The Monad.Reader/Issue5/Number Param Types
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Number-parameterized types
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Abstract
This paper describes practical programming with types
parameterized by numbers: e.g., an array type parameterized by the
array’s size or a modular group type `Zn`
parameterized by the modulus. An attempt to add, for example, two
integers of different moduli should result in a compile-time error
with a clear error message. Number-parameterized types let the
programmer capture more invariants through types and eliminate some
run-time checks.
We review several encodings of the numeric
parameter but concentrate on the phantom type representation of a
sequence of decimal digits. The decimal encoding makes
programming with number-parameterized types convenient and error
messages more comprehensible. We implement arithmetic on
decimal number-parameterized types, which lets us statically
typecheck operations such as array concatenation.
Overall we demonstrate a practical
dependent-type-like system that is just a Haskell library. The basics
of the number-parameterized types are written in Haskell98.
Keywords:
Haskell, number-parameterized types, type arithmetic, decimal types, type-directed programming.
Contents
Introduction
Discussions about types parameterized by values — especially types of arrays or finite groups parameterized by their size — reoccur every couple of months on functional programming languages newsgroups and mailing lists. The often expressed wish is to guarantee that, for example, we never attempt to add two vectors of different lengths. As one poster said, “This [feature] would be helpful in the crypto library where I end up having to either define new length Words all the time or using lists and losing the capability of ensuring I am manipulating lists of the same length.” Number-parameterized types as other more expressive types let us tell the typechecker our intentions. The typechecker may then help us write the code correctly. Many errors (which are often trivial) can be detected at compile time. Furthermore, we no longer need to litter the code with array boundary match checks. The code therefore becomes more readable, reliable, and fast. Number-parameterized types when expressed in signatures also provide a better documentation of the code and let the invariants be checked across module boundaries.
In this paper, we develop realizations of number-parameterized
types in Haskell that indeed have all the above advantages. The
numeric parameter is specified in decimal rather than in
binary, which makes types smaller and far easier to read. Type error
messages also become more comprehensible. The programmer may write or
the compiler can infer equality constraints (e.g., two argument
vectors of a function must be of the same size), arithmetic
constraints (e.g., one vector must be larger by some amount), and
inequality constraints (e.g., the size of the argument vector must be
at least one). The violations of the constraints are detected at
compile time. We can remove run-time tag checks in functions like
vhead, which are statically assured to receive a non-empty
vector.
Although we come close to the dependent-type programming, we do not extend either a compiler or the language. Our system is a regular Haskell library. In fact, the basic number-parameterized types can be implemented entirely in Haskell98. Advanced operations such as type arithmetic require commonly supported Haskell98 extensions to multi-parameter classes with functional dependencies and higher-ranked types.
Our running example is arrays parameterized over their size. The
parameter of the vector type is therefore a non-negative integer
number. For simplicity, all the vectors in the paper are indexed from
zero. In addition to vector constructors and element accessors, we
define a zipWith-like operation to map two vectors onto
the third, element by element. An attempt to map vectors of different
sizes should be reported as a type error. The typechecker will also
guarantee that there is no attempt to allocate a vector of a negative
size. In Section sec:arithmetic we introduce operations vhead, vtail and vappend on number-parameterized vectors.
The types of these operations exhibit arithmetic and inequality
constraints.
The present paper describes several gradually more sophisticated number-parameterized Haskell libraries. We start by paraphrasing the approach by Chris Okasaki, who represents the size parameter of vectors in a sequence of data constructors. We then switch to the encoding of the size in a sequence of type constructors. The resulting types are phantom and impose no run-time overhead. Section sec:unary-type describes unary encoding of numerals in type constructors, Sections sec:decimal-fixed and sec:decimal-arb discuss decimal encodings. Section sec:decimal-fixed introduces a type representation for fixed-precision decimal numbers. Section sec:decimal-arb removes the limitation on the maximal size of representable numbers, at a cost of a more complex implementation and of replacing commas with unsightly dollars signs. The decimal encoding is extendible to other bases, e.g., 16 or 64. The latter can be used to develop practical realizations of number-parameterized cryptographically interesting groups.
Section sec:arithmetic describes the first contribution of the paper. We develop addition and subtraction of “decimal types”, i.e., of the type constructor applications representing non-negative integers in decimal notation. The implementation is significantly different from that for more common unary numerals. Although decimal numerals are notably difficult to add, they make number-parameterized programming practical. We can now write arithmetic equality and inequality constraints on number-parameterized types.
Section sec:dynamic briefly describes working with number-parameterized types when the numeric parameter, and even its upper bound, are not known until run time. We show one, quite simple technique, which assures a static constraint by a run-time check — witnessing. The witnessing code, which must be trustworthy, is notably compact. The section uses the method of blending of static and dynamic assurances that was first described in stanamic-trees.
Section sec:related compares our approach with the phantom type programming in SML by Matthias Blume, with a practical dependent-type system of Hongwei Xi, with statically-sized and generic arrays in Pascal and C, with the shape inference in array-oriented languages, and with C++ template meta-programming. Section sec:conclusions concludes.
Encoding the number parameter in data constructors
The first approach to vectors parameterized by their size encodes the size as a series of data constructors. This approach has been used extensively by Chris Okasaki. For example, in Okasaki99 he describes square matrixes whose dimensions can be proved equal at compile time. He digresses briefly to demonstrate vectors of statically known size. A similar technique has been described by McBride. In this section, we develop a more naive encoding of the size through data constructors, for introduction and comparison with the encoding of the size via type constructors in the following sections.
Our representation of vectors of a statically checked size is reminiscent of the familiar representation of lists:
data List a = Nil | Cons a (List a)
List a is a recursive datatype. Lists of different sizes
have the same recursive type. To make the types different (so that
we can represent the size, too) we break the explicit recursion in the
datatype declaration. We introduce two data constructors:
module UnaryDS where
data VZero a = VZero deriving Show
infixr 3 :+:
data Vecp tail a = a :+: (tail a) deriving Show
The constructor VZero represents a vector of a zero
size. A value of the type Vecp tail a is a non-empty vector
formed from an element of the type a and (a smaller vector)
of the type tail a. We place our vectors into the class
Show for expository purposes. Thus vectors holding one
element have the type Vecp VZero a, vectors with two
elements have the type Vecp (Vecp VZero) a, with three elements
Vecp (Vecp (Vecp VZero)) a, etc. We should stress the
separation of the shape type of a vector, Vecp (Vecp VZero) in the last example, from the type of vector elements. The shape
type of a vector clearly encodes vector’s size, as repeated
applications of a type constructor Vecp to the type
constructor VZero, i.e., as a Peano numeral. We have indeed
designed a number-parameterized type.
To generically manipulate the family of differently-sized vectors, we define a class of polymorphic functions:
class Vec t where
vlength:: t a -> Int
vat:: t a -> Int -> a
vzipWith:: (a->b->c) -> t a -> t b -> t c
The method vlength gives us the size of a vector; the
method vat lets us retrieve a specific element, and the method
vzipWith produces a vector by an element-by-element
combination of two other vectors. We can use vzipWith to
add two vectors elementwise. We must emphasize the type of vzipWith: the two argument vectors may hold elements of different
types, but the vectors must have the same shape, i.e., size.
