Alternative Prelude with numeric and logic expressions typed bottom-up.

Please see the README on GitHub at https://github.com/michalkonecny/mixed-types-num#readme

mixed-types-num

This package provides a version of Prelude where unary and binary operations such as not, +, == have their result type derived from the parameter type(s) and thus supports mixed-type arithmetic and comparisons such as:

> a = [1..10]; b = [1..11]
> length a > 2^((length b)/3)
{?(prec 36): CertainFalse}

Partial operations such as division, sqrt and power do not throw exceptions even when errors such as division by zero occur. Instead, these errors are propagated bottom-up in a bespoke error-accumulating functor from package collect-errors.

This library (as well as collect-errors) arose while developing the AERN2 library for interval and exact real computation. Certain aspects are specifically tailored for interval or exact real arithmetics, including three-valued numerical comparisons and distinguishing potential and certain errors.

API documentation available on the Hackage page.

Table of contents

• 1. Examples
  • 1.1. Main idea
  • 1.2. Dealing with numerical errors
  • 1.3. The generalised power operator
  • 1.4. Undecided comparisons
  • 1.5. Fuzzy if-then-else
• 2. Type classes
• 3. Testable specifications
• 4. Limitations
• 5. Credits

1. Examples

To replicate the examples included below, start ghci as follows:

$ stack ghci mixed-types-num:lib --no-load --ghci-options MixedTypesNumPrelude
*MixedTypesNumPrelude>

1.1. Main idea

Literals have a fixed type:

...> :t 1
... Integer

...> :t 1.0
... Rational

...> 1 :: Rational
... Couldn't match type ‘Integer’ with ‘GHC.Real.Ratio Integer’ ...

Operations permit operands of mixed types, types inferred bottom-up:

...> :t 1/2
... :: Rational

...> :t 1.5 * (length [[]])
... :: Rational

1.2. Dealing with numerical errors

To avoid runtime exceptions, it is recommended to use the CN error-collecting wrapper from package collect-errors.

All arithmetic operations have been extended so that it is possible to have expressions that operate exclusively on CN-wrapped types:

...> f (n :: CN Integer) = 1/(1/(n-1) + 1/n) :: CN Rational
...> f (cn 0)
{{ERROR: division by 0}}
...> f (cn 1)
{{ERROR: division by 0}}
...> f (cn 2)
2 % 3

Note that the errors printed above are not exceptions, but special values. See the collect-errors documentation for more details.

1.3. The generalised power operator

...> :t 2^(-2)
... :: Rational

...> :t 2^2
... :: Rational

...> :t round (2^2)
... :: Integer

...> :t (double 2)^(1/2)
... :: Double

The following examples require also package aern2-real. To get access to this via stack, you can start ghci eg as follows:

$ stack ghci aern2-real:lib --no-load --ghci-options AERN2.Real
AERN2.Real> import MixedTypesNumPrelude

...> :t pi
...  :: CReal

...> :t sqrt 2
...  :: CReal

...> :t 2^(1/2)
... :: CReal

1.4. Undecided comparisons

Comparisons involving intervals are undecided when the intervals overlap:

> pi10 = pi ? (bits 10)
> pi10
[3.1416015625 ± ~9.7656e-4 ~2^(-10)]

> pi10 > 0
CertainTrue

> pi10 == pi10
TrueOrFalse

The above equality cannot be decided since pi10 is not a single number but a set of numbers spanning the interval and the comparison operator cannot tell if the two operands sets represent the same number or a different number.

Comparison involving real numbers are semi-decidable. The result of such a comparison is a lazy Kleenean, ie an infinite sequence of Kleeneans. Please see package aern2-real for further details.

1.5. Fuzzy if-then-else

This package generalises the Haskell if-then-else statement so that it admits Kleenean and lazy Kleenean conditions:

...> abs1 x = max 0 (if x < 0 then -x else x)
...> abs1 (pi10 - pi10)
[0.0009765625 ± ~9.7656e-4 ~2^(-10)]

Although the condition x < 0 cannot be decided for the interval pi10-pi10 = [0 ± ~1.9531e-3 ~2^(-9)], the if-then-else statement is resolved by computing both branches and unifying the resulting intervals. This makes sense only if both branches compute the same number whenever the condition cannot be decided, ie when x = 0 in this case, making the function continuous.

If we try to define a discontinuous function this way, we get an error as soon as it is detected:

...> bad1 x = if x < 0 then 1-x else x
...> bad1 (pi10 - pi10)
[0.5 ± ~0.5020 ~2^(-1)]{{ERROR: numeric error: union of enclosures: not enclosing the same value}}

The generalised if-then-else works also for real numbers with lazy Kleenean comparisons:

...> abs1 (pi - pi)
{?(prec 36): [0.00000000001455191522836685... ± ~1.4552e-11 ~2^(-36)]}

2. Type classes

Mixed-type arithmetic operations are provided via multi-parameter type classes and the result type is given by associated type families. For example:

(+) :: (CanAddAsymmetric t1 t2) => t1 -> t2 -> AddType t1 t2

The constraint CanAdd t1 t2 is a shortcut for both CanAddAsymmetric t1 t2 and CanAddAsymmetric t2 t1.

For convenience there are other aggregate type constraints such as CanAddThis t1 t2, which implies that the result is of type t1, and CanAddSameType t, which is a shortcut for CanAddThis t t.

Notably, there are convenience classes Ring and Field as well as OrderedRing and OrderedField.

For types that instantiate Prelude classes such as Num, one can define instances of the new classes using the default implementation, eg:

{-# LANGUAGE GeneralizedNewtypeDeriving #-}
import MixedTypesPrelude
import qualified Prelude as P

newtype II = II Integer deriving (P.Eq, P.Ord, P.Num) 
instance CanAddAsymmetric II II

Conversely, if one defines instances such as CanAddAsymmetric T T, one can then trivially define also instances Num T etc:

instance P.Num T where
  (+) = (+)
  ...

3. Testable specifications

The arithmetic type classes are accompanied by generic hspec test suites, which are specialised to concrete instance types for their testing. These test suites include the expected algebraic properties of operations, such as commutativity and associativity of addition.

4. Limitations

• Not all numerical operations are supported yet. Eg tan, atan are missing at the moment.

• Not all Prelude numerical types are supported yet. Eg Natural and Float are not supported at present, but Double is supported.

• Many common operations such as fromEnum, threadDelay give or require an Int value, which means we sometimes need to convert:

  threadDelay (int 1000000)
  integer (fromEnum True)
  

  Prelude functions such as take, !! and length that use Int in Prelude are shadowed in MixedTypesNumPrelude with more compatible/flexible versions. Beware that Data.List.length clashes with length in MixedTypesNumPrelude.

• Inferred types can be very large. Eg for f a b c = sqrt (a + b * c + 1) the inferred type is:

  f :: (CanSqrt (AddType (AddType t2 (MulType t3 t4)) Integer),
      CanAddAsymmetric (AddType t2 (MulType t3 t4)) Integer,
      CanAddAsymmetric t2 (MulType t3 t4), CanMulAsymmetric t3 t4) =>
      t2
      -> t3
      -> t4
      -> SqrtType (AddType (AddType t2 (MulType t3 t4)) Integer)
  

5. Credits

The idea of having numeric expressions in Haskell with types derived bottom-up was initially suggested and implemented by Pieter Collins. This version is a fresh rewrite by Michal Konečný.