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.
[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
ncopies of a given value could have the following signature
replicate :: (n::Nat) -> a -> List n a
List n ais the type of lists with elements of type
aand the length exactly
nis a non-negative integer
Nat. To be more precise,
List n ais a family of types (of lists of various lengths with elements of type
a); the natural number
nselects one member of that family: ``indexes within the family''. Such an indexing is one of the common manifestations of dependent types. The index
nis the argument of
replicate: its value will be known only when an application of
replicateis 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) awhere
n+mis the ordinary addition of two natural numbers
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 :: List (succ n) a -> a tail :: List (succ n) a -> List n athat 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 listexceptions or wonder how to deal with it in the production code: the above
taildo 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
polymorphic (or, technically, parametric), in
a. In Agda,
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
-- 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
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 l1The type checker has to verify that
append l1 l2and
append l2 l1in the conditional branches have the same type. Recalling the signature of
append, the type checker has to ascertain that
n+mis 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,
mare 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+mis equal to
m+nwithout knowing the concrete values of
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.
tailof the empty list in Haskell are equivalent to the dereferencing of the zero pointer in C/C++ or
NullPointerExceptionin Java. These errors occur because the domain of the function is smaller than the function's type suggests. For example, the type of
headsays 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
taila 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 aAll the above functions are total: in particular,
tailS. The operation
fromFLlets 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_expressionwill also fail to type check unless we can statically assure
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
regular_reverse :: [a] -> [a] regular_reverse l = loop l  where 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  where 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
indeedFLto 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
lis non-empty and so the applications of
tailSare 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) = xThe other functions of our interface are equally straightforward, see the source code for details. The undisputed advantage of the implementation is its obvious correctness:
FullListrepresents a non-empty list by its very construction, without any pre-conditions and reservations.
FullListtruly 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)) = xWe 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
FullListrepresents 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
respect the invariant and ensure the truth of the non-emptiness proposition.
Once we have verified these exported constructors, all operations
NList or outside,
can take this non-emptiness proposition for granted. Therefore, we are
justified in using unsafe-head operations in implementing
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
NList, ensure the non-emptiness invariant by construction and do not
need a proof. The advantage of the second implementation is the easy
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
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:
FullList a; furthermore, the result of
FullList a, required by
headS. The first problem is easy to solve:
FullList acan 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
fromFLas an analogue of
fromIntegral-- which, too, we have to write explicitly, in any code with more than one sort of integral numbers (e.g.,
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
fooindeed 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
footurns out empty. If we succeed, we would be given permission to add to the module
NListthe following definition:
nfoo (FullList x) = FullList $ foo xafter 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.
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.
tis either a type or a type variable ranging over types. Seemingly, we cannot express a proposition
nranges 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 aHere, 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.
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.
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.
term4'on page 9 should be read as
The complete code for the paper, updated for the current GHC
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.
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
readfunction to read the trees of only that shape.
We have implemented Cardelli's example, the type function
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
Fib we obtain the type
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) 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
(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
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