BCS The Weakest Precondition

An extension of Dijkstra's guarded command language is studied, including unbounded demonic choice and a backtrack operator. We consider three orderings on this language: a reenement ordering deened by Back, a new deadlock ordering, and an approximation ordering of Nelson. The deadlock ordering is in between the two other orderings. All operators are monotonic in the Nelson ordering, but backtracking is not monotonic in the Back ordering and sequential composition is not monotonic for the deadlock ordering. At rst sight recursion can only be added using the Nelson ordering. We show that, under certain circumstances , least xed points for non-monotonic functions can be obtained by iteration from the least element. This permits the addition of recursion even using the Back ordering or the deadlock ordering in a fully compositional way. In order to give a semantic characterization of the three orderings in terms of semantics that relate initial states to possible outcomes of the computation, the relation between predicate transformers and discrete power domains is studied. We consider (two versions of) the Smyth power domain and the Egli-Milner power domain.