Approximate Dependent-Type Programming

Haskell is not a dependently-typed language -- at least it was not designed that way. Nevertheless, even Haskell98 admits some form of dependent types, enough to statically prevent the annoying ``head of empty list'' error. After multi-parameter type classes with functional dependencies were introduced, it was quickly (Hallgren, 2001) realized that we can compute with types. HList (2004) has lifted the entire list library to the type level, opening the floodgates of type computation, all the way to SkewLists (Martinez et al., PEPM 2013) and RSA. Later on, type families have made the programming with types deceptively more functional. 2004 brought another key realization: polymorphic recursion lets us dynamically select among the family of types and hence makes the selected type in effect depend on a run-time value. Singleton programming and GADTs perfected this pattern. ``This much is clear: many programmers are already finding practical uses for the approximants to dependent types which mainstream functional languages (especially Haskell) admit, by hook or by crook.'' (Altenkirch et al., 2005)

This page is a panorama of approximations to dependently-typed programming. We will see how closely Haskell comes to dependent types, yet not reaching them. No worry: cruder approximations turn out just as, if not more, practically useful as the overweight approaches.


What is a dependent type

An ordinary type such as [a] may depend on other types -- in our case, the type of list elements -- but not on the values of those elements or their number. A dependent type does depend on such dynamic values. For example, in the hypothetical dependent Haskell, the function to produce a list with n copies of a given value could have the following signature
    replicate :: (n::Nat) -> a -> List n a
List n a is the type of lists with elements of type a and the length exactly n, where n is a non-negative integer Nat. To be more precise, List n a is a family of types (of lists of various lengths with elements of type a); the natural number n selects one member of that family: ``indexes within the family''. Such an indexing is one of the common manifestations of dependent types. The index n is the argument of replicate: its value will be known only when an application of replicate is carried out, at run-time.

The function to append length-indexed lists List n a naturally has the signature:

    append :: List n a -> List m a -> List (n+m) a
where n+m is the ordinary addition of two natural numbers n and m. Thus not only values may appear in types but also arbitrary expressions (terms). The pay-off for including the length of the list in its type is being able to give head and tail the signatures:
    head :: List (succ n) a -> a
    tail :: List (succ n) a -> List n a
that make their applications to the empty list ill-typed. Taking the head/tail of an empty list -- the functional programming equivalent of the infamous NullPointerException -- is hence statically prevented. We no longer have to puzzle out the location of the Prelude.head: empty list exceptions or wonder how to deal with it in the production code: the above head and tail do not raise any exceptions.

The Curry-Howard correspondence regards types as propositions and programs as proofs. In simply-typed systems, types (propositions) are made of type constants such as Int (atomic propositions), combined by the function arrow (implication), pairing (conjunction), etc. Simply-typed systems hence correspond to propositional logic. First-order predicate logic distinguishes between terms (denoting ``things'': elements of some domain) and propositions, stating properties and relations among terms. Propositions like Prime(n) naturally contain terms, n in our case. Dependent types thus are the Curry-Howard interpretation of the first-order predicate logic. Therefore, any program specification written in first-order logic can be expressed as a dependent type, to be checked by the compiler. ``Dependently typed programs are, by their nature, proof carrying code.'' (Altenkirch et al.)

One should distinguish a dependent type (which depends on a dynamic value) from a polymorphic type such as Maybe a. The type List n a from our running example is indexed by the value (the list length) and by the type (of its elements): it is both dependent (in n) and polymorphic (or, technically, parametric), in a. In Agda, List would have the type Nat -> Set -> Set, which clearly shows the distinction between the dependency and the parametric polymorphism. The first argument of List is a natural number, an element of the domain Nat. The second argument of List is a Set -- itself a proposition. Parametric polymorphism hence has the Curry-Howard correspondence to (a fragment of a) second-order logic. The different nature of the two arguments in List n a has many consequences: since the list length n is a natural number, one may apply to it any operation on natural numbers such as addition, increment, multiplication, etc. -- as we have seen in the append example. On the other hand, there is much less one can do with types. List concatenation and many similar operations keep the type of list elements but change its length.

