Dexter Kozen. Some Results in Dynamic Model Theory. Science of Computer Programming 51:1-2, May 2004, 3-22. Special issue, Mathematics of Program Construction (MPC 2002). Ed. E. Boiten and B. Moller.

First-order structures over a fixed signature Sgive rise to a family of trace-based and relational Kleene algebras with tests defined in terms of Tarskian frames. A Tarskian frame is a Kripke frame whose states are valuations of program variables and whose atomic actions are state changes effected by variable assignments x := e, where e is a S-term. The Kleene algebras with tests that arise in this way play a role in dynamic model theory akin to the role played by Lindenbaum algebras in classical first-order model theory. Given a first-order theory T over S, we exhibit a Kripke frame U whose trace algebra Tr_U is universal for the equational theory of Tarskian trace algebras over Ssatisfying T, although U itself is not Tarskian in general. The corresponding relation algebra Rel_U is not universal for the equational theory of relation algebras of Tarskian frames, but it is so modulo observational equivalence.