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.

For two parallel languages with recursion a compositional weakest precondition semantics is given using two new metric resumption domains. The underlying domains are characterized by domain equations involving functors that deliver `observable' and `safety' predicate transformers. Further a refinement relation is defined for this domains and illustrated by rules dealing with concurrent composition. It turns out, by extending the classical duality of predicate vs. state transformers, that the weakest precondition semantics for the parallel languages is isomorphic to the standard metric state transformers semantics. Moreover, the proposed refinement relation on the predicate transformer domain will correspond to the familiar notion of simulation in the state transformer domain. Contents 1 Introduction 1 2 Mathematical Preliminaries 3 3 Four Languages with Recursion 5 4 Domains for Predicate Transformers 8 5 Predicate Transformer Semantics 14 6 Refinement, Simulation and State Transforme...

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. 1991 Mathematics Subject Classification: 68Q55, 68Q10, 68Q60, 06A06. 1991 CR Categories: D.3.1, F.1.2, F.3.1, F.3.2. Keywords and Phrases: Predicates, liveness, safety, predicate transformers, state transformers, weakest preconditions, refinement, Hoare power domain, Smyth power domain, Plotkin power domain. Note: The research of Marcello Bonsangue was supported by a grant of the Centro Nazionale delle Ricerche (CNR), Italy, announcement no. 203.15.3 of 15/2/90. 1 Contents 1