# numeric-prelude: An experimental alternative hierarchy of numeric type classes

Revisiting the Numeric Classes

The Prelude for Haskell 98 offers a well-considered set of numeric
classes which cover the standard numeric types
(`Integer`

, `Int`

, `Rational`

, `Float`

, `Double`

, `Complex`

) quite well.
But they offer limited extensibility and have a few other flaws.
In this proposal we will revisit these classes, addressing the following concerns:

- 1
- The current Prelude defines no semantics for the fundamental operations. For instance, presumably addition should be associative (or come as close as feasible), but this is not mentioned anywhere.
- 2
- There are some superfluous superclasses.
For instance,
`Eq`

and`Show`

are superclasses of`Num`

. Consider the data type`data IntegerFunction a = IF (a -> Integer)`

One can reasonably define all the methods of`Algebra.Ring.C`

for`IntegerFunction a`

(satisfying good semantics), but it is impossible to define non-bottom instances of`Eq`

and`Show`

. In general, superclass relationship should indicate some semantic connection between the two classes. - 3
- In a few cases, there is a mix of semantic operations and
representation-specific operations.
`toInteger`

,`toRational`

, and the various operations in`RealFloating`

(`decodeFloat`

, ...) are the main examples. - 4
- In some cases, the hierarchy is not finely-grained enough: Operations that are often defined independently are lumped together. For instance, in a financial application one might want a type "Dollar", or in a graphics application one might want a type "Vector". It is reasonable to add two Vectors or Dollars, but not, in general, reasonable to multiply them. But the programmer is currently forced to define a method for '(*)' when she defines a method for '(+)'.

In specifying the semantics of type classes, I will state laws as follows:

(a + b) + c === a + (b + c)

The intended meaning is extensional equality:
The rest of the program should behave in the same way
if one side is replaced with the other.
Unfortunately, the laws are frequently violated by standard instances;
the law above, for instance, fails for `Float`

:

(1e20 + (-1e20)) + 1.0 = 1.0 1e20 + ((-1e20) + 1.0) = 0.0

For inexact number types like floating point types, thus these laws should be interpreted as guidelines rather than absolute rules. In particular, the compiler is not allowed to use them. Unless stated otherwise, default definitions should also be taken as laws.

Thanks to Brian Boutel, Joe English, William Lee Irwin II, Marcin Kowalczyk, Ketil Malde, Tom Schrijvers, Ken Shan, and Henning Thielemann for helpful comments.

Scope & Limitations/TODO: * It might be desireable to split Ord up into Poset and Ord (a total ordering). This is not addressed here.

In some cases, this hierarchy may not be fine-grained enough. For instance, time spans ("5 minutes") can be added to times ("12:34"), but two times are not addable. ("12:34 + 8:23") As it stands, users have to use a different operator for adding time spans to times than for adding two time spans. Similar issues arise for vector space et al. This is a consciously-made tradeoff, but might be changed. This becomes most serious when dealing with quantities with units like

`length/distance^2`

, for which`(*)`

as defined here is useless. (One way to see the issue: should`f x y = iterate (x *) y`

have principal type`(Ring.C a) => a -> a -> [a]`

or something like`(Ring.C a, Module a b) => a -> b -> [b]`

?)I stuck with the Haskell 98 names. In some cases I find them lacking. Neglecting backwards compatibility, we have renamed classes as follows: Num --> Ring Fractional --> Field Floating --> Algebraic, Transcendental RealFloat --> RealTranscental

It's slightly unfortunate that

`abs`

can no longer be used for complex numbers, since it is standard mathematically.`magnitude`

or more generally`Algebra.NormedSpace.Euclidean.norm`

can be used. But it had the wrong type before, and I couldn't see how to fit it in without complicating the hierarchy.

Additional standard libraries might include Enum, IEEEFloat (including the bulk of the functions in Haskell 98's RealFloat class), VectorSpace, Ratio.T, and Lattice.

Versions | 0.0.2, 0.0.4, 0.0.5, 0.1, 0.1.1, 0.1.2, 0.1.3, 0.1.3.1, 0.1.3.2, 0.1.3.3, 0.1.3.4, 0.2, 0.2.1, 0.2.2, 0.2.2.1, 0.3, 0.3.0.1, 0.3.0.2, 0.4, 0.4.0.1, 0.4.0.2, 0.4.0.3, 0.4.1, 0.4.2, 0.4.3 |
---|---|

Dependencies | array, base (>=1.0), containers, HUnit (==1.*), non-negative (>=0.0.2 && <0.1), parsec (>=1), QuickCheck (==1.*), random [details] |

License | LicenseRef-GPL |

Author | Dylan Thurston <dpt@math.harvard.edu>, Henning Thielemann <numericprelude@henning-thielemann.de>, Mikael Johansson |

Maintainer | Henning Thielemann <numericprelude@henning-thielemann.de> |

Category | Math |

Home page | http://www.haskell.org/haskellwiki/Mathematical_prelude_discussion |

Uploaded | by HenningThielemann at Fri Sep 26 08:06:57 UTC 2008 |

Distributions | LTSHaskell:0.4.3, NixOS:0.4.3, Stackage:0.4.3 |

Executables | testsuite, test |

Downloads | 9630 total (28 in the last 30 days) |

Rating | 2.0 (votes: 1) [estimated by rule of succession] |

Your Rating | |

Status | Docs uploaded by user Build status unknown [no reports yet] Hackage Matrix CI |

## Modules

[Index]

*Algebra*- Algebra.Additive
- Algebra.Algebraic
- Algebra.Differential
- Algebra.DimensionTerm
- Algebra.DivisibleSpace
- Algebra.Field
- Algebra.Indexable
- Algebra.IntegralDomain
- Algebra.Lattice
- Algebra.Laws
- Algebra.Module
- Algebra.ModuleBasis
- Algebra.Monoid
- Algebra.NonNegative
*NormedSpace*- Algebra.OccasionallyScalar
- Algebra.PrincipalIdealDomain
- Algebra.Real
- Algebra.RealField
- Algebra.RealIntegral
- Algebra.RealTranscendental
- Algebra.RightModule
- Algebra.Ring
- Algebra.ToInteger
- Algebra.ToRational
- Algebra.Transcendental
- Algebra.Units
- Algebra.Vector
- Algebra.VectorSpace
- Algebra.ZeroTestable

*MathObj*- MyPrelude
*Number*- NumericPrelude
- PreludeBase

## Flags

Name | Description | Default | Type |
---|---|---|---|

splitbase | Choose the new smaller, split-up base package. | Enabled | Automatic |

buildtests | Build test executables | Disabled | Automatic |

Use -f <flag> to enable a flag, or -f -<flag> to disable that flag. More info

## Downloads

- numeric-prelude-0.0.4.tar.gz [browse] (Cabal source package)
- Package description (as included in the package)