A necessary and sufficient condition for the existence of hoare logics

The important paper of Cook[3] on the existence of a relative complete and sound Hoare logic for a fragment of Algol has led many researchers to consider the existence of such logics for other programming languages[Z,5,7,8,9,10,11]. In particular, Clarke[Z] and Lipton-Snyder[9] have shown that a programming language P can have no relative complete and sound Hoare logic unless the programs of P have a decidable halting problem over fini te interpretations. (This result was first observed implictly in [2].) :Here our main result is the extension of this result to