Non-uniformity of dynamic logic

Abstract

Harel proved that the dynamic logic is complete in the class of arithmetical interpretations. The set of natural numbers stands for primitive notion in arithmetical interpretations, i.e. the following condition holds: "there exists a unary relation symbol nat such that for every arithmetical interpretation I, nat_I is the set of natural numbers". It is proved that the completeness is lost when this condition is relaxed to the following one: "for every interpretation^I involved, the set of natural numbers is first-order definable in I". Thus we prove that the dynamic logic is not relatively complete in the sense of Cook.