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.

For a growing number of people, animal testing conjures up images of small, defenceless creatures being poisoned by chemicals, or having caustic substances sprayed in their eyes. But for others, testing on animals is a necessary step in finding cures for diseases that cause extensive human suffering. It's one thing to argue that animals should not suffer for the sake of an improved hair dye, but more difficult to object to animal testing in the quest for, say, a cure for cancer. Fortunately, there is some room for compromise. It is possible to find viable alternatives to tests on live animals and, thanks to technology, at least some of them can saved without abandoning important research.