Non-associative Kleene Algebra and Temporal Logics

We introduce new variants of Kleene star and omega iteration for the case where the iterated operator is neither associative nor has a neutral element. The associated repetition algebras are used to give closed semantic expressions for the Until and While operators of the temporal logic \(\mathsf {CTL}^*\) and its sublogics \(\mathsf {CTL}\) and \(\mathsf {LTL}\). Moreover, the relation between the semantics of these logics can be expressed by homomorphisms between repetition algebras, which is a more systematic and compact approach than the ones taken in earlier papers.