Some results in dynamic model theory

Abstract

First-order structures over a fixed signature Σ give 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 Σ-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 Σ, we exhibit a Kripke frame U whose trace algebra Tr U is universal for the equational theory of Tarskian trace algebras over Σ satisfying 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.