The implementation of the class Vec has only two
instances:
instance Vec VZero where
vlength = const 0
vat = error "null array or index out of range"
vzipWith f a b = VZero
instance (Vec tail) => Vec (Vecp tail) where
vlength (_ :+: t) = 1 + vlength t
vat (a :+: _) 0 = a
vat (_ :+: ta) n = vat ta (n-1)
vzipWith f (a :+: ta) (b :+: tb) =
(f a b) :+: (vzipWith f ta tb)
The second instance makes it clear that a value of a type Vecp tail a is a vector Vec if and only if
tail a is a vector Vec. Our vectors,
instances of the class Vec, are recursively defined too. Unlike
lists, our vectors reveal their sizes in their types.
That was the complete implementation of the number-parameterized vectors. We can now define a few sample vectors:
v3c = 'a' :+: 'b' :+: 'c' :+: VZero
v3i = 1 :+: 2 :+: 3 :+: VZero
v4i = 1 :+: 2 :+: 3 :+: 4 :+: VZero
and a few simple tests:
test1 = vlength v3c
test2 = [vat v3c 0, vat v3c 1, vat v3c 2]
We can load the code into a Haskell system and run the tests. Incidentally, we can ask the Haskell system to tell us the inferred type of a sample vector:
*UnaryDS> :t v3c
Vecp (Vecp (Vecp VZero)) Char
The inferred type indeed encodes the size of the vector as a Peano numeral. We can try more complex tests, of element-wise operations on two vectors:
test3 = vzipWith (\c i -> (toEnum $ fromEnum c + fromIntegral i)::Char)
v3c v3i
test4 = vzipWith (+) v3i v3i
*UnaryDS> test3
'b' :+: ('d' :+: ('f' :+: VZero))
In particular, test3 demonstrates an operation on two
vectors of the same shape but of different element types.
An attempt to add, by mistake, two vectors of different sizes is revealing:
test5 = vzipWith (+) v3i v4i
Couldn't match `VZero' against `Vecp VZero'
Expected type: Vecp (Vecp (Vecp VZero)) a
Inferred type: Vecp (Vecp (Vecp (Vecp VZero))) a1
In the third argument of `vzipWith', namely `v4i'
In the definition of `test5': vzipWith (+) v3i v4i
We get a type error, with a clear error message (the quoted message, here and elsewhere in the paper, is by GHCi. The Hugs error message is essentially the same). The typechecker, at the compile time, has detected that the sizes of the vectors to add elementwise do not match. To be more precise, the sizes are off by one.
For vectors described in this section, the element access
operation, vat, takes O(n) time where
n is the size of the vector. Chris Okasaki has designed more sophisticated number-parameterized
vectors with element access time O(log n). Although this
is an improvement, the overhead of accessing an element adds up for
many operations. Furthermore, the overhead of data constructors,
:+: in our example, becomes noticeable for longer
vectors. When we encode the size of a vector as a sequence of data
constructors, the latter overhead cannot be eliminated.
Although we have achieved the separation of the shape type of a
vector from the type of its elements, we did so at the expense of a
sequence of data constructors, :+:, at the term
level. These constructors add time and space overheads, which
increase with the vector size. In the following sections we
show more efficient representations for number-parameterized
vectors. The structure of their type will still tell us the size of
the vector; however there will be no corresponding term structure,
and, therefore, no space overhead of storing it nor run-time overhead
of traversing it.
Encoding the number parameter in type constructors, in unary
To improve the efficiency of number-parameterized vectors, we choose a better run-time representation: Haskell arrays. The code in the present section is in Haskell98.
module UnaryT (..elided..) where
import Data.Array
First, we need a type structure (an infinite family of types) to encode non-negative numbers. In the present section, we will use an unary encoding in the form of Peano numerals. The unary type encoding of integers belongs to programming folklore. It is also described in Blume01 in the context of a foreign-function interface library of SML.
data Zero = Zero
data Succ a = Succ a
That is, the term Zero of the type Zero
represents the number 0. The term (Succ (Succ Zero)) of the type
(Succ (Succ Zero)) encodes the number two. We call these
numerals Peano numerals because the number n is
represented as a repeated application of n type (data)
constructors Succ to the type (term) Zero. We observe a one-to-one correspondence between the types of our
numerals and the terms. In fact, a numeral term looks precisely the
same as its type. This property is crucial as we shall see on many
occasions below. It lets us “lift” number computations to the type
level. The property also makes error messages lucid
[1].
We place our Peano numerals into a class Card, which
has a method c2num to convert a numeral into the
corresponding number.
class Card c where
c2num:: (Num a) => c -> a -- convert to a number
cpred::(Succ c) -> c
cpred = undefined
instance Card Zero where
c2num _ = 0
instance (Card c) => Card (Succ c) where
c2num x = 1 + c2num (cpred x)
The function cpred determines the predecessor for a
positive Peano numeral. The definition for that function may seem
puzzling: it is undefined. We observe that the callers do not need the value
returned by that function: they merely need the type of that
value. Indeed, let us examine the definitions of the method c2num in the above two instances. In the instance Card Zero, we are certain that the argument of c2num has
the type Zero. That type, in our encoding, represents the
number zero, which we return. There can be only one non-bottom value
of the type Zero: therefore, once we know the type, we do
not need to examine the value. Likewise, in the instance
Card (Succ c), we know that the type of the argument of c2num is (Succ c), where c is itself a
Card numeral. If we could convert a value of the type
c to a number, we can convert the value of the type (Succ c) as well. By induction we determine that c2num never examines the value of its argument. Indeed, not only c2num (Succ (Succ Zero)) evaluates to 2, but so does
c2num (undefined::(Succ (Succ Zero))).
The same correspondence between the types and the terms suggests that the numeral type alone is enough to describe the size of a vector. We do not need to store the value of the numeral. The shape type of our vectors could be phantom (as in Blume01).
newtype Vec size a = Vec (Array Int a) deriving Show
That is, the type variable size does not occur on the
right-hand size of the Vec declaration. More importantly,
at run-time our Vec is indistinguishable from an Array, thus incurring no additional overhead and providing
constant-time element access. As we mentioned earlier, for simplicity,
all the vectors in the paper are indexed from zero. The data
constructor Vec is not exported from the module, so one
has to use the following functions to construct vectors.
listVec':: (Card size) => size -> [a] -> Vec size a
listVec' size elems = Vec $ listArray (0,(c2num size)-1) elems
listVec:: (Card size) => size -> [a] -> Vec size a
listVec size elems | not (c2num size == length elems) =
error "listVec: static/dynamic sizes mismatch"
listVec size elems = listVec' size elems
vec:: (Card size) => size -> a -> Vec size a
vec size elem = listVec' size $ repeat elem
The private function listVec' constructs the vector
of the requested size initialized with the given values. The function
makes no check that the length of the list of the initial values
elems is equal to the length of the vector. We use this
non-exported function internally, when we have proven that elems has the right length, or when truncating such a list is
appropriate. The exported function listVec is a safe
version of listVec'. The former assures that the
constructed vector is consistently initialized. The function vec initializes all elements to the same value. For example, the
following expression creates a boolean vector of two elements with the
initial values True and False.
