In
computer science and
logic, a
dependent type is a
type which depends on a value. Dependent types play a central role in
intuitionistic type theory and in the design of experimental
functional programming languages like
Dependent ML and
Epigram.
An example is the type of n-tuples of real numbers, which we may denote as 
. This is a dependent type because the type depends on the value 
. Deciding 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 becomes undecidable.
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 then 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.
The system ?P 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.
