A Theory of Indirection via Approximation Aquinas Hobor ∗† Robert Dockins† Andrew W. Appel † National University of Singapore Princeton University Princeton University
[email protected] [email protected] [email protected] Abstract Consider general references in the polymorphic λ-calculus. Building semantic models that account for various kinds of indirect Here is a flawed semantic model of types for this calculus: reference has traditionally been a difficult problem. Indirect refer- value ≡ loc of address + num of N + ... ence can appear in many guises, such as heap pointers, higher-order type ≡ (memtype × value) → T (1) functions, object references, and shared-memory mutexes. memtype ≈ address * type We give a general method to construct models containing indi- rect reference by presenting a “theory of indirection”. Our method Values are a tagged disjoint union, with the tag loc indicating can be applied in a wide variety of settings and uses only simple, a memory address. T is some notion of truth values (e.g., the elementary mathematics. In addition to various forms of indirect propositions of the metalogic or a boolean algebra) and a natural reference, the resulting models support powerful features such as interpretation of the type A → T is the characteristic function for a impredicative quantification and equirecursion; moreover they are set of A. We write ≈ to mean “we wish we could define things this compatible with the kind of powerful substructural accounting re- way” and A*B to indicate a finite partial function from A to B. quired to model (higher-order) separation logic. In contrast to pre- The typing judgment has the form ψ ` v : τ, where ψ is vious work, our model is easy to apply to new settings and has a a memory typing (memtype), v is a value, and τ is a type; the simple axiomatization, which is complete in the sense that all mod- semantic model for the typing judgment is ψ ` v : τ ≡ τ(ψ, v).