*UnaryT> listVec (Succ (Succ Zero)) [True,False]
Vec (array (0,1) [(0,True),(1,False)])
A Haskell interpreter created the requested value, and printed it out. We can confirm that the inferred type of the vector encodes its size:
*UnaryT> :type listVec (Succ (Succ Zero)) [True,False]
Vec (Succ (Succ Zero)) Bool
We can now introduce functions to operate on our vectors. The functions are similar to those in the previous section. As before, they are polymorphic in the shape of vectors (i.e., their sizes). This polymorphism is expressed differently however. In the present section we use just the parametric polymorphism rather than typeclasses.
vlength_t:: Vec size a -> size
vlength_t _ = undefined
vlength:: Vec size a -> Int
vlength (Vec arr) = let (0,last) = bounds arr in last+1
velems:: Vec size a -> [a]
velems (Vec v) = elems v
vat:: Vec size a -> Int -> a
vat (Vec arr) i = arr ! i
vzipWith:: Card size =>
(a->b->c) -> Vec size a -> Vec size b -> Vec size c
vzipWith f va vb =
listVec' (vlength_t va) $ zipWith f (velems va) (velems vb)
The functions vlength_t and vlength tell
the size of their argument vector. The function vat
returns the element of a vector at a given zero-based index. The function
velems, which gives the list of vector’s elements, is the
left inverse of listVec. The function
vzipWith elementwise combines two vectors into the third
one by applying a user-specified function f to the
corresponding elements of the argument vectors. The polymorphic types
of these functions indicate that the functions generically operate on
number-parameterized vectors of any size. Furthermore,
the type of vzipWith expresses the constraint that the
two argument vectors must have the same size. The result will be a
vector of the same size as that of the argument vectors. We rely on
the fact that the function zipWith, when applied to two
lists of the same size, gives the list of that size. This justifies our
use of listVec'.
We have introduced two functions that yield the size of their
argument vector. One is the function vlength_t: it
returns a value whose type represents the size of the vector. We are
interested only in the type of the return value — which we extract
statically from the type of the argument vector. The function vlength_t is a compile-time function. Therefore, it is
no surprise that its body is undefined. The type of the
function is its true definition. The function vlength in contrast retrieves vector’s size from the run-time
representation as an array. If we export listVec from the
module UnaryT but do not export the constructor Vec, we can guarantee that c2num . vlength_t is
equivalent to vlength: our number-parameterized vector
type is sound.
From the practical point of view, passing terms such as
(Succ (Succ Zero)) to the functions vec or listVec to construct vectors is inconvenient. The previous
section showed a better approach. We can implement it here too: we let
the user enumerate the values, which we accumulate into a list,
counting them at the same time:
infixl 3 &+
data VC size a = VC size [a]
vs:: VC Zero a; vs = VC Zero []
(&+):: VC size a -> a -> VC (Succ size) a
(&+) (VC size lst) x = VC (Succ size) (x:lst)
vc:: (Card size) => VC size a -> Vec size a
vc (VC size lst) = listVec' size (reverse lst)
The counting operation is effectively performed by a typechecker
at compile time. Finally, the function vc will allocate
and initialize the vector of the right size — and of the right
type. Here are a few sample vectors and operations on them:
v3c = vc $ vs &+ 'a' &+ 'b' &+ 'c'
v3i = vc $ vs &+ 1 &+ 2 &+ 3
v4i = vc $ vs &+ 1 &+ 2 &+ 3 &+ 4
test1 = vlength v3c; test1' = vlength_t v3c
test2 = [vat v3c 0, vat v3c 1, vat v3c 2]
test3 = vzipWith (\c i -> (toEnum $ fromEnum c + fromIntegral i)::Char)
v3c v3i
test4 = vzipWith (+) v3i v3i
We can run the tests as follows:
*UnaryT> test3
Vec (array (0,2) [(0,'b'),(1,'d'),(2,'f')])
*UnaryT> :type test3
Vec (Succ (Succ (Succ Zero))) Char
The type of the result bears the clear indication of the size of the vector. If we attempt to perform an element-wise operation on vectors of different sizes, for example:
test5 = vzipWith (+) v3i v4i
Couldn't match `Zero' against `Succ Zero'
Expected type: Vec (Succ (Succ (Succ Zero))) a
Inferred type: Vec (Succ (Succ (Succ (Succ Zero)))) a1
In the third argument of `vzipWith', namely `v4i'
In the definition of `test5': vzipWith (+) v3i v4i
we get a message from the typechecker that the sizes are off by one.
Fixed-precision decimal types
Peano numerals adequately represent the size of a vector in vector’s type. However, they make the notation quite verbose. We want to offer a programmer a familiar, decimal notation for the terms and the types representing non-negative numerals. This turns out possible even in Haskell98. In this section, we describe a fixed-precision notation, assuming that a programmer will never need a vector with more than 999 elements. The limit is not hard and can be readily extended. The next section will eliminate the limit altogether.
We again will be using Haskell arrays as the run-time representation for our vectors. In fact, the implementation of vectors is the same as that in the previous section. The only change is the use of decimal rather than unary types to describe the sizes of our vectors.
module FixedDecT (..export list elided..) where
import Data.Array
Since we will be using the decimal notation, we need the terms and the types for all ten digits:
data D0 = D0
data D1 = D1
...
data D9 = D9
For clarity and to save space, we elide repetitive code
fragments. The full code is available. To manipulate the digits uniformly (e.g., to find out the
corresponding integer), we put them into a class Digit. We also introduce a class for non-zero digits. The latter has no
methods: we use NonZeroDigit as a constraint on allowable
digits.
class Digit d where -- class of digits
d2num:: (Num a) => d -> a -- convert to a number
instance Digit D0 where d2num _ = 0
instance Digit D1 where d2num _ = 1
...
instance Digit D9 where d2num _ = 9
class Digit d => NonZeroDigit d
instance NonZeroDigit D1
instance NonZeroDigit D2
...
instance NonZeroDigit D9
We define a class of non-negative numerals. We make all single-digit numerals the members of that class:
class Card c where
c2num:: (Num a) => c -> a -- convert to a number
-- Single-digit numbers are non-negative numbers
instance Card D0 where c2num _ = 0
instance Card D1 where c2num _ = 1
...
instance Card D9 where c2num _ = 9
We define a two-digit number, a tuple (d1,d2)
where d1 is a non-zero digit, a member of the class Card. The class NonZeroDigit makes expressing
the constraint lucid. We also introduce three-digit decimal
numerals (d1,d2,d3):
instance (NonZeroDigit d1,Digit d2) => Card (d1,d2) where
c2num c = 10*(d2num $ t12 c) + (d2num $ t22 c)
instance (NonZeroDigit d1,Digit d2,Digit d3) =>
Card (d1,d2,d3) where
c2num c = 100*(d2num $ t13 c) + 10*(d2num $ t23 c)
+ (d2num $ t33 c)
The instance constraints of the Card instances
guarantee the uniqueness of our representation of numbers: the
major decimal digit of a multi-digit number is not zero. It will be a
type error to attempt to form such an number:
*FixedDecT> vec (D0,D1) 'a'
<interactive>:1:
No instance for (NonZeroDigit D0)
The auxiliary compile-time functions t12...t33 are tuple selectors. We could have avoided them in GHC with
Glasgow extensions, which supports local type variables. We feel
however that keeping the code Haskell98 justifies the extra hassle:
t12::(a,b) -> a; t12 = undefined
t22::(a,b) -> b; t22 = undefined
...
