Local Completeness Logic on Kleene Algebra with Tests

Local Completeness Logic (LCL) has been put forward as a program logic for proving both the correctness and incorrectness of specifications. LCL is an abstract logic, parameterized by domain that allows combining over- under-approximations behaviors. It turns out instantiated to trivial singleton abstraction boils down O’Hearn which us prove presence bugs. recently proved suitable extensions Kleene algebra with tests (KAT) allow representing Hoare logics within same equational framework. In this work, we generalize result showing how KATs extended either modal diamond operator or top element are able represent local completeness LCL. This achieved studying these can be endowed so define validity correctness/incorrectness triples show proof system logically sound and, under some hypotheses, complete.