Algebra Simple

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Table of Contents

• Overview
• Algebraic Typeclasses
  • Motivation
  • Solution
• Miscellaneous

Overview

Algebra-Simple intends to provide a simple, reasonably principled interface to typical operations (i.e. addition, subtraction, multiplication, division). This package is organized into three sections: Numeric.Algebra, Numeric.Data, and Numeric.Literal.

Algebraic Typeclasses

Motivation

The primary interface to numerical operations in Haskell is Num. Unfortunately, Num has a key limitation: it is "too large". For example, if we want to opt-in to addition, we must also opt-in to subtraction, multiplication, and integer literal conversions. These may not make sense for the type at hand (e.g. naturals), so we are stuck either providing an invariant-breaking dangerous implementation (e.g. defining subtraction for arbitrary naturals) or throwing runtime errors.

Solution

algebra-simple's approach is to split this functionality into multiple typeclasses, so types can opt-in to exactly as much functionality as they want. The typeclasses are inspired by abstract algebra. The algebraic hierarchy can be found in the following diagram. An arrow A -> B should be read as "B is an A". For example, a Module is both a Semimodule and an AGroup.

A longer description can be found in the table below, along with the Num functionality they are intended to replace:

Typeclass	Description	New	Num
ASemigroup	Addition.	(.+.)	(+)
AMonoid	ASemigroup with identity.	zero
AGroup	Subtraction.	(.-.)	(-)
MSemigroup	Multiplication.	(.*.)	(*)
MMonoid	MSemigroup with identity.	one
MGroup	Division.	(.%.)	div, (/)
MEuclidean	Euclidean division.	mmod	mod
Normed	Types that support a "norm".
Semiring	AMonoid and MMonoid.
Ring	AGroup and MMonoid.
Semifield	AMonoid and MGroup.
Field	Ring and Semifield.
MSemiSpace	Scalar multiplication.	(.*), (*.)
MSpace	Scalar division.	(.%), (%.)
Semimodule	AMonoid and MSemiSpace.
Module	Semimodule and AGroup.
SemivectorSpace	Semimodules that are "scalar division"	(.%)
VectorSpace	Module and SemivectorSpace

We have the following guiding principles:

1. Simplicity

   This is not a comprehensive implementation of abstract algebra, merely the classes needed to replace the usual Num-like functionality. For the former, see algebra.

2. Practicality

   When there is tension between practicality and theoretical "purity", we favor the former. To wit:

   • We provide two semigroup/monoid/group hierarchies: ASemigroup/AMonoid/AGroup and MSemigroup/MMonoid/MGroup. Formally this is clunky, but it allows us to:

     • Reuse the same operator for ring multiplication and types that have sensible multiplication but cannot be rings (e.g. naturals).

     • Provide both addition and multiplication without an explosion of newtype wrappers.

   • Leniency vis-à-vis algebraic laws

     For instance, integers cannot satisfy the field laws, and floats do not satisfy anything, as their equality is nonsense. Nevertheless, we provide instances for them. Working with technically unlawful numerical instances is extremely common, so we take the stance that it is better to provide such instances (albeit with known limitations) than to forgo them completely (read: integer division is useful). The only instances we disallow are those likely to cause runtime errors (e.g. natural subtraction) or break expected invariants.

   • Division classes (i.e. MGroup, VectorSpace) have their own division function that must be implemented. Theoretically this is unnecessary, as we need only a function inv :: a -> a and we can then define division as x .%. d = x .*. inv d. But this will not work for many types (e.g. integers), so we force users to define a (presumably sensible) (.%.), so there is no chance of accidentally using a nonsensical inv.

3. Safety

   Instances that break the type's invariants (instance Ring Natural), are banned. Furthermore, instances that are highly likely to go wrong (e.g. Rational with bounded integral types) are also forbidden.

4. Ergonomics

   We choose new operators that do not clash with prelude.

We provide instances for built-in numeric types where it makes sense.

Miscellaneous

Finally, there are typeclasses in Numeric.Convert for conversions.