t33::(a,b,c) -> c; t33 = undefined
The rest of the code is as before, e.g.:
newtype Vec size a = Vec (Array Int a) deriving Show
listVec':: Card size => size -> [a] -> Vec size a
listVec' size elems = Vec $ listArray (0,(c2num size)-1) elems
The implementations of the polymorphic functions listVec, vec, vlength_t, vlength, vat,
velems, and vzipWith are precisely the same
as those in Section sec:unary-type. We elide the code for
the sake of space. We introduce a few sample vectors, using the
decimal notation this time:
v12c = listVec (D1,D2) $ take 12 ['a'..'z']
v12i = listVec (D1,D2) [1..12]
v13i = listVec (D1,D3) [1..13]
The decimal notation is so much convenient. We can now define long vectors without pain. As before, the type of our vectors — the size part of the type — looks precisely the same as the corresponding size term expression:
*FixedDecT> :type v12c
Vec (D1, D2) Char
We can use the sample vectors in the tests like those of the previous section. If we attempt to elementwise add two vectors of different sizes, we get a type error:
test5 = vzipWith (+) v12i v13i
Couldn't match `D2' against `D3'
Expected type: Vec (D1, D2) a
Inferred type: Vec (D1, D3) a1
In the third argument of `vzipWith', namely `v13i'
In the definition of `test5': vzipWith (+) v12i v13i
The error message literally says that 12 is not equal to 13: the typechecker expected a vector of size 12 but found a vector of size 13 instead.
Arbitrary-precision decimal types
From the practical point of view, the fixed-precision number-parameterized vectors of the previous section are sufficient. The imposition of a limit on the width of the decimal numerals — however easily extended — is nevertheless intellectually unsatisfying. One may wish for an encoding of arbitrarily large decimal numbers within a framework that has been set up once and for all. Such an SML framework has been introduced in Blume01, to encode the sizes of arrays in their types. It is interesting to ask if such an encoding is possible in Haskell. The present section demonstrates a representation of arbitrary large decimal numbers in Haskell98. We also show that typeclasses in Haskell have made the encoding easier and precise: our decimal types are in bijection with non-negative integers. As before, we use the decimal types as phantom types describing the shape of number-parameterized vectors.
We start by defining the types for the ten digits:
module ArbPrecDecT (..export list elided..) where
import Data.Array
data D0 a = D0 a
data D1 a = D1 a
...
data D9 a = D9 a
Unlike the code in the previous section, D0 through D9 are type constructors of one argument. We
use the composition of the constructors to represent sequences of
digits. And so we introduce a class for arbitrary sequences of
digits:
class Digits ds where
ds2num:: (Num a) => ds -> a -> a
with a method to convert a sequence to the corresponding
number. The method ds2num is designed in the
accumulator-passing style: its second argument is the accumulator. We
also need a type, which we call Sz, to represent an empty
sequence of digits:
data Sz = Sz -- zero size (or the Nil of the sequence)
instance Digits Sz where
ds2num _ acc = acc
We now inductively define arbitrarily long sequences of digits:
instance (Digits ds) => Digits (D0 ds) where
ds2num dds acc = ds2num (t22 dds) (10*acc)
instance (Digits ds) => Digits (D1 ds) where
ds2num dds acc = ds2num (t22 dds) (10*acc + 1)
...
instance (Digits ds) => Digits (D9 ds) where
ds2num dds acc = ds2num (t22 dds) (10*acc + 9)
t22::(f x) -> x; t22 = undefined
The type and the term Sz is an empty sequence;
D9 Sz — that is, the application of the constructor D9 to Sz — is a sequence of one digit, digit 9. The
application of the constructor D1 to the latter sequence
gives us D1 (D9 Sz), a two-digit sequence of digits one
and nine. Compositions of data/type constructors indeed encode
sequences of digits. As before, the terms and the types look precisely
the same. The compositions can of course be arbitrarily long:
*ArbPrecDecT> :type D1$ D2$ D3$ D4$ D5$ D6$ D7$ D8$ D9$ D0$ D9$
D8$ D7$ D6$ D5$ D4$ D3$ D2$ D1$ Sz
D1 (D2 (D3 (D4 (D5 (D6 (D7 (D8 (D9 (D0 (D9 (D8 (D7
(D6 (D5 (D4 (D3 (D2 (D1 Sz))))))))))))))))))
*ArbPrecDecT> ds2num (D1$ D2$ D3$ D4$ D5$ D6$ D7$ D8$ D9$ D0$ D9$
D8$ D7$ D6$ D5$ D4$ D3$ D2$ D1$ Sz) 0
1234567890987654321
We should point out a notable advantage of Haskell typeclasses in
designing of sophisticated type families — in particular, in
specifying constraints. Nothing prevents a programmer from using our
type constructors, e.g., D1, in unintended ways. For
example, a programmer may form a value of the type D1 Bool: either by applying a data constructor D1 to a boolean
value, or by casting a polymorphic value, undefined,
into that type:
*ArbPrecDecT> :type D1 True
D1 Bool
*ArbPrecDecT> :type (undefined::D1 Bool)
D1 Bool
However, such types do not represent decimal
sequences. Indeed, an attempt to pass either of these values to
ds2num will result in a type error:
*ArbPrecDecT> ds2num (undefined::D1 Bool) 0
No instance for (Digits Bool)
arising from use of `ds2num' at <interactive>:1
In the definition of `it': ds2num (undefined :: D1 Bool) 0
In contrast, the approach in Blume01 prevented the
user from constructing (non-bottom) values of these types by a careful
design and export of value constructors. That approach relied on SML’s
module system to preclude the overt mis-use of the decimal type
system. Yet the user can still form a (latent, in SML) bottom value of
the “bad” type, e.g., by attaching an appropriate type signature to
an empty list, error function or other suitable polymorphic value. In
a non-strict language like Haskell such values would make our approach,
which relies on phantom types, unsound. Fortunately, we are able to
eliminate ill-formed decimal types at the type level rather than at
the term level. Our class Digits admits those and only those types that represent sequences of digits.
