Dependent type

In computer science and logic, a dependent type is a type whose definition depends on a value. It is an overlapping feature of type theory and type systems. In intuitionistic type theory, dependent types are used to encode logic's quantifiers like "for all" and "there exists". In functional programming languages like Agda, ATS, Coq, F*, Epigram, and Idris, dependent types may help reduce bugs by enabling the programmer to assign types that further restrain the set of possible implementations.
Two common examples of dependent types are dependent functions and dependent pairs. The return type of a dependent function may depend on the "value" (not just type) of one of its arguments. For instance, a function that takes a positive integer formula_1 may return an array of length formula_1, where the array length is part of the type of the array. (Note that this is different from polymorphism and generic programming, both of which include the type as an argument.) A dependent pair may have a second value of which the type depends on the first value. Sticking with the array example, a dependent pair may be used to pair an array with its length in a type-safe way.
Dependent types add complexity to a type system. Deciding the equality of dependent types in a program may require computations. If arbitrary values are allowed in dependent types, then deciding type equality may involve deciding whether two arbitrary programs produce the same result; hence type checking may become undecidable.
History.
In 1934, Haskell Curry noticed that the types used in typed lambda calculus, and in its combinatory logic counterpart, followed the same pattern as axioms in propositional logic. Going further, for every proof in the logic, there was a matching function (term) in the programming language. One of Curry's examples was the correspondence between simply typed lambda calculus and intuitionistic logic.
Predicate logic is an extension of propositional logic, adding quantifiers. Howard and de Bruijn extended lambda calculus to match this more powerful logic by creating types for dependent functions, which correspond to "for all", and dependent pairs, which correspond to "there exists".
Formal definition.
Π type.
Loosely speaking, dependent types are similar to the type of an indexed family of sets. More formally, given a type formula_3 in a universe of types formula_4, one may have a family of types formula_5, which assigns to each term formula_6 a type formula_7. We say that the type "B(a)" varies with "a".
A function whose type of return value varies with its argument (i.e. there is no fixed codomain) is a dependent function and the type of this function is called dependent product type, pi-type or dependent function type. From a family of types formula_8 we may construct the type of dependent functions formula_9, whose terms are functions which take a term formula_10 and return a term in formula_11. For this example, the dependent function type is typically written as formula_12 formula_13 or formula_14
If formula_5 is a constant function, the corresponding dependent product type is equivalent to an ordinary function type. That is, formula_16 is judgmentally equal to formula_17 when "B" does not depend on "x".
The name 'pi-type' comes from the idea that these may be viewed as a Cartesian product of types. Pi-types can also be understood as models of universal quantifiers.
For example, if we write formula_18 for "n"-tuples of real numbers, then formula_19 would be the type of a function which, given a natural number "n", returns a tuple of real numbers of size "n". The usual function space arises as a special case when the range type does not actually depend on the input. E.g. formula_20 is the type of functions from natural numbers to the real numbers, which is written as formula_21 in typed lambda calculus.
Σ type.
The dual of the dependent product type is the dependent pair type, dependent sum type, sigma-type, or (confusingly) dependent product type. Sigma-types can also be understood as existential quantifiers. Continuing the above example, if, in the universe of types formula_4, there is a type formula_3 and a family of types formula_5, then there is a dependent pair type formula_25. (The alternate notations are similar to that of Π types.)
The dependent pair type captures the idea of an ordered pair where the type of the second term is dependent on the value of the first. If formula_26 then formula_6 and formula_28. If "B" is a constant function, then the dependent pair type becomes (is judgementally equal to) the product type, that is, an ordinary Cartesian product formula_29.
Example as existential quantification.
Let formula_3 be some type, and let formula_5. By the Curry–Howard correspondence, "B" can be interpreted as a logical predicate on terms of "A". For a given formula_6, whether the type "B(a)" is inhabited indicates whether "a" satisfies this predicate. The correspondence can be extended to existential quantification and dependent pairs: the proposition formula_33 is true if and only if the type formula_34 is inhabited.
For example, formula_35 is less than or equal to formula_36 if and only if there exists another natural number formula_37 such that "m" + "k" = "n". In logic, this statement is codified by existential quantification: formula_38 This proposition corresponds to the dependent pair type: formula_39 That is, a proof of the statement that "m" is less than "n" is a pair that contains both a positive number "k", which is the difference between "m" and "n", and a proof of the equality "m" + "k" = "n".
Systems of the lambda cube.
Henk Barendregt developed the lambda cube as a means of classifying type systems along three axes. The eight corners of the resulting cube-shaped diagram each correspond to a type system, with simply typed lambda calculus in the least expressive corner, and calculus of constructions in the most expressive. The three axes of the cube correspond to three different augmentations of the simply typed lambda calculus: the addition of dependent types, the addition of polymorphism, and the addition of higher kinded type constructors (functions from types to types, for example). The lambda cube is generalized further by pure type systems.
First order dependent type theory.
The system formula_40 of pure first order dependent types, corresponding to the logical framework LF, is obtained by generalising the function space type of the simply typed lambda calculus to the dependent product type.
Second order dependent type theory.
The system formula_41 of second order dependent types is obtained from formula_40 by allowing quantification over type constructors. In this theory the dependent product operator subsumes both the formula_43 operator of simply typed lambda calculus and the formula_44 binder of System F.
Higher order dependently typed polymorphic lambda calculus.
The higher order system formula_45 extends formula_41 to all four forms of abstraction from the lambda cube: functions from terms to terms, types to types, terms to types and types to terms. The system corresponds to the calculus of constructions whose derivative, the calculus of inductive constructions is the underlying system of the Coq proof assistant.
Simultaneous programming language and logic.
The Curry–Howard correspondence implies that types can be constructed that express arbitrarily complex mathematical properties. If the user can supply a constructive proof that a type is "inhabited" (i.e., that a value of that type exists) then a compiler can check the proof and convert it into executable computer code that computes the value by carrying out the construction. The proof checking feature makes dependently typed languages closely related to proof assistants. The code-generation aspect provides a powerful approach to formal program verification and proof-carrying code, since the code is derived directly from a mechanically verified mathematical proof.