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