To guarantee the bijection between non-negative numbers and sequences of digits, we need to impose an additional restriction: the first, i.e., the major, digit of a sequence must be non-zero. Expressing such a restriction is surprisingly straightforward in Haskell, even Haskell98.
class (Digits c) => Card c where
c2num:: (Num a) => c -> a
c2num c = ds2num c 0
instance Card Sz
instance (Digits ds) => Card (D1 ds)
instance (Digits ds) => Card (D2 ds)
...
instance (Digits ds) => Card (D9 ds)
As in the previous sections, the class Card
represents non-negative integers. A non-negative integer is realized
here as a sequence of decimal digits — provided, as the instances
specify, that the sequence starts with a digit other than zero. We can
now define the type of our number-parameterized vectors:
newtype Vec size a = Vec (Array Int a) deriving Show
which looks precisely as before, and polymorphic functions vec, listVec, vlength_t, vlength, velems, vat, and vzipWith — which are identical to those in Section sec:unary-type. We can define a few sample vectors:
v12c = listVec (D1 $ D2 Sz) $ take 12 ['a'..'z']
v12i = listVec (D1 $ D2 Sz) [1..12]
v13i = listVec (D1 $ D3 Sz) [1..13]
we should note a slight change of notation compared to the corresponding vectors of Section sec:decimal-fixed. The tests are not changed and continue to work as before:
test4 = vzipWith (+) v12i v12i
*ArbPrecDecT> :type test4
Vec (D1 (D2 Sz)) Int
*ArbPrecDecT> test4
Vec (array (0,11) [(0,2),(1,4),(2,6),...(11,24)])
The compiler has been able to infer the size of the result of the
vzipWith operation. The size is lucidly spelled in
decimal in the type of the vector. Again, an attempt to elementwise
add vectors of different sizes leads to a type error:
test5 = vzipWith (+) v12i v13i
Couldn't match `D2 Sz' against `D3 Sz'
Expected type: Vec (D1 (D2 Sz)) a
Inferred type: Vec (D1 (D3 Sz)) a1
In the third argument of `vzipWith', namely `v13i'
In the definition of `test5': vzipWith (+) v12i v13i
The typechecker complains that 2 is not equal to 3: it found the vector of size 13 whereas it expected a vector of size 12. The decimal types make the error message very clear.
We must again point out a significant difference of our approach
from that of Blume01. We were able to state that only
those types of digital sequences that start with a non-zero digit
correspond to a non-negative number. SML, as acknowledged in Blume01, is unable to express such a restriction directly. The
paper, therefore, prevents the user from building
invalid decimal sequences by relying on the module system: by
exporting carefully-designed value constructors. The latter use an
auxiliary phantom type to keep track of “nonzeroness” of the major
digit. Our approach does not incur such a complication. Furthermore,
by the very inductive construction of the classes Digits
and Card, there is a one-to-one correspondence between
types, the members of Card, and the integers
in decimal notation. In Blume01, the similar mapping
holds only when the family of decimal types is restricted to the types
that correspond to constructible values. A user of that system may
still form bottom values of invalid decimal types, which will cause
run-time errors. In our case, when the digit constructors are
misapplied, the result will no longer be in the class Card, and so the error will be detected statically by the
typechecker:
*ArbPrecDecT> vec (D1$ D0$ D0$ True) 0
No instance for (Digits Bool)
arising from use of `vec' at <interactive>:1
In the definition of `it': vec (D1 $ (D0 $ (D0 $ True))) 0
*ArbPrecDecT> vec (D0$ D1$ D0 Sz) 0
No instance for (Card (D0 (D1 (D0 Sz))))
arising from use of `vec' at <interactive>:1
In the definition of `it': vec (D0 $ (D1 $ (D0 Sz))) 0
Computations with decimal types
The previous sections gave many examples of functions such as
vzipWith that take two vectors statically
known to be of equal size. The signature of these functions states
quite detailed invariants whose violations will be reported at
compile-time. Furthermore, the invariants can be inferred by the
compiler itself. This use of the type system is not particular to
Haskell: Matthias Blume has derived a similar
parameterization of arrays in SML, which can express such equality of
size constraints. Matthias Blume however cautions one not to overstate
the usefulness of the approach because the type system can express
only fairly simple constraints: “There is still no type that, for
example, would force two otherwise arbitrary arrays to differ in size
by exactly one.” That was written in the context of SML however. In
Haskell with common extensions we can define vector
functions whose type contains arithmetic constraints on the sizes of
the argument and the result vectors. These constraints can be verified
statically and sometimes even inferred by a compiler. In this section,
we consider the example of vector concatenation. We shall see that the
inferred type of vappend manifestly affirms that the size
of the result is the sum of the sizes of two argument vectors. We also
introduce the functions vhead and vtail,
whose type specifies that they can only be applied to non-empty
vectors. Furthermore, the type of vtail says that the
size of the result vector is less by one than the size of the argument
vector. These examples are quite unusual and almost cross into the
realm of dependent types.
We must note however that the examples in this section require the
Haskell98 extension to multi-parameter classes with functional
dependencies. That extension is activated by flags -98 of
Hugs and -fglasgow-exts -fallow-undecidable-instances of
GHCi.
We will be using the arbitrary precision decimal types introduced
in the previous section. We aim to design a ‘type addition’ of decimal
sequences. Our decimal types spell the corresponding non-negative
numbers in the conventional (i.e., big-endian) decimal notation: the
most-significant digit first. However, it is more convenient to add
such numbers starting from the least-significant digit. Therefore, we
need a way to reverse digital sequences, or more precise, types of the
class Digits. We use the conventional sequence reversal
algorithm written in the accumulator-passing style.
class DigitsInReverse' df w dr | df w -> dr
instance DigitsInReverse' Sz acc acc
instance (Digits (d drest), DigitsInReverse' drest (d acc) dr)
=> DigitsInReverse' (d drest) acc dr
We introduced the class DigitsInReverse' df w dr where
df is the source sequence, dr is the
reversed sequence, and w is the accumulator. The three
digit sequence types belong to DigitsInReverse' if
the reverse of df appended to w gives the
digit sequence dr. The functional dependency and the two
instances spell this constraint out. We can now introduce a class that
relates a sequence of digits with its reverse:
class DigitsInReverse df dr | df -> dr, dr -> df
instance (DigitsInReverse' df Sz dr, DigitsInReverse' dr Sz df)
=> DigitsInReverse df dr
Two sequences of digits df and dr belong
to the class DigitsInReverse if they are the reverse of
each other. The functional dependencies make the “each other” part
clear: one sequence uniquely determines the other. The typechecker
will verify that given df, it can find dr so
that both DigitsInReverse' df Sz dr and DigitsInReverse' dr Sz df are satisfied. To test the reversal
process, we define a function digits_rev:
digits_rev:: (Digits ds, Digits dsr, DigitsInReverse ds dsr)
=> ds -> dsr
digits_rev = undefined
It is again a compile-time function specified entirely by its type. Its body is therefore undefined. We can now run a few examples:
*ArbArithmT> :t digits_rev (D1$D2$D3 Sz)
D3 (D2 (D1 Sz))
*ArbArithmT> :t (\v -> digits_rev v `asTypeOf` (D1$D2$D3 Sz))
D3 (D2 (D1 Sz)) -> D1 (D2 (D3 Sz))
Indeed, the process of reversing sequences of decimal digits works
both ways. Given the type of the argument to digits_rev,
the compiler infers the type of the result. Conversely, given the type
of the result the compiler infers the type of the argument.
