Typed Adapton: Refinement types for nominal memoization

Nominal memoization combines memoized functional programming with a controlled form of imperative cache effects. By leveraging these imperative effects, nominal memoization can dramatically outperform traditional (“structural”) memoization. However, the nominal memoization programming model is error-prone: when the programmer unintentionally misuses names, their incremental program ceases to correspond to its purely functional semantics. This paper develops a refinement type system for nominal memoization that enforces a program’s correspondence with its purely functional semantics. Our type system employs set-indexed types in the style of DML (Xi and Pfenning 1999), extended with polymorphism over kinds and index functions. We prove that our type system enforces the dynamic side conditions required by Hammer et al. (2015). Past work shows that these conditions suffice for realistic uses of nominal memoization while also guaranteeing from-scratch consistency of the incremental programs. Furthermore, we propose a decidable form of name-level computations for expressing generic naming strategies in type-generic incremental code.