The main issue of dependently-typed systems is illustrated by the following simple example (which will be our running example):

    pcomm :: Bool -> List n a -> List m a -> List (n+m) a
    pcomm b l1 l2 = if b then append l1 l2 else append l2 l1
The type checker has to verify that append l1 l2 and append l2 l1 in the conditional branches have the same type. Recalling the signature of append, the type checker has to ascertain that n+m is equal to m+n. Now, deciding if two types are the same involves determining if two expressions are equal, which is generally undecidable (think of functions or recursive expressions). There is a bigger problem: in our example, n and m are just variables, whose values will be known only at run-time. The type-checker, which runs at compile-time, therefore has to determine that n+m is equal to m+n  without knowing the concrete values of n and m. We know that the natural addition is commutative, but the type-checker does not. It is usually not so smart to figure out the commutativity from the definition of addition. Therefore, we have to somehow supply the proof of the commutativity to the type-checker. Programming with dependent types involves a great deal of theorem proving.
Thorsten Altenkirch, Conor McBride, and James McKinna: Why Dependent Types Matter. April 2005


Non-empty lists

Errors such as taking head or tail of the empty list in Haskell are equivalent to the dereferencing of the zero pointer in C/C++ or NullPointerException in Java. These errors occur because the domain of the function is smaller than the function's type suggests. For example, the type of head says that the function applies to any list. In reality, it can be meaningfully applied only to a non-empty list. One can eliminate such errors by giving functions head and tail a more precise type, such as FullList a. Languages like Cyclone and Cw do exactly that.

We stress that the head-of-empty-list errors can be eliminated now, without any modification to the Haskell type system, without developing any new tool. Already Haskell98 can do that. The same technique applies to OCaml and even Java and C++. The only required advancement is in our thinking and programming style.

Thinking of full lists as a separate type from ordinary, potentially empty lists does affect our programming style -- but it does not have to break the existing code. The new style is easy to introduce gradually. Besides safety, its explicitness makes list processing algorithms more insightful, separating out algorithmically meaningful empty list checks from the redundant safety checks. Let us see some examples.

Assume the following interface

    type FullList a -- abstract; for an implementation, see below
    fromFL :: FullList a -> [a]
    indeedFL :: [a] -> w -> (FullList a -> w) -> w  -- an analogue of `maybe'
    headS :: FullList a -> a
    tailS :: FullList a -> [a]
    -- Adding something to a general list surely gives a non-empty list
    infixr 5 !:
    class Listable l where
        (!:) :: a -> l a -> FullList a
All the above functions are total: in particular, headS and tailS. The operation fromFL lets us forget that the list is FullList, giving the ordinary list. The application headS [] obviously does not type-check. Less obvious is that the application headS some_expression will also fail to type check unless we can statically assure some_expression produces a FullList. To see how easy or difficult it may be to obtain that assurance, let's take a couple of examples. First is the accumulating list reversal function, explicitly using head and tail:
    regular_reverse :: [a] -> [a]
    regular_reverse l = loop l []
       loop [] accum = accum
       loop l  accum = loop (Prelude.tail l) (Prelude.head l : accum)
Let us re-write it with the safe head and tail functions:
    safe_reverse :: [a] -> [a]
    safe_reverse l = loop l [] 
       loop l accum = indeedFL l accum $
                         (\l -> loop (tailS l) (headS l : accum))
    test1 = safe_reverse [1,2,3]
That was straightforward: we relied on indeedFL to perform the case analysis on the argument list (if it is empty or not). We had to do this analysis anyway, according to the algorithm. In the case of a non-empty list, we statically know that l is non-empty and so the applications of headS and tailS are well-typed. Even the type-checker can see that. Here is another example, which should be self-explanatory.
    safe_append :: [a] -> FullList a -> FullList a
    safe_append [] l    = l
    safe_append (h:t) l = h !: safe_append t l
    l1 :: FullList Int
    l1 = 1 !: 2 !: []
    -- We can apply safe_append on two FullList without any problems
    test5 = tailS $ safe_append (fromFL l1) l1
    -- [2,1,2]
Before further discussing this programming style and the problem of backwards compatibility (``do we have to re-write all the code?''), let us see the implementation. Recall, our goal is to define the type of FullList, of lists that are guaranteed to be non-empty. One can see two approaches: one particular and insightful, and the other is generic and insightful. We start with the first one: a non-empty list for sure has at least one element, to be carried around explicitly:
    data FullList a = FullList a [a]  -- carry the list head explicitly
    fromFL :: FullList a -> [a]
    fromFL (FullList x l) = x : l
    headS :: FullList a -> a
    headS (FullList x _) = x
    tailS :: FullList a -> [a]
    tailS (FullList _ x) = x
The other functions of our interface are equally straightforward, see the source code for details. The undisputed advantage of the implementation is its obvious correctness: FullList represents a non-empty list by its very construction, without any pre-conditions and reservations. FullList truly is a non-empty list. There is a deep satisfaction in finding a data structure that ensures a desired property by its very construction, where the property is built-in. In this respect, one can't help but think of Chris Okasaki's work.

