Foundational certification of data-flow analyses

Data-flow analyses, such as live variables analysis, available expressions analysis etc., are usefully specifiable as type systems. These are sound and, in the case of distributive analysis frameworks, complete wrt. appropriate natural semantics on abstract properties. Applications include certification of analyses and "optimization" of functional correctness proofs alongside programs. On the example of live variables analysis, we show that analysis type systems are applied versions of more foundational Hoare logics describing either the same abstract property semantics as the type system (liveness states) or a more concrete natural semantics on transition traces of a suitable kind (future defs and uses). The rules of the type system are derivable in the Hoare logic for the abstract property semantics and those in turn in the Hoare logic for the transition trace semantics. This reduction of the burden of trusting the certification vehicle can be compared to foundational proof-carrying code, where general-purpose program logics are preferred to special-purpose type systems and universal logic to program logics. We also look at conditional liveness analysis to see that the same foundational development is also possible for conditional data-flow analyses proceeding from type systems for combined "standard state and abstract property" semantics.