## Isomorphisms between Predicate and State Transformers (1993)

Venue: | In Proc., MFCS '93, Springer LNCS 711 |

Citations: | 4 - 0 self |

### BibTeX

@INPROCEEDINGS{Bonsangue93isomorphismsbetween,

author = {Marcello Bonsangue and Joost N. Kok},

title = {Isomorphisms between Predicate and State Transformers},

booktitle = {In Proc., MFCS '93, Springer LNCS 711},

year = {1993},

pages = {301--310},

publisher = {Springer}

}

### OpenURL

### Abstract

We study the relation between state transformers based on directed complete partial orders and predicate transformers. Concepts like `predicate', `liveness', `safety' and `predicate transformers' are formulated in a topological setting. We treat state transformers based on the Hoare, Smyth and Plotkin powerdomains and consider continuous, monotonic and unrestricted functions. We relate the transformers by isomorphisms thereby extending and completing earlier results and giving a complete picture of all the relationships.

### Citations

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Citation Context ...Plo81, Abr91] power domains, we also add the empty set in all the three power domains as a separate element, comparable only with itself and with the bottom. Predicate transformers were introduced in =-=[Dij76]-=- with a series of healthness conditions. Back and von Wright [Bac80, vW90] use only the monotonicity restriction on predicate transformers. They use predicate transformers for refinement and provide a... |

410 |
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Citation Context ...ogy as G ffi sets: countable intersections of open sets) while we take arbitrary intersections of open sets as liveness predicates. In [Kwi91] liveness predicates are also G ffi -sets. We differ from =-=[AS85]-=- where liveness predicates are the dense sets (the complement does not contain non-empty open sets). To provide some more intuition we show that taking different topologies corresponds to different re... |

336 | Proving the correctness of multiprocess programs
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(Show Context)
Citation Context ...state transformers we use an extension of the standard power domains. For predicate transformers we start from the (informal) classification of predicates in liveness and safety predicates of Lamport =-=[Lam77]-=-. Later Smyth [Smy83] followed by [AS85, Kwi91] used topology to formalize this classification. Also we use topology for defining predicates and safety and liveness predicate transformers. We consider... |

226 | A powerdomain construction
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Citation Context ...ll picture of all the relationships filling several gaps that were present in the literature. 2 Next we discuss how this paper is related to previous work. Power domains for dcpo's were introduced in =-=[Plo76]-=-, [Smy78] and [Plo81]. Our power domains are slightly more general in the sense that we do no restrict to non-empty (Scott-) compact sets. Besides the standard ways of adding the empty set to the Smyt... |

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Citation Context ...k and von Wright [Bac80, vW90] use only the monotonicity restriction on predicate transformers. They use predicate transformers for refinement and provide a nice lattice theoretical framework. Nelson =-=[Nel89] has (for -=-the flat case) used "compatible" pairs of predicate transformers for giving semantics to a language with backtracking. Smyth [Smy83] introduced predicate transformers (with the Dijkstra heal... |

124 |
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(Show Context)
Citation Context ... the power domains. These extended power domains can then be used in state transformers. The empty set is usually added to the Plotkin power domain by means of a smash-product [MM79],[Plo81] and also =-=[Abr91]-=-. The same is done [BK92] for the Smyth power domain in the flat case. We have the following: Definition 6.1 Let X be an algebraic dcpo. Define 1. the Hoare-deadlock power domain Hffi(X ) = hfAjA ` Xs... |

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Citation Context ...s of predicate transformers is given. Isomorphisms between state and predicate transformers have been given for the flat case of the Smyth power domain in [Plo79] (and for countable nondeterminism in =-=[AP86]-=-), and for the flat case of the Hoare power domain in [Plo81]. Also De Bakker and De Roever [Bak80, Roe76] studied (from a semantical point of view) for the flat case the relation between state transf... |

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(Show Context)
Citation Context ...he relationships filling several gaps that were present in the literature. Next we discuss how this paper is related to previous work. Power domains for dcpo's were introduced in [Plo76], [Smy78] and =-=[Plo81]-=-. Our power domains are slightly more general in the sense that we do no restrict to non-empty (Scott) compact sets. Besides the standard ways of adding the empty set to the Smyth [Smy83] and to the P... |

47 |
General Topology, Polish Scientific
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Citation Context ... 2 Mathematical Preliminaries We introduce some basic notions on domain theory and topology. For a more detailed discussions on domain theory consult for example [Plo81], and for topology we refer to =-=[Eng77]. A p-=-artial order v on a set P is a reflexive, transitive and antisymmetric relation in P \Theta P . Let P be ordered by the partial order v, x 2 P and A a subset of P . Define x "= fy jy 2 Psx v yg a... |

44 |
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Citation Context ... to add the empty set to the power domains. These extended power domains can then be used in state transformers. The empty set is usually added to the Plotkin power domain by means of a smash-product =-=[MM79]-=-,[Plo81] and also [Abr91]. The same is done [BK92] for the Smyth power domain in the flat case. We have the following: Definition 6.1 Let X be an algebraic dcpo. Define 1. the Hoare-deadlock power dom... |

28 |
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(Show Context)
Citation Context ...fixed point of f always exists and it can be calculated by iteration: there exists an ordinalssuch that :f = f , where the ff-iteration of f is defined by f ff = f ( F k!ff f k ) for every ordinal ff =-=[HP72]-=-. If f is also continuous thens! 0 . Via a fixed point transformation technique we can show that in certain cases also non-monotonic functions have least fixed points that can be calculated by iterati... |

