Diamond formulas in the dynamic logic of recursively enumerable programs

Abstract

Dynamic logic QDL as presented in |3| provides a comprehensive logical framework for the study of the before-after behaviour of deterministic and non-deterministic programs. While the set of all valid QDL-formulas is highly complex (π ¹₁ -complete) and hence not axiomatizable, the subset of valid termination assertions was shown to be axiomatizable in |5|. In |8|, this result was generalized to the effect that the much larger QDL-fragment of diamond formulas is still axiomatizable and satisfies a compactness theorem. The proofs were based on a rather delicate proof-theoretical treatment of consistency properties. We show how results of this kind can be obtained in the general framework of recursively enumerable dynamic logic by a very flexible approach that uses only the compactness and completeness of first-order logic and saturated structures. The method is also applicable to the dynamic logic involving undeclared global procedures and recursive procedure calls studied in |6|.