In computer science and logic, a dependent type is a type that depends on a value. Dependent types play a central role in intuitionistic type theory and in the design of functional programming languages like ATS, Agda and Epigram.

An example is the type of n-tuples of real numbers. This is a dependent type because the type depends on the value n.

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 may become 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 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.

Systems of the lambda cube [edit]

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 [edit]

The system 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.

Writing for -tuples of real numbers, as above, stands for the type of functions 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. is the type of functions from natural numbers to the real numbers, written as in the simply typed lambda calculus.

Second order dependent type theory [edit]

The system of second order dependent types is obtained from by allowing quantification over type constructors. In this theory the dependent product operator subsumes both the operator of simply typed lambda calculus and the binder of System F.

Higher order dependently typed polymorphic lambda calculus [edit]

The higher order system extends 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.

Object-oriented programming [edit]

Some recent research^[1] has been directed at combining dependent type theory with object-oriented programming.

Comparison [edit]

See also [edit]

• Typed lambda calculus
• Design by contract

Footnotes [edit]

1. ^ This refers to the core language, not to any tactic or code generation sublanguage.
2. ^ Subject to semantic constraints, such as universe constraints
3. ^ Ring solver^[3]
4. ^ Optional universes, optional universe polymorphism, and optional explicitly specified universes
5. ^ Universes, automatically inferred universe constraints (not the same as Agda's universe polymorphism) and optional explicit printing of universe constraints
6. ^ Has been superseded by ATS

References [edit]

Further reading [edit]

• Martin-Löf, Per (1984). Intuitionistic Type Theory. Bibliopolis. 
• Nordström, Bengt; Petersson, Kent; Smith, Jan M. (1990). Programming in Martin-Löf's Type Theory: An Introduction. Oxford University Press. 
• Barendregt, Henk (1992). "Lambda calculi with types". In S. Abramsky, D. Gabbay and T. Maibaum. Handbook of Logic in Computer Science. Oxford Science Publications. 
• McBride, Conor; McKinna, James (January 2004). "The view from the left". Journal of Functional Programming 14 (1): 69–111. 
• Altenkirch, Thorsten; McBride, Conor; McKinna, James (April 2005). Why dependent types matter. 
• Norell, Ulf. Towards a practical programming language based on dependent type theory. PhD thesis, Department of Computer Science and Engineering, Chalmers University of Technology, SE-412 96 Göteborg, Sweden, September 2007.
• Oury, Nicolas and Swierstra, Wouter (2008). "The Power of Pi". Accepted for presentation at ICFP, 2008.
• Norell, Ulf (2008). Dependently Typed Programming in Agda.

External [edit]

• Dependently Typed Programming 2008
• Dependently Typed Programming 2010
• Dependently Typed Programming 2011
• "Dependent type" at the Haskell Wiki