Dependent type
From HaskellWiki
EndreyMark (Talk  contribs) (Links on CurryHoward isomorphism added to ,,Concept'' section / ,,TypeTheory'' subsection) 
(→Cayenne: links updated; link added for constructive type theory) 
(36 intermediate revisions by 14 users not shown) 
Revision as of 20:29, 8 August 2012
Contents 
1 The concept of dependent types
1.1 General
 Wikipedia
 Dependent Types in Programming abstract in APPSEM'2000
 Do we need dependent types? by Daniel Fridlender and Mia Indrika, 2001.
1.2 Type theory
Simon Thompson: Type Theory and Functional Programming. Section 6.3 deals with dependent types, but because of the strong emphasis on CurryHoward isomorphism and the connections between logic and programming, the book seemed cathartic for me even from its beginning.
Another interesting approach to CurryHoward isomorphism and the concept of dependent type: Lecture 9. Semantics and pragmatics of text and dialogue dicsusses these concepts in the context of linguistics. Written by Arne Ranta, see also his online course and other linguistical materials on the Linguistics wikipage.
1.3 Illative combinatory logic
To see how Illative Combinatory logic deals with dependent types, see combinator G described in Systems of Illative Combinatory Logic complete for firstorder propositional and predicate calculus by Henk Barendregt, Martin Bunder, Wil Dekkers. It seems to me that the dependent type construct of Epigram corresponds to in Illative Combinatory Logic. I think e.g. the followings should correspond to each other:
2 Dependently typed languages
2.1 Epigram
Epigram is a full dependently typed programming language, see especially
 Epigram Tutorial by Conor McBride
 and Why dependent types matter by Thorsten Altenkirch, Conor McBride and James McKinna).
Dependent types (of this language) also provide a notforgetful concept of views (already mentioned in the Haskell Future of Haskell#Extensions of Haskell; the connection between these concepts is described in p. 32 of Epigram Tutorial (section 4.6 Patterns Forget; Matching Is Remembering).
See Epigram also as theorem prover.
2.2 Agda
Agda is a system for incrementally developing proofs and programs. Agda is also a functional language with dependent types. This language is similar to Epigram but has a more Haskelllike syntax.
People who are interested also in theorem proving may see the theorem provers page.
2.3 Cayenne
Cayenne is influenced also by constructive type theory.
Dependent types make it possible not to have a separate module language and a core language. This idea may concern Haskell too, see Firstclass module page.
Dependent types make it useful also as a theorem prover.
3 Dependent types in Haskell programming
3.1 Lightweight Dependent Typing
This web page describes the lightweight approach and its applications, e.g., statically safe head/tail functions and the elimination of array bound check (even in such complex algorithms as KnuthMorrisPratt string search). The page also briefly describes `singleton types' (Hayashi and Xi).
3.2 Library
Ivor is type theory based theorem proving library  written by Edwin Brady (see also the author's homepage, there are a lot of materials concerning dependent type theory there).
3.3 Proposals
John Hughes: Dependent Types in Haskell (some ideas).
3.4 Simulating them
 Faking it: Simulating Dependent Types in Haskell, by Conor McBride

SimulatingDependentTypes of HaWiki(404 Error)  The See also section of Type page contains links to many related idioms. Especially type arithmetic seems to me also a way yielding some tastes from dependent type theory.
 On the usefulness of such idioms in practice, see HaskellDB's pages
 updated page (see Papers subsection on Documentation)
 which presupposes reading also paper on the original page (see Documentation subpage, PostScript version)