There is another approach of representing FullList, which easily generalizes to other structures.

    module NList (FullList, fromFL, headS, tailS, ...) where
    newtype FullList a = FullList [a]  -- data constructor is not exported!
    fromFL (FullList x) = x
    -- The following are _total_ functions
    -- They are guaranteed to be safe, and so we could have used
    -- unsafeHead# and unsafeTail# if GHC provided them.
    headS :: FullList a -> a
    headS (FullList (x:_)) = x
    tailS :: FullList a -> [a]
    tailS (FullList (_:x)) = x
We introduce the abstract data type FullList -- abstract in the sense that its constructor, also named FullList, is not exported. The only way to construct and manipulate the values of that type is to use the operations exported by the module. All the exported operations ensure that FullList represents a non-empty list.

One may regard the abstract type FullList as standing for the proposition (invariant) that the represented list is non-empty. Now we have something to prove: we have to verify, manually or semi-automatically, that all operations within the module NList whose return type is FullList respect the invariant and ensure the truth of the non-emptiness proposition. Once we have verified these exported constructors, all operations that consume FullList, within NList or outside, can take this non-emptiness proposition for granted. Therefore, we are justified in using unsafe-head operations in implementing headS. Compared to the first implementation, the fact that FullList represents a non-empty list is no longer obvious and has to be proven. Fortunately, we only have to prove the operations within the NList module, that is, the ones that make use of the data constructor FullList. All other functions, which produce FullList merely by invoking the operations of NList, ensure the non-emptiness invariant by construction and do not need a proof. The advantage of the second implementation is the easy generalization to ByteStrings, Text, and other sequences and collections. The data constructor FullList is merely a newtype wrapper, with no run-time overhead. Thus the second implementation provides the safety of head and tail operations without sacrificing efficiency.

In the old (2006) discussion of non-empty lists on Haskell-Cafe, Jan-Willem Maessen wrote: ``In addition, we have this rather nice assembly of functions which work on ordinary lists. Sadly, rewriting them all to also work on NonEmptyList or MySpecialInvariantList is a nontrivial task.'' Backwards compatibility is indeed a serious concern: no matter how better a new programming style may be, the mere thought of re-writing the existing code is a deterrent. Suppose we have a function foo:: [a] -> [a] (whose code, if available, we'd rather not change) and we want to write something like

    \l -> [head l, head (foo l)]
The first attempt of using the safe functions
    \l -> indeedFL l onempty (\l -> [headS l, headS (foo l)])
does not type: foo applies to [a] rather than FullList a; furthermore, the result of foo is not FullList a, required by headS. The first problem is easy to solve: FullList a can always be cast into the general list, with fromFL. We insist on writing the latter function explicitly, which keeps the type system simple, free of subtyping and implicit coercions. One may regard fromFL as an analogue of fromIntegral -- which, too, we have to write explicitly, in any code with more than one sort of integral numbers (e.g., Int and Integer, or Int and CInt).

