Smart constructors

Smart constructors

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.

Runtime checking : smart constructors

A first attempt

Code up 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

for example.

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:

> metalResistor 4
  Metal 4

> metalResistor 7
  Metal 7

> metalResistor 9
  *** Exception: Invalid number of resistor bands

> 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

 ...

Using assertions

Hand-coding error messages can be tedious when used often. Instead we can use the assert function, provided (from Control.Exception). 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:

> 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.

Compile-time checking : the type system

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

Examples

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

Great!

> :t resistor d4
resistor d4 :: Resistor (S (S (S (S Z))))

And its 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.

Summary

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).

Extensions

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).

Related work

• These ideas are also discussed in Dimensionalized numbers and on the old wiki here (for compile-time unit analysis error catching at the type level).

• There is also Liquid Haskell, which allows you to annotate your functions with invariants ("the list that this function produces has to be sorted", etc) that are run through a SMT solver at compile time.

• "refined" library, which abstracts over smart constructors and provides compile-time checking, essentially implementing refinement types.

• Recently migrated are the pages on worker wrapper and factory function.

In general, the more information you place on the type level, the more static checks you get -- and thus less chance for bugs.

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]