A sequence of digits belongs to the class Card only
if the most-significant digit is not a zero. To convert an arbitrary
sequence to Card we need a way to strip leading zeros:
class NoLeadingZeros d d0 | d -> d0
instance NoLeadingZeros Sz Sz
instance (NoLeadingZeros d d') => NoLeadingZeros (D0 d) d'
instance NoLeadingZeros (D1 d) (D1 d)
...
instance NoLeadingZeros (D9 d) (D9 d)
We are now ready to build the addition machinery. We draw our inspiration from the computer architecture: the adder of an arithmetical-logical unit (ALU) of the CPU is constructed by chaining of so-called full-adders. A full-adder takes two summands and the carry-in and yields the result of the summation and the carry-out. In our case, the summands and the result are decimal rather than binary. Carry is still binary.
class FullAdder d1 d2 cin dr cout
| d1 d2 cin -> cout, d1 d2 cin -> dr,
d1 dr cin -> cout, d1 dr cin -> d2
where
_unused:: (d1 xd1) -> (d2 xd2) -> cin -> (dr xdr)
_unused = undefined
The class FullAdder establishes a relation among
three digits d1, d2, and dr and
two carry bits cin and cout: d1 + d2 + cin = dr + 10*cout. The digits are represented by the type
constructors D0 through D9. The sole purpose
of the method _unused is to cue the compiler that
d1, d2, and dr are type
constructors. The functional dependencies of the class tell us that
the summands and the input carry uniquely determine the result digit
and the output carry. On the other hand, if we know the result digit,
one of the summands, d1, and the input carry, we can
determine the other summand. The same relation FullAdder
can therefore be used for addition and for subtraction. In the latter
case, the carry bits should be more properly called borrow bits.
data Carry0
data Carry1
instance FullAdder D0 D0 Carry0 D0 Carry0
instance FullAdder D0 D0 Carry1 D1 Carry0
instance FullAdder D0 D1 Carry0 D1 Carry0
...
instance FullAdder D9 D8 Carry1 D8 Carry1
instance FullAdder D9 D9 Carry0 D8 Carry1
instance FullAdder D9 D9 Carry1 D9 Carry1
The full code indeed contains 200 instances of
FullAdder. The exhaustive enumeration verifies the
functional dependencies of the class. The number of instances could be
significantly reduced if we availed ourselves to an overlapping
instances extension. For generality however we tried to use as few
Haskell98 extensions as possible. Although 200 instances seems like
quite many, we have to write them only once. We place the instances
into a module and separately compile it. Furthermore, we did not write
those instances by hand: we used Haskell itself:
make_full_adder
= mapM_ putStrLn
[unwords $ doit d1 d2 cin | d1<-[0..9],
d2<-[0..9], cin<-[0..1]]
where
doit d1 d2 cin
= ["instance FullAdder", tod d1, tod d2, toc cin,
tod d12, toc cout]
where
(d12,cout) = let sum = d1 + d2 + cin
in if sum >= 10 then (sum-10,1) else (sum,0)
tod n | (n >= 0 && 9 >= n) = "D" ++ (show n)
toc 0 = "Carry0"; toc 1 = "Carry1"
That function is ready for Template Haskell. Currently we used a low-tech approach of cutting and pasting from an Emacs buffer with GHCi into the Emacs buffer with the code.
We use FullAdder to build the full adder of two
little-endian decimal sequences ds1 and ds2.
The relation DigitsSum ds1 ds2 cin dsr holds if ds1+ds2+cin = dsr. We add the digits from the least significant
onwards, and we propagate the carry. If one input sequence turns out
shorter than the other, we pad it with zeros. The correctness of the
algorithm follows by simple induction.
class DigitsSum ds1 ds2 cin dsr | ds1 ds2 cin -> dsr
instance DigitsSum Sz Sz Carry0 Sz
instance DigitsSum Sz Sz Carry1 (D1 Sz)
instance (DigitsSum (D0 Sz) (d2 d2rest) cin (d12 d12rest)) =>
DigitsSum Sz (d2 d2rest) cin (d12 d12rest)
instance (DigitsSum (d1 d1rest) (D0 Sz) cin (d12 d12rest)) =>
DigitsSum (d1 d1rest) Sz cin (d12 d12rest)
instance (FullAdder d1 d2 cin d12 cout,
DigitsSum d1rest d2rest cout d12rest) =>
DigitsSum (d1 d1rest) (d2 d2rest) cin (d12 d12rest)
We also need the inverse relation: DigitsDif ds1 ds2 cin dsr holds on precisely the same condition as DigitsSum. Now, however, the sequences ds1, dsr and
the input carry cin determine one of the summands,
ds2. The input carry actually means the input borrow
bit. The relation DigitsDif is defined only if the output
sequence dsr has at least as many digits as ds1 — which is the necessary condition for the result of the
subtraction to be non-negative.
class DigitsDif ds1 ds2 cin dsr | ds1 dsr cin -> ds2
instance DigitsDif Sz ds Carry0 ds
instance (DigitsDif (D0 Sz) (d2 d2rest) Carry1 (d12 d12rest)) =>
DigitsDif Sz (d2 d2rest) Carry1 (d12 d12rest)
instance (FullAdder d1 d2 cin d12 cout,
DigitsDif d1rest d2rest cout d12rest) =>
DigitsDif (d1 d1rest) (d2 d2rest) cin (d12 d12rest)
The class CardSum with a single instance puts it all
together:
class (Card c1, Card c2, Card c12) =>
CardSum c1 c2 c12 | c1 c2 -> c12, c1 c12 -> c2
instance (Card c1, Card c2, Card c12,
DigitsInReverse c1 c1r,
DigitsInReverse c2 c2r,
DigitsSum c1r c2r Carry0 c12r,
DigitsDif c1r c2r' Carry0 c12r,
DigitsInReverse c2r' c2', NoLeadingZeros c2' c2,
DigitsInReverse c12r c12)
=> CardSum c1 c2 c12
The class establishes the relation between three Card
sequences c1, c2, and c12 such
that the latter is the sum of the formers. The two summands determine
the sum, or the sum and one summand determine the other. The class can
be used for addition and subtraction of sequences. The dependencies of
the sole CardSum instance spell out the algorithm. We
reverse the summand sequences to make them little-endian, add them
together with the zero carry, and reverse the result. We also make
sure that the subtraction and summation are the exact inverses. The
addition algorithm DigitsSum never produces a sequence
with the major digit zero. The subtraction algorithm however may
result in a sequence with zero major digits, which have to be stripped
away, with the help of the relation NoLeadingZeros. We
introduce a compile-time function card_sum so we can try
the addition out:
card_sum:: CardSum c1 c2 c12 => c1 -> c2 -> c12
card_sum = undefined
*ArbArithmT> :t card_sum (D1 Sz) (D9$D9 Sz)
D1 (D0 (D0 Sz))
*ArbArithmT> :t \v -> card_sum (D1 Sz) v `asTypeOf` (D1$D0$D0 Sz)
D9 (D9 Sz) -> D1 (D0 (D0 Sz))
*ArbArithmT> :t \v -> card_sum (D9$D9 Sz) v `asTypeOf` (D1$D0$D0 Sz)
D1 Sz -> D1 (D0 (D0 Sz))
The typechecker can indeed add and subtract with carry and
borrow. Now we define the function vappend to
concatenate two vectors.
vappend va vb = listVec (card_sum (vlength_t va) (vlength_t vb))
$ (velems va) ++ (velems vb)
We could have used the function listVec'; for illustration,
we chose however to perform a run-time check and avoid proving the theorem
about the size of the list concatenation result. We did not declare
the type of vappend; still the compiler is able to infer it:
*ArbArithmT> :t vappend
vappend :: (CardSum size size1 c12) =>
Vec size a -> Vec size1 a -> Vec c12 a
which literally says that the size of the result vector is the sum
of the sizes of the argument vectors. The constraint is spelled out
patently, as part of the type of vappend. The sizes may
be arbitrarily large decimal numbers: for example, the following
expression demonstrates the concatenation of a vector of 25 elements
and a vector of size 979:
*ArbArithmT> :t vappend (vec (D2$D5 Sz) 0) (vec (D9$D7$D9 Sz) 0)
(Num a) => Vec (D1 (D0 (D0 (D4 Sz)))) a
We introduce the deconstructor functions vhead and
vtail. The type of the latter is exactly what was listed in
Blume01 as an unattainable wish.
