published on July 31, 2013

Dependent types, that is type systems where types may depend on values, are the hot new thing. Dependent types seem like a logical succession to Haskell, which people much smarter than me argue is becoming a dependently typed language itself. I am interested in learning more about dependent types and want to list some resources here that I have used or am planing on using.

Why bother learning languages with dependent types? Given that I’ve argued that it is beneficial to encode a program’s invariants in the type system, it is only natural to want a type system that allows you to do more of this. Dependent types give you this possibilitye, however there is a trade off: writing programs in dependently typed languages is more complex (you may have to write some proofs!) and we don’t have that much experience with it.

I can’t really answer the question if the more powerful type system is worth the additional difficulties, but it seems like things are in motion and that now is an interesting time to find out:

New dependently typed languages that are more or less intended to be used as real programming languages have started to appear in recent years. These languages include Agda, Idris and ATS. At least the latter two are clearly intended to be used as

*real programming languages*(as opposed to theorem provers) and they do appear to be quite usuable to the casual observer.New books have come out which make heavy use of the theorem prover Coq and are intended to teach the use of dependently typed languages. These books make the use of Coq accessible to a much larger audience (including me ;).

Domain specific languages (DSLs) and specifically embedded domain specific languages (EDSLs) are becomming more and more important. Dependent types allow you to typecheck these languages with the build-in type checker.

Haskell is moving towards dependent types and so the smart people behind Haskell seem to think this is a good idea. Who am I to disagree?

It is of course quite possible that these indications mean nothing or that it simply looks like a trend to me since I have only recently started to look at this topic. However, if Haskell has taught me one thing then it is that great ideas, however different, may eventually become successful when pursued with the necessary tenacity and that things that look like huge inconveniences (purity!) may actually turn out to be great advantages once we get accustomed to them. Non-total functions have always felt like a wart in Haskell, and that is why I am willing to bet on dependently typed languages now. I think there will be a lot of exploring and a lot of learning before these languages will be anything near mainstream (like where Haskell is now), but now seems like an exciting time to be part of this development.

There are now a number of interesting languages with dependent types. This list makes no attempt to be exhaustive and is slanted towards the things that interest me.

Coq is the 800-pound Gorilla in dependent type land. Coq is first and foremost a theorem prover, but at its heart sits a dependently typed language called Gallina, which itself is an extensions of the

*calculus of indicutive constructions*.Agda is also a theorem prover based on the

*intuitionistic type theory*develop by Martin-Löf. The syntax is heavily influenced by Haskell (as opposed to Coq whose syntax closer to ML). A major difference between Agda and Coq is that Agda has no tactics language for proving theorems.Idris is the new kid on the block, having appeared only in 2011. It is also heavily influenced by Haskell (the introducing paper asked the question

*“What if Haskell had full dependent types?”*). It differs from Coq and Agda in that it is not described as a theorem prover but as a general programming language. Indeed, functions must be annotated if one wants them to be checked for totality. This is a kind of escape hatch that will make developing regular programs easier in Idris. While it has (appart from dependent types) a lot in common with Haskell, it defaults to eager evaluation (with optional lazy evaluation available with special annotations).ATS (which stands for

*applied type system*) looks like a fusion of C and ML with dependent types thrown in for good measure. I am not really sure what to think of this language but at first sight it feels very different from the other ones listed here. What makes this language really interesting is that it is intended for systems programming, i.e. for the domain where one would usually use C or C++. I think this is great because those two have very little competition in their fields.^{1}Furthermore, when programming in C it is so easy to make mistakes that the dependent types and linear types that ATS has could be a real boon. That said, from my very limited impression the language seems a bit messy and therefore I think it is not the best place to start learning about dependent types.

There are also a few books that have come out recently-ish that make dependently typed languages (primarily Coq) much more accessible. This is of course of huge importance to an autodidact like me.

Software foundations by Benjamin Pierce teaches Coq, functional programming, basic typing theory and the universe. There are basically no prerequisites (except for being able to install Emacs ;)) and lots and lots of little excercises. It seems a bit slow at the start but working through all of the excercises will probably give a lot of familiarity with Coq, so that may be worth it.

Certified programming with dependent types by Adam Chlipala. This book seems much more advanced than software foundations. It states in the introduction that it wants to initiate a discussion on best practices for developing certified programs in dependently typed languages. The author argues that every proof should be automated so that no manual steps are required (once the right lemmata have been developed).

Types and programming languages by Benjamin Pierce. This book is not really on dependent types but introduces the foundations for programming language theory such as the typing rules, operational semantics, the (simply typed) lambda calculus, subtyping and a few more. I’ve meant to read it completely but I am stuck half way. It is certainly a very accessible book and a fun read.

A tutorial implmentation of a dependently typed lambda calculus by Andres Löh et al. I am not sure if I can give you a better summary than the title. Doesn’t it just make you want to read the paper?

Martin Löf’s type theory is the foundation for Agda. I think learning the theory might not acutally be necessary for a working understanding of dependent types and to get an idea of what you can do with them, but it would sure be nice to know more about the foundations.

HoTT or

*Homotopy Type Theory*refers to a new interpretation of Martin-Löf’s system of intensional, constructive type theory into abstract homotopy theory. The book’s authors believe that univalent foundations will eventually become a viable alternative to set theory as the “implicit foundation” for the unformalized mathematics done by most mathematicians.^{2}I am not sure I’ll ever make it this far, but it seems like a very interesting theory. Advanced Haskellers seem to get a lot out of category theory, maybe the same will be said for HoTT and dependent types.

There are also a few videos and screencasts which revolve around dependent types or some programming language that features dependent types. First of all there is a four day course on Idris with videos and excercises held by the creator of Idris Edwin Brady. I have found the video and the excercises to be a good way to get started with Idris. There is also an introduction to Agda with nine lectures by Conner McBride.

While I do appreciate some theoretical background, I am not sure that I have the stamina to work through a huge amount of theory without also seeing some applications. I have thus decided to try an approach that combines theory with practice. First of all, I would like to work through *Sofware Foundations*. While this is theoretically a book, there are so many exercises that it nicely combines theory and practice. Once I am done with that I would like to work my way through *Certified Programming with Dependent Types*. At the same time I am going to try to port some nice Haskell program to Idris and attempt to prove the totality of as many functions as possible. Since Wouter Swierstra has already ported Xmonad to Coq, this seems like an interesting candidate. Besides, I am running xmonad and like it a lot, so what could be a better opportunity to learn more about it?

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