Values Of Data Types Research Articles

Sums of uncertainty: refinements go gradual

A long-standing shortcoming of statically typed functional languages is that type checking does not rule out pattern-matching failures (run-time match exceptions). Refinement types distinguish different values of datatypes; if a program annotated with refinements passes type checking, pattern-matching failures become impossible. Unfortunately, refinement is a monolithic property of a type, exacerbating the difficulty of adding refinement types to nontrivial programs. Gradual typing has explored how to incrementally move between static typing and dynamic typing. We develop a type system of gradual sums that combines refinement with imprecision. Then, we develop a bidirectional version of the type system, which rules out excessive imprecision, and give a type-directed translation to a target language with explicit casts. We prove that the static sublanguage cannot have match failures, that a well-typed program remains well-typed if its type annotations are made less precise, and that making annotations less precise causes target programs to fail later. Several of these results correspond to criteria for gradual typing given by Siek et al. (2015).

Denotational cost semantics for functional languages with inductive types

A central method for analyzing the asymptotic complexity of a functional program is to extract and then solve a recurrence that expresses evaluation cost in terms of input size. The relevant notion of input size is often specific to a datatype, with measures including the length of a list, the maximum element in a list, and the height of a tree. In this work, we give a formal account of the extraction of cost and size recurrences from higher-order functional programs over inductive datatypes. Our approach allows a wide range of programmer-specified notions of size, and ensures that the extracted recurrences correctly predict evaluation cost. To extract a recurrence from a program, we first make costs explicit by applying a monadic translation from the source language to a complexity language, and then abstract datatype values as sizes. Size abstraction can be done semantically, working in models of the complexity language, or syntactically, by adding rules to a preorder judgement. We give several different models of the complexity language, which support different notions of size. Additionally, we prove by a logical relations argument that recurrences extracted by this process are upper bounds for evaluation cost; the proof is entirely syntactic and therefore applies to all of the models we consider.