Program invariants as fixed points

We argue that relative soundness and completeness theorems for Floyd-Hoare Axiom Systems ([6], [5], [18]) are really fixed point theorems. We give a characterization of program invariants as fixed points of functionals which may be obtained in a natural manner from the text of a program. We show that within the framework of this fixed point theory, relative soundness and completeness results have a particularly simple interpretation. Completeness of a Floyd-Hoare Axiom system is equivalent to the existence of a fixed point for an appropriate functional, and soundness follows from the maximality of this fixed point, The functionals associated with regular procedure declarations are similar to predicate transformers of Dijkstra; for non-regular recursions it is necessary to use a generalization of the predicate transformer concept which we call a relational transformer.

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