A Correctness and Incorrectness Program Logic

Abstract interpretation is a well-known and extensively used method to extract over-approximate program invariants by a sound program analysis algorithm. Soundness means that no program errors are lost and it is, in principle, guaranteed by construction. Completeness means that the abstract interpreter reports no false alarms for all possible inputs, but this is extremely rare because it needs a very precise analysis. We introduce a weaker notion of completeness, called local completeness, which requires that no false alarms are produced only relatively to some fixed program inputs. Based on this idea, we introduce a program logic, called Local Completeness Logic for an abstract domain A, for proving both the correctness and incorrectness of program specifications. Our proof system, which is parameterized by an abstract domain A, combines over- and under-approximating reasoning. In a provable triple ⊦A [p] 𝖼 [q], 𝖼 is a program, q is an under-approximation of the strongest post-condition of 𝖼 on input p such that their abstractions in A coincide. This means that q is never too coarse, namely, under some mild assumptions, the abstract interpretation of 𝖼 does not yield false alarms for the input p iff q has no alarm. Therefore, proving ⊦A [p] 𝖼 [q] not only ensures that all the alarms raised in q are true ones, but also that if q does not raise alarms, then 𝖼 is correct. We also prove that if A is the straightforward abstraction making all program properties equivalent, then our program logic coincides with O’Hearn’s incorrectness logic, while for any other abstraction, contrary to the case of incorrectness logic, our logic can also establish program correctness.