Functional Program Correctness Through Types

Doctoral thesis, 2007

This thesis addresses the problem of avoiding errors in functional programs. The thesis has three parts, discussing different aspects of program correctness, with the unifying theme that types are an integral part of the methods used to establish correctness. The first part validates a common, but not obviously correct, method for reasoning about functional programs. In this method, dubbed "fast and loose reasoning", programs written in a language with non-terminating functions are treated as if they were written in a total language. It is shown that fast and loose reasoning is sound when the programs are written in a given total subset of the language, and the resulting properties are translated back to the partial setting using certain partial equivalence relations which capture the concept of totality. The second part discusses a method for ensuring that functions meet specified time bounds. The method is aimed at implementations of purely functional data structures, which often make essential use of lazy evaluation to ensure good time complexity in the presence of persistence. The associated complexity analyses are often complicated and hence error-prone, but by annotating the type of every function with its time complexity, using an annotated monad to combine time complexities of subexpressions, it is ensured that no details are forgotten. The last part of the thesis is a case study in programming with strong invariants enforced by the type system. A dependently typed object language is represented in the meta language, which is also dependently typed, in such a way that it is impossible to form ill-typed terms. An interpreter is then implemented for the object language by using normalisation by evaluation. By virtue of the strong types used this implementation is a proof that every term has a normal form, and hence normalisation is proved. This also seems to be the first formal account of normalisation by evaluation for a dependently typed language.