On relative completeness of Hoare logics

In this paper a generalization of a certain theorem of Lipton (“Proc. 18th IEEE Sympos. Found. of Comput Sci.” (1977), pp. 1–6) is presented. Namely, we show that for a wide class of programming languages the following holds: the set of all partial correctness assertions true in an expressive interpretation I is uniformly dedicable (in I) in the theory of I iff the halting problem is decidable for finite interpretations. In the effect we show that such limitations as effectiveness or Herbrand-definability of interpretation (they are relevant in the previous proofs) can be removed in the case of partial correctness