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
andShow
are superclasses ofNum
. Consider the data typedata IntegerFunction a = IF (a -> Integer)
One can reasonably define all the methods ofAlgebra.Ring.C
forIntegerFunction a
(satisfying good semantics), but it is impossible to define non-bottom instances ofEq
andShow
. 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 inRealFloating
(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: shouldf 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 generallyAlgebra.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.
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
Automatic Flags
Name | Description | Default |
---|---|---|
splitbase | Choose the new smaller, split-up base package. | Enabled |
buildtests | Build test executables | Disabled |
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 (revised from the package)
Note: This package has metadata revisions in the cabal description newer than included in the tarball. To unpack the package including the revisions, use 'cabal get'.
Maintainer's Corner
For package maintainers and hackage trustees
Candidates
Versions [RSS] | 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, 0.4.3.1, 0.4.3.2, 0.4.3.3, 0.4.4 |
---|---|
Dependencies | array, base (>=1.0 && <4.3), containers, HUnit (>=1 && <2), non-negative (>=0.0.2 && <0.1), parsec (>=1), QuickCheck (>=1 && <2), 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> |
Revised | Revision 1 made by HerbertValerioRiedel at 2019-01-19T13:37:12Z |
Category | Math |
Home page | http://www.haskell.org/haskellwiki/Mathematical_prelude_discussion |
Uploaded | by HenningThielemann at 2008-09-26T08:06:57Z |
Distributions | LTSHaskell:0.4.4, NixOS:0.4.4, Stackage:0.4.4 |
Reverse Dependencies | 32 direct, 22 indirect [details] |
Executables | testsuite, test |
Downloads | 24261 total (140 in the last 30 days) |
Rating | 2.0 (votes: 1) [estimated by Bayesian average] |
Your Rating | |
Status | Docs uploaded by user Build status unknown [no reports yet] |