Dependent type

In computer science and logic, a dependent type is a type whose definition depends on a value. A "pair of integers" is a type. A "pair of integers where the second is greater than the first" is a dependent type because of the dependence on the 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, Epigram and Idris, dependent types prevent bugs by allowing extremely expressive types.

Two common examples of dependent types are dependent functions and dependent pairs. A dependent function's return type may depend on the value (not just type) of an argument. A function that takes a positive integer "n" may return an array of length "n". (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 that depends on the first. It can be used to encode a pair of integers where the second one is greater than the first.

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

Dependent types were created to deepen the connection between programming and logic.

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".

(Because of this and other work by Howard, propositions-as-types is known as the Curry-Howard correspondence.)

Formal definition

Loosely speaking, dependent types are similar to the type of an indexed family of sets. More formally, given a type $A:\mathcal{U}$ in a universe of types $\mathcal{U}$, one may have a family of types $B:A\to\mathcal{U}$, which assigns to each term $a:A$ a type $B(a):\mathcal{U}$. 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 simply dependent type. For this example, the dependent type would be written as

$\Pi_{(x:A)}B(x)$

or as

$\Pi (x:A),B(x)$

If $B:A\to\mathcal{U}$ is a constant function, the corresponding dependent product type is equivalent to an ordinary function type. That is, $\Pi_{(x:A)}B$ is judgementally equal to $A\to B$.

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, writing $\mbox{Vec}({\mathbb R},n)$ for $n$-tuples of real numbers, then $\Pi_{(n:{\mathbb N})} \mbox{Vec}({\mathbb R},n)$ would be the type of functions which, given a natural number n, return 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. $\Pi_{(n:{\mathbb N})}\; {\mathbb R}$ is the type of functions from natural numbers to the real numbers, which is written as ${\mathbb N}\to{\mathbb R}$ in the simply typed lambda calculus.

Polymorphic functions are an important example of dependent functions, that is, functions having dependent type. Given a type, these functions act on elements of that type (or on elements of a type constructed (derived, inherited) from that type). A polymorphic function returning elements of type C would have a polymorphic type written as

$\Pi_{(A:\mathcal{U})} A\to C$

Dependent pair type

The opposite of the dependent product type is the dependent pair type, dependent sum type or sigma-type. It is analogous to the coproduct or disjoint union. Sigma-types can also be understood as existential quantifiers. Notationally, it is written as

$\Sigma_{(x:A)} B(x)$

The dependent pair type captures the idea of an indexed pair, where the type of the second term is dependent on the first. Thus, if

$(a,b):\Sigma_{(x:A)} B(x)$

then $a:A$ and $b:B(a)$. If B is a constant, then the dependent pair type becomes (is judgementally equal to) the product type, that is, an ordinary Cartesian product $A\times B$.

Example as existential quantification

Let $\Sigma A B_a: \mathcal{U}$ be sigma-type quantifying over type $A: \mathcal{U}$ with predicate $B: A \rarr \mathcal{U}$. There exists an $A$ such that $B_a$ holds if and only if $\Sigma A B_a$ is inhabited. For example, a is less than or equal to b iff there exists a natural number n and a proof that a+n=b.

$a\le b \iff \Sigma \mathbb{N} (\lambda n \rarr a+n=b)$

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 $\lambda \Pi$ 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 $\lambda \Pi 2$ of second order dependent types is obtained from $\lambda \Pi$ by allowing quantification over type constructors. In this theory the dependent product operator subsumes both the $\to$ operator of simply typed lambda calculus and the $\forall$ binder of System F.

Higher order dependently typed polymorphic lambda calculus

The higher order system $\lambda \Pi \omega$ extends $\lambda \Pi 2$ 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.

Comparison of languages with dependent types

See also: Proof assistant § Comparison

Language	Actively developed	Paradigm	Tactics	Proof terms	Termination checking	Types can depend on	Universes	Proof irrelevance	Program extraction	Extraction erases irrelevant terms
Agda	Yes	Purely functional	Few/limited	Yes	Yes (optional)	Any term	Yes (optional)	Proof-irrelevant arguments (experimental)	Haskell, Javascript	Yes
ATS	Yes	Functional / imperative	No	Yes	Yes	?	?	?	Yes	?
Cayenne	No	Purely functional	No	Yes	No	Any term	No	No	?	?
Gallina (Coq)	Yes	Purely functional	Yes	Yes	Yes	Any term	Yes	No	Haskell, Scheme and OCaml	Yes
Dependent ML	No	?	?	Yes	?	Natural numbers	?	?	?	?
F*	Yes	Functional and imperative	No	Yes	Yes (optional)	Any pure term	Yes	Yes	OCaml and F#	Yes
Guru	No	Purely functional	hypjoin	Yes	Yes	Any term	No	Yes	Carraway	Yes
Idris	Yes	Purely functional	Yes	Yes	Yes (optional)	Any term	Yes	No	Yes	Yes, aggressively
Matita	Yes	Purely functional	Yes	Yes	Yes	Any term	Yes	Yes	OCaml	Yes
NuPRL	Yes	Purely functional	Yes	Yes	Yes	Any term	Yes	?	Yes	?
Perl 6	Yes	Functional and imperative	No	No	No	Any term	No	No	No	No
PVS	Yes	?	Yes	?	?	?	?	?	?	?
Sage	No	Purely functional	No	No	No	?	No	?	?	?
Twelf	Yes	Logic programming	?	Yes	Yes (optional)	Any (LF) term	No	No	?	?
Xanadu	No	Imperative	?	?	?	?	?	?	?	?

See also

• Typed lambda calculus
• Intuitionistic type theory

Further reading

• 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 links

• Dependently Typed Programming 2008
• Dependently Typed Programming 2010
• Dependently Typed Programming 2011
• "Dependent type" at the Haskell Wiki
• dependent type theory in nLab
• dependent type in nLab
• dependent product type in nLab
• dependent sum type in nLab
• dependent product in nLab
• dependent sum in nLab