24 |
domains and predicate transformers: a topological view , Techincal monograph 126
- Power
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(Show Context)
Citation Context ... use an extension of the standard power domains. For predicate transformers we start from the (informal) classification of predicates in liveness and safety predicates of Lamport [Lam77]. Later Smyth =-=[Smy83]-=- followed by [AS85, Kwi91] used topology to formalize this classification. Also we use topology for defining predicates and safety and liveness predicate transformers. We consider predicate transforme... |

21 |
Dijkstra’s predicate transformers and Smyth’s powerdomains
- Plotkin
- 1980
(Show Context)
Citation Context ...ion to the non-flat case of "compatible" pairs of predicate transformers is given. Isomorphisms between state and predicate transformers have been given for the flat case of the Smyth power =-=domain in [Plo79]-=- (and for countable nondeterminism in [AP86]), and for the flat case of the Hoare power domain in [Plo81]. Also De Bakker and De Roever [Bak80, Roe76] studied (from a semantical point of view) for the... |

11 | A lattice-theoretical basis for program refinement - Wright - 1990 |

7 | Dijkstra’s predicate transformer, non-determinism, recursion, and terminations - Roever - 1976 |

6 |
Convex powerdomains I
- Hrbacek
- 1987
(Show Context)
Citation Context ...braic dcpo then P + co (X ) is also an !-algebraic dcpo with finite elements the -closures of finite subsets of BX (cf. [Plo76], [Plo81]). For a treatment of more general algebraic dcpo's we refer to =-=[Hrb87]-=- and [Hrb89]. Definition 4.3 Let X ; Y be two algebraic dcpo's. Define the Hoare state transformers as functions (ordered pointwise) in X ! H(Y ), the Smyth state transformers as functions (ordered po... |

6 | Semantics, orderings and recursion in the weakest precondition calculus
- Bonsangue, Kok
- 1993
(Show Context)
Citation Context ...point of view) for the flat case the relation between state transformer and predicate transformer semantics. Moreover, for the flat case of the Plotkin power domain we have proposed an isomorphism in =-=[BK92]-=-. For the general case of the compact Smyth power domain in the paper [Smy83] an isomorphism is given for continuous state transformers. He uses a topological technique which Plotkin later used in [Pl... |

5 | Algebraic domains, chain completion and the plotkin powerdomain construction
- Knijnenburg
- 1993
(Show Context)
Citation Context ... for these relationships. Let X be an algebraic dcpo and A ` X . Define the set of minimal elements of A by min(A) = fx jx 2 As(8y 2 A : x vX y ) x = y)g: 26 Minimal elements are used in [Kni93b] and =-=[Kni93a] for -=-the construction of the power domains S + co (X ) and P + co (X ) from an !-algebraic dcpo's X . Define the set of maximal elements of A by max (A) = fx 2 Ajx 2 BX ) (8b 2 A " BX : b vX x ) b = x... |

4 |
On topological characterization of behavioural properties
- Kwiatowska
- 1989
(Show Context)
Citation Context ...rs A predicate P is a function from a set X to the boolean set ftt; ff g or, equivalently, is a subset of X . Topology provides an elegant way of selecting classes of predicates of programs ([Smy83], =-=[Kwi91]-=-) in which the open sets of a topological space X are the observable predicates, closed sets are the safety predicates and arbitrary intersections of open sets are the liveness predicates. In this pap... |

4 |
Isomorphisms between State and Predicate Transformers
- Bonsangue, Kok
- 1993
(Show Context)
Citation Context ...w no isomorphism was known for the non-flat Plotkin power domain (as for example is remarked in [Plo81] and in [Smy83]). For reasons of space, proofs are not given in this paper. They can be found in =-=[BK93]-=-. 2 Mathematical Preliminaries We introduce some basic notions on domain theory and topology. For a more detailed discussion on domain theory consult for example [Plo81], and for topology we refer to ... |

3 | Relating state transformation semantics and predicate transformer semantics for parallel programs
- Breugel
- 1993
(Show Context)
Citation Context ...state transformers. He uses a topological technique which Plotkin later used in [Plo81] for the continuous Hoare state transformers. A recent work includes an operational point of view in Van Breugel =-=[Bre93]-=-. In the present paper we give some new isomorphisms for the Hoare and the Smyth power domains, showing also how the previous ones can be obtained as combinations of the new isomorphisms. Our definiti... |

2 |
orderings and recursion in the weakest precondition calculus
- Semantics
- 1992
(Show Context)
Citation Context ...point of view) for the flat case the relation between state transformer and predicate transformer semantics. Moreover, for the flat case of the Plotkin power domain we have proposed an isomorphism in =-=[BK92]-=-. For the general case of the compact Smyth power domain in the paper [Smy83] an isomorphism is given for continuous state transformers. He uses a topological technique which Plotkin later used in [Pl... |

2 | A Note on the Smyth Powerdomain Construction
- Knijnenburg
- 1993
(Show Context)
Citation Context ...paves the way for these relationships. Let X be an algebraic dcpo and A ` X . Define the set of minimal elements of A by min(A) = fx jx 2 As(8y 2 A : x vX y ) x = y)g: 26 Minimal elements are used in =-=[Kni93b] and -=-[Kni93a] for the construction of the power domains S + co (X ) and P + co (X ) from an !-algebraic dcpo's X . Define the set of maximal elements of A by max (A) = fx 2 Ajx 2 BX ) (8b 2 A " BX : b... |