Constructing Hereditary Worlds Within Worlds

Although they appear unrelated, the type system of the polymorphic λ-calculus with references and the assertions of concurrent separation logic with first-class locks share a critical feature: an unsound contravariant circularity in their naı̈ve semantic model. We developed indirection theory to automatically construct, and cleanly axiomatize, step-indexed approximations to these naı̈ve models, as well as a large number of others [HDA10b]. Unfortunately, the previous axiomatization had a flaw. One is usually only interested in using hereditary predicates: those which are closed under the action of approximation. As previously presented, indirection theory allows nonhereditary predicates to exist in certain parts of the construction. Although not fatal, this flaw requires workarounds that are not entirely obvious to the uninitiated. We correct this flaw by presenting a new axiomatization of indirection theory that only permits heredity predicates and show that the new interface is sound by constructing a model. The new axiomatization is somewhat more subtle than the previous one, but it retains the same flavor, cleanliness, and metatheoretic properties. In contrast, the new construction is markedly more complex, especially in a mechanized context. Indeed, our Coq mechanization is one of our key contributions, and accordingly we present it in considerable detail.