If we are not sure if our foo maps non-empty lists to non-empty lists, we really should handle the empty list case:

    \l -> indeedFL l onempty $
   	\l -> [headS l, indeedFL (foo $ fromFL l) onempty' headS]
If we have a hunch that foo indeed maps non-empty lists to non-empty lists, but we are too busy to verify it, we can write
    \l -> indeedFL l onempty $
           \l -> [headS l, indeedFL (foo $ fromFL l) (error msg) headS]
      where msg = "I'm quite sure foo maps non-empty lists to " ++
                  "non-empty lists. I'll be darned if it doesn't."
That would get the code running. Possibly at some future date (during the code review?) we will be called to justify the hunch, to whatever required degree of formality (informal argument, formal proof). If we fail at this justification, we'd better think what to do if the result of foo turns out empty. If we succeed, we would be given permission to add to the module NList the following definition:
    nfoo (FullList x) = FullList $ foo x
after which we can write
    \l -> indeedFL l onempty (\l -> [headS l, headS (nfoo l)])
with no extra empty list checks.

In conclusion, we have demonstrated the programming style that ensures safety without sacrificing efficiency. The key idea is that an abstract data type ensures (possibly quite sophisticated) propositions about the data -- so long as the very limited set of basic constructors satisfy the propositions. This main idea is very old, advocated by Milner and Morris in the mid-1970s. If there is a surprise in this, it is in the triviality of approach. One can't help but wonder why we do not program in this style.

The current version is August 2015; Original: November 2006
NList0.hs [2K]
NList.hs [2K]
NListTest.hs [2K]
The two complete implementations of the library and the tests

James H. Morris Jr.: Protection in Programming Languages
Comm. of the ACM, 1973, V16, N1, pp. 15-21

Lightweight static guarantees Longer explanations of the technique, justification, formalization, and more complex examples

The FullList library was first presented and discussed on Haskell-Cafe in November 2006 and later summarized on Haskell Wiki
The present article is the expanded and elaborated version.


Singleton types

The type system of Haskell or ML lets us express propositions of the form P(t) where t is either a type or a type variable ranging over types. Seemingly, we cannot express a proposition P(n) where n ranges over individual members of a set (e.g., individual integers) rather than over sets (i.e., types of integers). There is a way around this limitation however: using types that are populated by only a single value. For example:
    data Zero   = Zero
    data Succ a = Succ a
Here, there is only one (non-bottom) value of the type Zero, there is only one (non-bottom) value of the type Succ (Succ Zero), etc. The two declarations thus induce the family of types that are in the straightforward bijection with non-negative integers. We are thus able to express in types propositions about integers: see the references below.

Singleton types have been introduced by Hayashi (1991) and first used for dependent type programming by Xi.

Matthias Blume: No-Longer-Foreign: Teaching an ML compiler to speak C ``natively''
In BABEL'01: First workshop on multi-language infrastructure and interoperability, September 2001, Firenze, Italy.

Hongwei Xi: Dependent Types in Practical Programming
Ph.D thesis, Carnegie Mellon University, September 1998

Number-parameterized types with binary and decimal arithmetic

Implicit configurations -- or, type classes reflect the values of types
The paper uses the a family of singleton types to ensure that modulo-n operations within an expression all use the same value of the modulus. The modulus itself is not passed around explicitly, and its value is only known at run-time.


Implicit configurations -- or, type classes reflect the values of types

The configurations problem is to propagate run-time preferences throughout a program, allowing multiple concurrent configuration sets to coexist safely under statically guaranteed separation. This problem is common in all software systems, but particularly acute in Haskell, where currently the most popular solution relies on unsafe operations and compiler pragmas.

We solve the configurations problem in Haskell using only stable and widely implemented language features like the type-class system. In our approach, a term expression can refer to run-time configuration parameters as if they were compile-time constants in global scope. Besides supporting such intuitive term notation and statically guaranteeing separation, our solution also helps improve the program's performance by transparently dispatching to specialized code at run-time. We can propagate any type of configuration data -- numbers, strings, IO actions, polymorphic functions, closures, and abstract data types. No previous approach to propagating configurations implicitly in any language provides the same static separation guarantees.

The enabling technique behind our solution is to propagate values via types, with the help of polymorphic recursion and higher-rank polymorphism. The technique essentially emulates local type-class instance declarations while preserving coherence. Configuration parameters are propagated throughout the code implicitly as part of type inference rather than explicitly by the programmer. Our technique can be regarded as a portable, coherent, and intuitive alternative to implicit parameters. It motivates adding local instances to Haskell, with a restriction that salvages principal types.

Joint work with Chung-chieh Shan.

