## Lower Bag Domains (1995)

Venue: | FUNDAMENTA INFORMATICAE |

Citations: | 7 - 3 self |

### BibTeX

@ARTICLE{Heckmann95lowerbag,

author = {Reinhold Heckmann},

title = {Lower Bag Domains},

journal = {FUNDAMENTA INFORMATICAE},

year = {1995},

volume = {24},

pages = {259--281}

}

### OpenURL

### Abstract

Two lower bag domain constructions are introduced: the initial construction which gives free lower monoids, and the final construction which is defined explicitly in terms of second order functions. The latter is analyzed closely. For sober dcpo's, the elements of the final lower bag domains can be described concretely as bags. For continuous domains, initial and final lower bag domains coincide. They are continuous again and can be described via a basis which is constructed from a basis of the argument domain. The lower bag domain construction preserves algebraicity and the properties I and M, but does not preserve bounded completeness, property L, or bifiniteness.

### Citations

456 | Domain Theory - Abramsky, Jung - 1994 |

212 | A Powerdomain Construction
- Plotkin
- 1976
(Show Context)
Citation Context ...The lower bag domain construction preserves algebraicity and the properties I and M, but does not preserve bounded completeness, property L, or bifiniteness. 1 Introduction Power domain constructions =-=[13, 15, 16]-=- were introduced to describe the denotational semantics of non-deterministic programming languages. A power domain construction is a domain constructor P , which maps domains to domains, together with... |

136 |
Probabilistic Non-determinism
- Jones
- 1990
(Show Context)
Citation Context ...m the DCPO product in general. 4 The Final Construction in terms of Modular Functions In this section, we present a simpler description of the final lower bag domains. This description is inspired by =-=[8, 9]-=-, where a probabilistic power construction P is defined by PX = [WX mod ! I], the dcpo of continuous modular functions from WX, the lattice of open sets of X, to I, the unit interval of the reals with... |

125 |
A probabilistic powerdomain of evaluations
- Jones, Plotkin
- 1989
(Show Context)
Citation Context ...m the DCPO product in general. 4 The Final Construction in terms of Modular Functions In this section, we present a simpler description of the final lower bag domains. This description is inspired by =-=[8, 9]-=-, where a probabilistic power construction P is defined by PX = [WX mod ! I], the dcpo of continuous modular functions from WX, the lattice of open sets of X, to I, the unit interval of the reals with... |

78 |
Power domains
- Smyth
- 1978
(Show Context)
Citation Context ...The lower bag domain construction preserves algebraicity and the properties I and M, but does not preserve bounded completeness, property L, or bifiniteness. 1 Introduction Power domain constructions =-=[13, 15, 16]-=- were introduced to describe the denotational semantics of non-deterministic programming languages. A power domain construction is a domain constructor P , which maps domains to domains, together with... |

25 | Matching Theory, volume 29 of Annals of Discrete Mathematics - Lovasz, Plummer - 1986 |

23 | Power domain constructions
- Heckmann
- 1991
(Show Context)
Citation Context ... ! P R f Y) If P is any R-construction, then E : [X ! P1] ! [PX ! P1] can be rearranged into H : PX ! [[X ! R] ! R], which indicates why P R f is the final R-construction. The details can be found in =-=[3, 4, 5]-=-. 3 Lower Monoids The lower power domain construction produces free lower semilattices, i.e., semilattices in DCPO where 0 is the least element (or equivalently, the inherent semilattice order coincid... |

23 |
domains and predicate transformers: a topological view
- Power
- 1983
(Show Context)
Citation Context ...The lower bag domain construction preserves algebraicity and the properties I and M, but does not preserve bounded completeness, property L, or bifiniteness. 1 Introduction Power domain constructions =-=[13, 15, 16]-=- were introduced to describe the denotational semantics of non-deterministic programming languages. A power domain construction is a domain constructor P , which maps domains to domains, together with... |

17 |
is not always sober
- Johnstone
- 1981
(Show Context)
Citation Context ...each elementsof L m X can be represented by some bag A ass= A , or equivalently can be written as a (possibly infinite) sum of singletons' (point measures). This is not true for the non-sober dcpo of =-=[6]-=-. For continuous X, L m X is again continuous, and is isomorphic to L i X (Section 8). In this special case, we introduce the common name L for L i and L m . Given a basis B of X, a basis of L X consi... |

17 |
Cartesian closed categories of algebraic CPO’s
- Jung
- 1990
(Show Context)
Citation Context ...L 1 \Theta L 1 = N 1 0 \Theta N 1 0 . The domain X satisfies the hypothesis of Prop. 9.4 with a = (1; 0), b = (0; 1), and c = (1; 1). Thus, Y = L X is neither bifinite nor an L-domain. By Cor. 3.7 in =-=[10]-=-, the function space [Y ! Y] is not algebraic. 9.5 Any Alternatives? Maybe there is another lower bag domain construction which preserves bifiniteness or property L. To investigate this, let us assume... |

17 | Geometric theories and databases
- Vickers
- 1992
(Show Context)
Citation Context ...r power domain construction with characteristic semiring N 1 0 that preserves these domain classes (Subsection 9.5). In Section 10, we compare our approach with various other proposals of bag domains =-=[11, 2, 17, 7]-=-. R. Heckmann / Lower Bag Domains 2 Mathematical Background After fixing some set-theoretical notation (Subsection 2.1), we briefly recall the definition of the category DCPO of dcpo's (domains) and c... |

13 | Power domains and second order predicates
- Heckmann
- 1993
(Show Context)
Citation Context ...ns) and continuous functions (Subsection 2.2), introduce some algebraic structures in this category (Subsection 2.3), and recall the definition and basic properties of power domain constructions from =-=[4, 5]-=- (Subsection 2.4). 2.1 Some Set-Theoretic Notation We only mention some slightly non-standard notation. If f : X ! Y is a function, then we denote the image of a set A ` X by f + A = ffx j x 2 Ag, and... |

8 |
Call-by-value and non-determinism
- Sieber
- 1993
(Show Context)
Citation Context ... computationally adequate. However, only the lower power domain semantics is fully abstract for its kind of observable behavior in case of a simple callby -value non-deterministic functional language =-=[14]-=-. The other two power domain semantics are not fully abstract for the behavior they ought to describe, neither in case of the simple language nor in any reasonable extension thereof. Mathematically, t... |

7 |
A Fixed Point Approach to Applicative Multiprogramming
- Broy
- 1981
(Show Context)
Citation Context ...r power domain construction with characteristic semiring N 1 0 that preserves these domain classes (Subsection 9.5). In Section 10, we compare our approach with various other proposals of bag domains =-=[11, 2, 17, 7]-=-. R. Heckmann / Lower Bag Domains 2 Mathematical Background After fixing some set-theoretical notation (Subsection 2.1), we briefly recall the definition of the category DCPO of dcpo's (domains) and c... |

7 | Query languages for bags
- Libkin, Wong
- 1910
(Show Context)
Citation Context ...r power domain construction with characteristic semiring N 1 0 that preserves these domain classes (Subsection 9.5). In Section 10, we compare our approach with various other proposals of bag domains =-=[11, 2, 17, 7]-=-. R. Heckmann / Lower Bag Domains 2 Mathematical Background After fixing some set-theoretical notation (Subsection 2.1), we briefly recall the definition of the category DCPO of dcpo's (domains) and c... |

2 |
Partial products and bagdomains
- Johnstone
- 1992
(Show Context)
Citation Context |