Dependent type

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

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.

Writing ${\mbox{Vec}}({\mathbb {R} },n)$ for $n$ -tuples of real numbers, as above, $\Pi n:{\mathbb {N} }.{\mbox{Vec}}({\mathbb {R} },n)$ 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. $\Pi n:{\mathbb {N} }.{\mathbb {R} }$ is the type of functions from natural numbers to the real numbers, written as ${\mathbb {N} }\to {\mathbb {R} }$ in the simply typed lambda calculus.

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.

Object-oriented programming

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

Comparison

Language	Actively developed	Paradigm^{[fn 1]}	Tactics	Proof terms	Termination checking	Types can depend on^{[fn 2]}	Universes	Proof irrelevance	Program extraction	Extraction erases irrelevant terms
Agda	Yes^[2]	Purely functional	Few/limited^{[fn 3]}	Yes	Yes (optional)	Any term	Yes (optional)^{[fn 4]}	Proof-irrelevant arguments (experimental)^[4]	Haskell, Javascript	Yes^[4]
ATS	Yes^[5]	Functional / imperative	No^[6]	Yes	Yes	?	?	?	Yes	?
Cayenne	No	Purely functional	No	Yes	No	Any term	No	No	?	?
Coq	Yes^[7]	Purely functional	Yes	Yes	Yes	Any term	Yes^{[fn 5]}	No	Haskell, Scheme and OCaml	Yes
Dependent ML	No^{[fn 6]}	?	?	Yes	?	Natural numbers	?	?	?	?
Epigram 2	Yes^[8]	Purely functional	No	Coming soon^{[needs update]}	By construction	Any term	Coming soon^{[needs update]}	Coming soon^{[needs update]}	Coming soon^{[needs update]}	Coming soon^{[needs update]}
Guru	Yes^[9]	Purely functional^[10]	hypjoin^[11]	Yes^[10]	Yes	Any term	No	Yes	Carraway	Yes
Idris	Yes^[12]	Purely functional^[13]	Yes^[14]	Yes	Coming soon^{[needs update]}	Any term	No	No	Yes	Yes, aggressively^[14]
Matita	Yes^[15]	Purely functional	Yes	Yes	Yes	Any term	Yes	?	OCaml	?
NuPRL	No	Purely functional	Yes	Yes	Yes	Any term	Yes	?	Yes	?
F*	Yes	Functional /imperative	?	?	?	?	?	?	?	?
PVS	Yes	?	Yes	?	?	?	?	?	?	?
Typed Racket	Yes	No	?	?	?	?	?	?	?	?
Sage	?	Hybrid typechecking	?	?	?	?	?	?	?	?
Twelf	Yes	Logic programming	?	Yes	Yes (optional)	Any (LF) term	No	No	?	?
Xanadu	No^[16]	Imperative	?	?	?	?	?	?	?	?

Footnotes

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

References

^ Anton Setzer (2007). "Object-oriented programming in dependent type theory". In Henrik Nilsson (ed.). Trends in Functional Programming, vol. 7 (PDF). Intellect. pp. 91–108.
^ "Agda download page".
^ "Agda Ring Solver".
^ ^a ^b "Announce: Agda 2.2.8".
^ "ATS Changelog".
^ "email from ATS inventor Hongwei Xi".
^ "Coq CHANGES in Subversion repository".
^ "Epigram homepage".
^ "Guru SVN (1.0 branch)".
^ ^a ^b Aaron Stump (6 April 2009). "Verified Programming in Guru" (PDF). Retrieved 28 September 2010.
^ Adam Petcher (1 April 2008). "Deciding Joinability Modulo Ground Equations in Operational Type Theory" (PDF). Retrieved 14 October 2010.
^ "Idris git repository".
^ "Idris, a language with dependent types - extended abstract" (PDF).
^ ^a ^b Edwin Brady. "How does Idris compare to other dependently-typed programming languages?".
^ "Matita SVN".
^ "Xanadu home page".

External

[2] This refers to the core language, not to any tactic or code generation sublanguage.

[3] Subject to semantic constraints, such as universe constraints

[6] Ring solver^[3]

[7] Optional universes, optional universe polymorphism, and optional explicitly specified universes

[12] Universes, automatically inferred universe constraints (not the same as Agda's universe polymorphism) and optional explicit printing of universe constraints

[13] Has been superseded by ATS

[setzer-oo-1] Anton Setzer (2007). "Object-oriented programming in dependent type theory". In Henrik Nilsson (ed.). Trends in Functional Programming, vol. 7 (PDF). Intellect. pp. 91–108.

[4] "Agda download page".

[5] "Agda Ring Solver".

[agda-2.2.8-8] "Announce: Agda 2.2.8".

[9] "ATS Changelog".

[10] "email from ATS inventor Hongwei Xi".

[11] "Coq CHANGES in Subversion repository".

[14] "Epigram homepage".

[15] "Guru SVN (1.0 branch)".

[guru-book-16] Aaron Stump (6 April 2009). "Verified Programming in Guru" (PDF). Retrieved 28 September 2010.

[hypjoin-paper-17] Adam Petcher (1 April 2008). "Deciding Joinability Modulo Ground Equations in Operational Type Theory" (PDF). Retrieved 14 October 2010.

[18] "Idris git repository".

[19] "Idris, a language with dependent types - extended abstract" (PDF).

[idris-compare-20] Edwin Brady. "How does Idris compare to other dependently-typed programming languages?".

[21] "Matita SVN".

[22] "Xanadu home page".

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