The current version is August 2004
tr-15-04.pdf [252K]
Technical report TR-15-04, Division of Engineering and Applied Sciences, Harvard University
This is the expanded version of the paper, with extra appendices.
The original paper was published in the Proceedings of the ACM SIGPLAN 2004 workshop on Haskell. Snowbird, Utah, USA -- September 22, 2004 -- ACM Press, pp. 33 - 44
Typo: several occurrences of term4' on page 9 should be read as test4'.

Prepose.hs [10K]
The complete code for the paper, updated for the current GHC


Strongly typed heterogeneous collections

A heterogeneous collection is a datatype that is capable of storing data of different types, while providing operations for look-up, update, iteration, and others. There are various kinds of heterogeneous collections, differing in representation, invariants, and access operations. We describe HList -- a Haskell library for strongly typed heterogeneous collections including open, extensible records with first-class, reusable, and compile-time only labels. HList also includes the dual of records: extensible polymorphic variants, or open unions. HList lets the user formulate statically checkable constraints: for example, no two elements of a collection may have the same type (so the elements can be unambiguously indexed by their type).

We and others have used HList for type-safe database access in Haskell. HList-based Records form the basis of OOHaskell. HList is being used in AspectAG, typed EDSL of attribute grammars, and in HaskellDB. The HList library relies on common extensions of Haskell 2010. Our exploration raises interesting issues regarding Haskell's type system, in particular, avoidance of overlapping instances, and reification of type equality and type unification.

Joint work with Ralf Laemmel and Keean Schupke, and, recently, Adam Vogt.

The October 2012 version marks the beginning of the significant re-write to take advantage of the fancier types offered by GHC 7.4+. HList now relies on native type-level booleans, natural numbers and lists, and on kind polymorphism. A number of operations are implemented as type functions. Another notable addition is unfold for heterogeneous lists, used to implement projections and splitting. More computations are moved to type-level, with no run-time overhead.

The current version is October 2012
HList on Hackage

HList-ext.pdf [166K]
CWI Technical report SEN-E0420, ISSN 1386-369X, CWI, Amsterdam, August 2004
This is the expanded version (with Appendices) of the paper originally published in the Proceedings of the ACM SIGPLAN 2004 workshop on Haskell. Snowbird, Utah, USA -- September 22, 2004 -- ACM Press, pp. 96 - 107


Applications of computable types

We have previously introduced a functional language for computing types and implemented it with Haskell type classes. The implementation is trivial because it relies on the type checker to do all the work. We now demonstrate the applications of computable types, to ascribe signatures to terms and to drive the selection of overloaded functions. One example computes a complex XML type, and instantiate the read function to read the trees of only that shape.

We have implemented Cardelli's example, the type function Prop where Prop(n) is the type of n-ary propositions. For example, Prop(3) is the type Bool -> Bool -> Bool -> Bool. By composing the type functions Prop and Fib we obtain the type function StrangeProp of the kind NAT ->Type: that is, StrangeProp(n) is the type of propositions whose arity is fib(n). We use not only (a->) but (->a) as unary type functions. The former is just (->) a. The latter was considered impossible to express in Haskell. We can now write it simply as (flip (->) a).

We demonstrate two different ways of defining type-level abstractions: as `lambda-terms' in our type-level calculus (i.e., types of the form (F t)) and as polymorphic types, Haskell's native type abstractions. The two ways are profoundly related, by the correspondence between the type abstraction/instantiation and the functional abstraction/application.

The current version is 1.1, Sep 14, 2006
TypeFN.lhs [8K]
The literate Haskell code of type functions and their applications
It was posted as On computable types. II. Flipping the arrow on the Haskell mailing list on Thu, 14 Sep 2006 19:37:19 -0700 (PDT)

TypeLC.lhs [9K]
Lambda-calculator on types: writing and evaluating type-level functions
A type-level functional language with the notation that resembles lambda-calculus with case distinction, fixpoint recursion, etc. Modulo some syntactic tart, the language of the type functions quite like the pure Haskell.

Luca Cardelli: Phase Distinctions in Type Theory. "Unpublished manuscript, 1988

Simon Peyton Jones and Erik Meijer: Henk: A Typed Intermediate Language