vhead:: CardSum (D1 Sz) size1 size => Vec size a -> Vec (D1 Sz) a
vhead va = listVec (D1 Sz) $ [head (velems va)]
vtail:: CardSum (D1 Sz) size1 size => Vec size a -> Vec size1 a
vtail va = result
where result = listVec (vlength_t result) $ tail (velems va)
Although the body of vtail seem to refer to that
function result, the function is not divergent and not
recursive. Recall that vlength_t is a compile-time,
‘type’ function. Therefore the body of vtail refers to
the statically known type of result rather than to its
value. The type of vhead is also noteworthy: it
essentially specifies an inequality constraint: the input
vector is non-empty. The constraint is expressed via an implicitly
existentially quantified variable size1: the type of
vhead says that there must exist a non-negative number
size1 such that incrementing it by one should give the
size of the input vector.
We can now run a few examples. We note that the compiler could correctly infer the type of the result, which includes the size of the vector after appending or truncating it.
*ArbArithmT> let v = vappend (vec (D9 Sz) 0) (vec (D1 Sz) 1)
*ArbArithmT> :t v
Vec (D1 (D0 Sz)) Integer
*ArbArithmT> v
Vec (array (0,9) [(0,0),(1,0),...,(8,0),(9,1)])
*ArbArithmT> :type vhead v
Vec (D1 Sz) Integer
*ArbArithmT> :type vtail v
Vec (D9 Sz) Integer
*ArbArithmT> vtail v
Vec (array (0,8) [(0,0),(1,0),...,(7,0),(8,1)])
*ArbArithmT> :type (vappend (vhead v) (vtail v))
Vec (D1 (D0 Sz)) Integer
The types of vhead and vtail embed a
non-empty argument vector constraint. Indeed, an attempt to apply
vhead to an empty vector results in a type error:
*ArbArithmT> vtail (vec Sz 0)
<interactive>:1:0:
No instances for (DigitsInReverse' c2' Sz c2r',
DigitsInReverse' c2r' Sz c2',
DigitsDif (D1 Sz) c2r' Carry0 Sz,
DigitsSum (D1 Sz) c2r Carry0 Sz,
DigitsInReverse' c2r Sz size1,
DigitsInReverse' size1 Sz c2r)
arising from use of `vtail' at <interactive>:1:0-4
The error message essentially says that there is no such decimal
type c2r such that DigitsSum (D1 Sz) c2r Carry0 Sz
holds. That is, there is no non-negative number that gives zero if
added to one.
We can form quite complex expressions from the functions vappend, vhead, and vtail, and the
compiler will infer and verify the corresponding
constraints on the sizes of involved vectors. For example:
testc1 =
let va = vec (D1$D2 Sz) 0
vb = vec (D5 Sz) 1
vc = vec (D8 Sz) 2
in vzipWith (+) va (vappend vb (vtail vc))
*ArbArithmT> testc1
Vec (array (0,11) [(0,1),...,(4,1),(5,2),(6,2),...,(11,2)])
The size of the vector va must be the sum of the
sizes of vb and vc minus one. Furthermore,
the vector vc must be non-empty. The compiler has
inferred this non-trivial constraint and checked it. Indeed, if we by
mistake write vc = vec (D9 Sz) 2, as we actually did when
writing the example, the compiler will instantly report a type
error:
Couldn't match `D9 Sz' against `D8 Sz'
Expected type: D9 Sz
Inferred type: D8 Sz
When using functional dependencies to combine
DigitsSum (D1 Sz) c2r Carry0 (D9 Sz),
arising from use of `vtail' at ArbArithmT.hs:420:34-38
DigitsSum (D1 Sz) c2r Carry0 (D8 Sz),
arising from use of `vtail' at ArbArithmT.hs:411:34-38
The result 12 - 5 + 1 is expected to be 8 rather than 9.
We can define other operations that extend or shrink our
vectors. For example, Section sec:unary-type introduced
the operator &+ to make the entering of vectors
easier. It is straightforward to implement such an operator for
decimally-typed vectors.
We must point out that the type system guarantees that vhead and vtail are applied to non-empty
vectors. Therefore, we no longer need the corresponding run-time
check. The bodies of vhead and vtail may
safely use unsafe versions of the library functions head and tail, and hence increase the performance
of the code without compromising its safety.
Statically-sized vectors in a dynamic context
In the present version of the paper, we demonstrate the simplest method of handling number-parameterized vectors in the dynamic context. The method involves run-time checks. The successful result of a run-time check is marked with the appropriate static type. Further computations can therefore rely on the result of the check (e.g., that the vector in question definitely has a particular size) and avoid the need to do that test over and over again. The net advantage is the reduction in the number of run-time checks. The complete elimination of the run-time checks is quite difficult (in general, may not even be possible) and ultimately requires a dependent type system.
For our presentation we use an example of dynamically-sized vectors: reversing a vector by the familiar accumulator-passing algorithm. Each iteration splits the source vector into the head and the tail, and prepends the head to the accumulator. The sizes of the vectors change in the course of the computation, to be precise, on each iteration. We treat vectors as if they were lists. Most of the vector processing code does not have such a degree of variation in vector sizes. The code is quite simple:
vreverse v = listVec (vlength_t v) $ reverse $ velems v
whose inferred type is obviously
*ArbArithmT> :t vreverse
vreverse :: (Card size) => Vec size a -> Vec size a
The use of listVec implies a dynamic test — as a
witness to ‘acquire’ the static type size, the size type
of the input vector. We do this test only once, at the conclusion of
the algorithm. We can treat the result as any other number-parameterized
vector, for example:
testv = let v = vappend (vec (D3 Sz) 1) (vec (D1 Sz) 4)
vr = vreverse v
in vhead (vtail (vtail vr))
using the versions of vhead and vtail
without any further run-time size checks.
Related work
This paper was inspired by Matthias Blume’s messages on the newsgroup comp.lang.functional in February 2002. Many ideas of this paper were first developed during the USENET discussion, and posted in a series of three messages at that time. In more detail Matthias Blume described his method in Blume01, although that paper uses binary rather than decimal types of array sizes for clarity. The approaches by Matthias Blume and ours both rely on phantom types to encode additional information about a value (e.g., the size of an array) in a manner suitable for a typechecker. The paper exhibits the most pervasive and thorough use of phantom types: to represent the size of arrays and the constness of imported C values, to encode C structure tag names and C function prototypes.
However, paper was written in the context of SML, whereas we use Haskell. The language has greatly influenced the method of specifying and enforcing complex static constraints, e.g., that digit sequences representing non-negative numbers must not have leading zeros. The SML approach in Blume01 relies on the sophisticated module system of SML to restrict the availability of value constructors so that users cannot build values of outlawed types. Haskell typeclasses on the other hand can directly express the constraint, as we saw in Section sec:decimal-arb. Furthermore, Haskell typeclasses let us specify arithmetic equality and inequality constraints — which, as admitted in Blume01, seems quite unlikely to be possible in SML.
