CS102: Introduction to Computer Science II

Especially since the development of Hindley–Milner type inference in the 1970s, functional programming languages have tended to use typed lambda calculus, rejecting all invalid programs at compilation time and risking false positive errors, as opposed to the untyped lambda calculus, that accepts all valid programs at compilation time and risks false negative errors, used in Lisp and its variants (such as Scheme), though they reject all invalid programs at runtime, when the information is enough to not reject valid programs. The use of algebraic datatypes makes manipulation of complex data structures convenient; the presence of strong compile-time type checking makes programs more reliable in absence of other reliability techniques like test-driven development, while type inference frees the programmer from the need to manually declare types to the compiler in most cases.

Some research-oriented functional languages such as Coq, Agda, Cayenne, and Epigram are based on intuitionistic type theory, which lets types depend on terms. Such types are called dependent types. These type systems do not have decidable type inference and are difficult to understand and program with. But dependent types can express arbitrary propositions in predicate logic. Through the Curry–Howard isomorphism, then, well-typed programs in these languages become a means of writing formal mathematical proofs from which a compiler can generate certified code. While these languages are mainly of interest in academic research (including in formalized mathematics), they have begun to be used in engineering as well. Compcert is a compiler for a subset of the C programming language that is written in Coq and formally verified.

A limited form of dependent types called generalized algebraic data types (GADT's) can be implemented in a way that provides some of the benefits of dependently typed programming while avoiding most of its inconvenience. GADT's are available in the Glasgow Haskell Compiler, in OCaml (since version 4.00) and in Scala (as "case classes"), and have been proposed as additions to other languages including Java and C#.