(An example where we perform a bounds check on the type level)
m (Turn on syntax highlighting)
Revision as of 04:15, 9 March 2006
This is an introduction to a programming idiom for placing extra constraints on the construction of values by using smart constructors.
Sometimes you need guarantees about the values in your program beyond what can be accomplished with the usual type system checks. Smart constructors can be used for this purpose.
Consider the following problem: we want to be able to specify a data type for electronic resistors. The resistors come in two forms, metal and ceramic. Resistors are labelled with a number of bands, from 4 to 8.
We'd like to be able to
- ensure only resistors with the right number of bands are constructed.
1 Runtime checking : smart constructors
1.1 A first attempt
Code up the a typical data type describing a resistor value:
data Resistor = Metal Bands | Ceramic Bands deriving Show type Bands = Int
This has a problem however, that the constructors of type Resistor are unable to check that only bands of size 4 to 8 are built. It is quite legal to say:
*Main> :t Metal 23 Metal 23 :: Resistor
1.2 Smart(er) constructors
Smart constructors are just functions that build values of the required type, but perform some extra checks when the value is constructed, like so:
metalResistor :: Bands -> Resistor metalResistor n | n < 4 || n > 8 = error "Invalid number of resistor bands" | otherwise = Metal n
This function behaves like the constructor Metal, but also performs a check. This check will be carried out at runtime, once, when the value is built.
Running this code:
*Main> metalResistor 4 Metal 4
*Main> metalResistor 7 Metal 7 *Main> metalResistor 9 *** Exception: Invalid number of resistor bands
*Main> metalResistor 0 *** Exception: Invalid number of resistor bands
One extra step has to be made though, to make the interface safe. When exporting the type Resistor we need to hide the (unsafe) constructors, and only export the smart constructors, otherwise a reckless user could bypass the smart constructor:
module Resistor ( Resistor, -- abstract, hiding constructors metalResistor, -- only way to build a metal resistor ) where ...
1.3 Using assertions
Hand-coding error messages can be tedious when used often. Instead we can use the assert function, provided (at least with GHC). We rewrite the smart constructor as:
metalResistor :: Bands -> Resistor metalResistor n = assert (n >= 4 && n <= 8) $ Metal n
And now obtain more detailed error messages, automatically generated for us:
*Main> metalResistor 0 *** Exception: A.hs:4:18-23: Assertion failed
We at least now are given the line and column in which the error occured.
2 Compile-time checking : the type system
2.1 Enforcing the constraint statically
There are other ways to obtain numerical checks like this. The most interesting are probably the static checks that can be done with Type arithmetic, that enforce the number of bands at compile time, rather than runtime, by lifting the band count into the type level.
In the following example, instead of checking the band count at runtime, we instead lift the resistor band count into the type level, and have the typecheck perform the check statically, using phantom types and Peano numbers.
We thus remove the need for a runtime check, meaning faster code. A consequence of this decision is that since the band count is now represented in the type, it is no longer necessary to carry it around at runtime, meaning less data has to be allocated.
Firstly, define some Peano numbers to represent the number of bands as types:
data Z = Z data S a = S a
Now specify a class for cardinal numbers.
class Card c where instance Card Z where instance (Card c) => Card (S c) where
Ok, now we're set. So encode a type-level version of the bounds check. Only resistors with bands >= 4 and <= 8 are valid:
class Card size => InBounds size where instance InBounds (S (S (S (S Z)))) where -- four instance InBounds (S (S (S (S (S Z))))) where -- five instance InBounds (S (S (S (S (S (S Z)))))) where -- six instance InBounds (S (S (S (S (S (S (S Z))))))) where -- seven instance InBounds (S (S (S (S (S (S (S (S Z)))))))) where -- eight
Now define a new resistor type. Note that since the bounds is represented in the type, we no longer need to store the bounds in the resistor value.
data Resistor size = Resistor deriving Show
And, finally, a convenience constructor for us to use, encoding the bounds check in the type:
resistor :: InBounds size => size -> Resistor size resistor _ = Resistor
First, define some convenience values:
d0 = undefined :: Z d3 = undefined :: S (S (S Z)) d4 = undefined :: S (S (S (S Z))) d6 = undefined :: S (S (S (S (S (S Z))))) d8 = undefined :: S (S (S (S (S (S (S (S Z))))))) d10 = undefined :: S (S (S (S (S (S (S (S (S (S Z)))))))))
Now try to construct some resistors:
> resistor d0 No instance for (InBounds Z)
So the value 0 isn't in bounds, as we want. And it is a compile-time error to try to create such a resistor.
> resistor d3 No instance for (InBounds (S (S (S Z))))
Ok, how about a valid resistor?
> resistor d4 Resistor
> :t resistor d4 resistor d4 :: Resistor (S (S (S (S Z))))
And it's type encodes the number of bands.
> resistor d6 Resistor > resistor d8 Resistor
> :t resistor d8 resistor d8 :: Resistor (S (S (S (S (S (S (S (S Z))))))))
Similar result for other valid resistors.
> resistor d10 No instance for (InBounds (S (S (S (S (S (S (S (S (S (S Z)))))))))))
And 10 is too big.
By using a standard encoding of numeric values on the type level we are able to encode a bounds check in the type of a value, thus removing a runtime check, and removing the need to store the numeric value at runtime. The code is safer, as it is impossible to compile the program unless all resistors have the correct number of bands.
An extension would be to use a decimal encoding for the integers (at the expense of longer code).
Further checks can be obtained by separating the metal and ceramic values on the type level, so no function that takes a metal resistor can be accidentally passed a ceramic one.
A newtype is useful for this:
newtype MetalResistor = Metal Bands newtype CeramicResistor = Ceramic Bands
now, a function of resistors must have either a MetalResistor type, or a CeramicResistor type:
foo :: MetalResistor -> Int foo (MetalResistor n) = n
You can't write a function over both resistor types (other than a purely polymorphic function).
2.5 Related work
In general, the more information you place on the type level, the more static checks you get -- and thus less chance for bugs.
3 Runtime Optimisation : smart constructors
Another use for smart constructors is to perform basic optimisations, often to obtain a normal form for constructed data. For example, consider a data structure representing addition and multiplication of variables.
data Expression = Variable String | Add [Expression] | Multiply [Expression]
In this data structure, it is possible to represent a value such as Add [Variable "a", Add [Variable "b", Variable "c"]] more compactly as Add [Variable "a", Variable "b", Variable "c"].
This can be done automatically with smart constructors such as:
add :: [Expression] -> Expression add xs = Add (concatMap fromAdd xs) multiply :: [Expression] -> Expression multiply xs = Multiply (concatMap fromMultiply xs) fromAdd (Add xs) = xs fromAdd x = [x] fromMultiply (Multiply xs) = xs fromMultiply x = [x]