Arrays of a statically known size — whose size is a part of their
type — are a fairly popular feature in programming languages. Such
arrays are present in Fortran, Pascal, C
[2].
Pascal has the most complete realization of statically sized
arrays. A Pascal compiler can therefore typecheck array functions like
our vzipWith. Statically sized arrays also contribute to
expressiveness and efficiency: for example, in Pascal we can copy one
instance of an array into another instance of the same type by a
single assignment, which, for small arrays, can be fully inlined by
the compiler into a sequential code with no loops or range
checks. However, in a language without the parametric polymorphism
statically sized arrays are a great nuisance. If the size of an array
is a part of its type, we cannot write generic functions that operate
on arrays of any size. We can only write functions dealing with arrays
of specific, fixed sizes. The inability to build generic
array-processing libraries is one of the most serious drawbacks of
Pascal. Therefore, Fortran and C introduce “generic” arrays whose
size type is not statically known. The compiler silently converts a
statically-sized array into a generic one when passing arrays as
arguments to functions. We can now build generic array-processing
libraries. We still need to know the size of the array. In Fortran and
C, the programmer must arrange for passing the size information to a
function in some other way, e.g., via an additional argument, global
variable, etc. It becomes then the responsibility of a programmer to
make sure that the size information is correct. The large number of
Internet security advisories related to buffer overflows and other
array-management issues testify that programmers in general are not to
be relied upon for correctly passing and using the array size
information. Furthermore, the silent, irreversible conversion of
statically sized arrays into generic ones negate all the benefits of
the former.
A different approach to array processing is a so-called shape-invariant programming, which is a key feature of array-oriented languages such as APL or SaC. These languages let a programmer define operations that can be applied to arrays of arbitrary shape/dimensionality. The code becomes shorter and free from explicit iterations, and thus more reusable, easier to read and to write. The exact shape of an array has to be known, eventually. Determining it at run-time is greatly inefficient. Therefore, high-performance array-oriented languages employ shape inference Scholz01, which tries to statically infer the dimensionalities or even exact sizes of all arrays in a program. Shape inference is, in general, undecidable, since arrays may be dynamically allocated. Therefore, one can either restrict the class of acceptable shape-invariant programs to a decidable subset, resort to a dependent-type language like Cayenne, or use “soft typing”. The latter approach is described in Scholz01, which introduces a non-unique type system based on a hierarchy of array types: from fully specialized ones with the statically known sizes and dimensionality, to a type of an array with the known dimensionality but not size, to a fully generic array type whose shape can only be determined at run-time. The system remains decidable because at any time the typechecker can throw up hands and give to a value a fully generic array type. Shape inference of SaC is specific to that language, whose type system is otherwise deliberately constrained: SaC lacks parametric polymorphism and higher-order functions. Using shape inference for compilation of shape-invariant array operations into a highly efficient code is presented in Kreye. Their compiler tries to generate as precise shape-specific code as possible. When the shape inference fails to give the exact sizes or dimensionalities, the compiler emits code for a dynamic shape dispatch and generic loops.
There is however a great difference in goals and implementation
between the shape inference of SaC and our approach. The former
aims at accepting more programs than can statically be inferred
shape-correct. We strive to express assertions about the array sizes
and enforcing the programming style that assures them. We have shown
the definitions of functions such as vzipWith whose the
argument and the result vectors are all of the same size. This
constraint is assured at compile-time — even if we do not statically
know the exact sizes of the vectors. Because SaC lacks parametric
polymorphism, it cannot express such an assertion and statically
verify it. If a SaC programmer applies a function such as vzipWith to vectors of unequal size, the compiler will not flag
that as an error but will compile a generic array code instead. The
error will be raised at run time during a range check.
The approach of the present paper comes close to emulating a dependent type system, of which Cayenne is the epitome. We were particularly influenced by a practical dependent type system of Hongwei Xi Xi98 XiThesis, which is a conservative extension of SML. In Xi98, Hongwei Xi et al. demonstrated an application of their system to the elimination of array bound checking and list tag checking. The related work section of that paper lists a number of other dependent and pseudo-dependent type systems. Using the type system to avoid unnecessary run-time checks is a goal of the present paper too.
C++ templates provide parametric polymorphism and indexing of
types by true integers. A C++ programmer can therefore define
functions like vzipWith and vtail with
equality and even arithmetic constraints on the sizes of the argument
vectors. Blitz++ was the first example of using a
so-called template meta-programming for generating efficient and safe
array code. The type system of C++ however presents innumerable
hurdles to the functional style. For example, the result type of a
function is not used for the overloading resolution, which significantly
restricts the power of the type inference. Templates were
introduced in C++ ad hoc, and therefore, are not well integrated with
its type system. Violations of static constraints expressed via
templates result in error messages so voluminous as to become
incomprehensible.
McBride gives an extensive survey of the
emulation of dependent type systems in Haskell. He also describes
number-parameterized arrays that are similar to the ones discussed in
Section sec:Okasaki. The
paper by Fridlender and Indrika shows another example of emulating dependent
types within the Hindley-Milner type system: namely, emulating
variable-arity functions such as generic zipWith. Their
technique relies on ad hoc codings for natural numbers which resemble
Peano numerals. They aim at defining more functions (i.e.,
multi-variate functions), whereas we are concerned with making
functions more restrictive by expressing sophisticated invariants in
functions’ types. Another approach to multivariate functions —
multivariate composition operator — is discussed in mcomp.
Conclusions
Throughout this paper we have demonstrated several realizations of number-parameterized types in Haskell, using arrays parameterized by their size as an example. We have concentrated on techniques that rely on phantom types to encode the size information in the type of the array value. We have built a family of infinite types so that different values of the vector size can have their own distinct type. That type is a decimal encoding of the corresponding integer (rather than the more common unary, Peano-like encoding). The examples throughout the paper illustrate that the decimal notation for the number-parameterized vectors makes our approach practical.
We have used the phantom size types to express non-trivial
constraints on the sizes of the argument and the result arrays in the
type of functions. The constraints include the size
equality, e.g., the type of a function of two arguments may indicate
that the arguments must be vectors of the same size. More
importantly, we can specify arithmetical constraints: e.g., that
the size of the vector after concatenation is the sum of the source
vector sizes. Furthermore, we can write inequality constraints by
means of an implicit existential quantification, e.g., the function
vhead must be applied to a non-empty vector. The
programmer should benefit from more expressive function signatures
and from the ability of the compiler to statically check complex
invariants in all applications of the vector-processing functions. The
compiler indeed infers and checks non-trivial constraints involving
addition and subtraction of sizes — and presents
readable error messages on violation of the constraints.
References
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- ↑ We could have declared
Succasnewtype Succ a = Succ aso thatSuccis just a tag and all non-bottom Peano numerals share the same run-time representation. As we shall see however, we hardly ever use the values of our numerals. - ↑ C does permit truly statically-sized arrays like those in Pascal. To achieve this, we should make a C array a member of a C structure. The compiler preserves the array size information when passing such a wrapped array as an argument. It is even possible to assign such “arrays”.
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