(X) ! L... |

3 | An expectation-transformer model for probabilistic temporal logic - Morgan, McIver - 1997 |