  Kleene algebra with tests (KAT) is a foundational equational framework for
reasoning about programs, which has found applications in program
transformations, networking and compiler optimizations, among many other areas.
In his seminal work, Kozen proved that KAT subsumes propositional Hoare logic,
showing that one can reason about the (partial) correctness of while programs
by means of the equational theory of KAT. In this work, we investigate the
support that KAT provides for reasoning about incorrectness, instead, as
embodied by Ohearn's recently proposed incorrectness logic. We show that KAT
cannot directly express incorrectness logic. The main reason for this
limitation can be traced to the fact that KAT cannot express explicitly the
notion of codomain, which is essential to express incorrectness triples. To
address this issue, we study Kleene Algebra with Top and Tests (TopKAT), an
extension of KAT with a top element. We show that TopKAT is powerful enough to
express a codomain operation, to express incorrectness triples, and to prove
all the rules of incorrectness logic sound. This shows that one can reason
about the incorrectness of while-like programs by means of the equational
theory of TopKAT.
