Hi Pat, you might be interested in Agda [1] and the tutorial "Dependently Typed Programming in Agda" [2]. I'm just reading the tutorial and it helps me a lot with understanding the concept. Also if you'd like to have a deeper understanding of the underlying theory, I'd suggest reading Barendregt's "Lambda Calculi with Types" [3]. Best regards, Petr [1] http://wiki.portal.chalmers.se/agda/ [2] http://www.cse.chalmers.se/~ulfn/papers/afp08/tutorial.pdf [3] http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.26.4391 and ftp://ftp.cs.kun.nl/pub/CompMath.Found/HBKJ.ps.Z On Fri, Oct 09, 2009 at 02:11:30PM +0100, pat browne wrote:
Hi, I am trying to understand the concept of dependent types as described in Haskell literature (Hallgren 2001),(Hinze,Jeuring et al. 2006). I am particularly interested in understanding two flavours of dependency mentioned by Hinze et al: 1) Types depending on values called dependent types 2) Types depending on types called parametric and type-indexed types
I think that ascending levels of abstraction are important here: values - types - classes (Hallgren 2001). Here is a Haskell example of the use of DT:
class Named object name | object -> name where name :: object -> name
instance (Eq name, Named object name) => Eq object where object1 == object2 = (name object1) == (name object2)
I think that the intended semantics are: 1) Objects have names and can be compared for equality using these names. 2) Two objects are considered equal (identical) if they have the same name. 3) The type of a name depends on the type of the object, which gets expressed as a type dependency (object -> name). I am not sure about the last semantic it might be interpreted as "the named object depends on..". This may indicate a flaw in my understanding of DT.
I am aware that dependent types can be used to express constraints on the size of lists and other collections. My understanding of Haskell's functional dependency is that object -> name indicates that fixing the type object should fix the type name (equivalently we could say that type name depends on type object). The class definition seems to show a *type-to-type* dependency (object to name), while the instance definition shows how a name value is used to define equality for objects which could be interpreted as a *value-to-type* dependency (name to object) in the opposite direction to that of the class.
My questions are: Question 1: Is my understanding correct? Question 2: What flavour of DT is this does this example exhibit?
Best regards, Pat
Hallgren, T. (2001). Fun with Functional Dependencies. In the Proceedings of the Joint CS/CE Winter Meeting, Varberg, Sweden, January 2001. Hinze, R., J. Jeuring, et al. (2006). Comparing Approaches to Generic Programming in Haskell. Datatype-generic programming: international spring school, SSDGP 2006.
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