Semantics, orderings and recursion in the weakest precondition calculus

Abstract

An extension of Dijkstra's guarded command language is studied, including sequential composition, demonic choice and a backtrack operator. To guide the intuition about this language we give an operational semantic that relates the initial states with possible outcome of the computations. Next we consider three orderings on this language: a refinement ordering defined 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 Nelson's ordering, but backtracking is not monotonic in Back's ordering and sequential composition is not monotonic for the deadlock ordering. At first sight recursion can only be added using Nelson's ordering. By extending the fixed point theory we show that, under certain circumstances, least fixed points for non monotonic functions can be obtained by iteration from the least element. This permits us the addition of recursion even using Back's ordering or the deadlock ordering. Furthermore, we give a semantic characterization of the three orderings above by extending the well known duality theory between predicate transformers and Smyth's powerdomain.