## Observable Modules and Power Domain Constructions (1993)

Venue: | Semantics of Programming Languages and Model Theory, volume 5 of Algebra, Logic, and Applications |

Citations: | 4 - 1 self |

### BibTeX

@INPROCEEDINGS{Heckmann93observablemodules,

author = {Reinhold Heckmann},

title = {Observable Modules and Power Domain Constructions},

booktitle = {Semantics of Programming Languages and Model Theory, volume 5 of Algebra, Logic, and Applications},

year = {1993},

pages = {159--187},

publisher = {Gordon and Breach Science Publishers}

}

### OpenURL

### Abstract

An R-module M is observable iff all its elements can be distinguished by observing them by means of linear morphisms from M to R. We show that free observable R-modules can be explicitly described as the cores of the final power domains with characteristic semiring R. Then, the general theory is applied to the cases of the lower and the upper semiring. All lower modules are observable, whereas there are non-observable upper modules. Accordingly, all known lower power constructions coincide, whereas there are at least three different upper power constructions. We show that they coincide for continuous ground domains, but differ on more general domains. 1 Introduction A power domain construction maps every domain X into a so-called power domain over X whose points represent sets of points of the ground domain. Power domain constructions were originally proposed to model the semantics of non-deterministic programming languages [Plo76, Smy78, HP79, Mai85]. Other motivations are the sema...

### Citations

94 | Full abstraction for a simple parallel programming language
- Hennessy, Plotkin
- 1979
(Show Context)
Citation Context ...any other set. Thus, the characteristic semiring of this construction is C = f0; 1g, where 0 and 1 are incomparable. If we define Plotkin's construction as free construction of semilattices following =-=[HP79]-=- and add a neutral element, then we arrive at the initial C-construction C i . In contrast to all the final constructions we met so far, the final C-construction is degenerated. The reason is that a d... |

26 | A semantics for complex objects and approximate queries
- Buneman, Davidson, et al.
- 1988
(Show Context)
Citation Context ...ere; all proofs will be omitted. They can be found in [Hec90a, Hec91, Hec93]. 8.1 The Boolean Semiring Recently, Buneman and Gunter defined two power domain constructions, the Sandwich construction S =-=[BDW88]-=- and the mixed construction M [Gun89, Gun90]. These constructions were defined for algebraic ground domains only via base and ideal completion. Although they differ even for small finite ground domain... |

24 | Probabilistic domains
- Heckmann
- 1994
(Show Context)
Citation Context ...structions were originally proposed to model the semantics of non-deterministic programming languages [Plo76, Smy78, HP79, Mai85]. Other motivations are the semantic representation of a set data type =-=[Hec90b]-=-, or of relational data bases [BDW88, Gun89]. In 1976, Plotkin [Plo76] proposed the convex power domain construction. A short time later, Smyth [Smy78] introduced a simpler construction, the upper pow... |

23 | Power domain constructions
- Heckmann
- 1991
(Show Context)
Citation Context ...om an existing one. The core of an R-construction P is the least subconstruction of P that still has characteristic semiring R. Definition and general properties of cores were published in the thesis =-=[Hec90a]-=-, but not elsewhere. 4.1 R-X-Modules As technical means to simplify reasoning, we introduce R-X-modules in this subsection. R-X-modules are R-modules together with a map from X. Definition 4.1 Let R b... |

13 | Power domains and second order predicates
- Heckmann
- 1993
(Show Context)
Citation Context ... mix algebras over X. In [Hec91], it is shown that mix algebras are nothing else than B-modules. Thus, M is the initial B-construction. On the other hand, S is shown to be the final B-construction in =-=[Hec93]-=-. Since (for algebraic ground domains) M is a subconstruction of S, the core of S coincides with M. - 19 - Our theory allows the extension of the two constructions M and S to arbitrary ground domains.... |

8 | Relating total and partial correctness interpretations of non-deterministic programs - Gunter - 1990 |

4 | The mixed powerdomain. Internal Report MS-CIS-89-77 - Gunter - 1989 |

4 | An upper power domain construction in terms of strongly compact sets
- Heckmann
- 1992
(Show Context)
Citation Context ... difference between the initial construction and the core of the final one (Sec. 7.6). The difference between the final construction and its core follows from an example that was already presented in =-=[Hec92]-=-. Thus, one has to be careful to avoid confusion when speaking about upper power domains and going beyond the class of continuous dcpo's. At the end, we briefly consider two further semirings: the one... |