algebraic-structures

Provides useful algebraic structures for programming using parametric modules in Chicken Scheme.

Install

Run chicken-install in the project's root directory.

$ cd algebraic-structures
$ chicken-install

Usage

Define Semigroup, Monoid and Group

(import (algebraic-structures semigroup)
        (algebraic-structures monoid)
        (algebraic-structures group)
        (algebraic-structures monoid fold)
        (algebraic-structures list foldable))

(module (mod7 semigroup) = (algebraic-structures semigroup)
  (import scheme
          (chicken module))
  (export op)

  (define (op x y)
    (assert (integer? x))
    (assert (integer? y))
    (assert (not (zero? x)))
    (assert (not (zero? y)))
    (modulo (* x y) 7)))

(module (mod7 monoid) = (algebraic-structures monoid)
  (import scheme
          (chicken module)
          (chicken base)
          (only (mod7 semigroup)))
  (reexport (mod7 semigroup))
  (export unit)

  (define unit 1))

(module (mod7 group) = (algebraic-structures group)
  (import scheme
          (chicken base)
          (chicken module)
          matchable
          (only (mod7 monoid)))
  (reexport (mod7 monoid))
  (export inv)

  (define (inv n)
    (assert (integer? n))
    (assert (not (zero? n)))
    (match (modulo n 7)
      (1 1)
      (2 4)
      (3 5)
      (4 2)
      (5 3)
      (6 6))))

(module (mod7 fold) = ((algebraic-structures monoid fold) (mod7 monoid) (algebraic-structures list foldable)))

(import (prefix (mod7 group) mod7:)
        (prefix (mod7 fold) mod7:))

In REPL:

> (map (cut mod7:pow 3 op) '(0 1 2 3 4 5 6 7 8 9 10 11))
(1 3 2 6 4 5 1 3 2 6 4 5)
> (mod7:fold '(1 2 3 4 5 6))
6

Monad Syntax

(import (srfi 41)
        (prefix (algebraic-structures stream monad) stream:)
        (prefix (algebraic-structures stream alternative) stream:))

(define (pythagorean-triples)
  (stream:do (b <- (stream-from 1))
             (a <- (stream-range 1 b))
             (let c^2 = (+ (* a a) (* b b)))
             (let-values (c r) = (exact-integer-sqrt c^2))
             (stream:guard (zero? r))
             (stream (list a b c))))

In REPL:

> (stream->list (stream-take 10 (pythagorean-triples)))
((3 4 5) (6 8 10) (5 12 13) (9 12 15) (8 15 17) (12 16 20) (15 20 25) (20 21 29) (7 24 25) (10 24 26))

Supported Features

• Semigroup: (algebraic-structures semigroup)
  • Number (product): (algebraic-structures number product semigroup)
  • Number (sum): (algebraic-structures number sum semigroup)
  • List: (algebraic-structures list semigroup)
  • Vector: (algebraic-structures vector semigroup)
  • Stream: (algebraic-structures stream semigroup)
• Semigroup.reduce: (algebraic-structures semigroup reduce)
• Monoid: (algebraic-structures monoid)
  • Number (product): (algebraic-structures number product monoid)
  • Number (sum): (algebraic-structures number sum monoid)
  • List: (algebraic-structures list monoid)
  • Vector: (algebraic-structures vector monoid)
  • Stream: (algebraic-structures stream monoid)
• Monoid.fold: (algebraic-structures monoid fold)
• Group
  • Number (product): (algebraic-structures number product group)
  • Number (sum): (algebraic-structures number sum group)
• Foldable: (algebraic-structures foldable)
  • List: (algebraic-structures list foldable)
  • Vector: (algebraic-structures vector foldable)
  • Stream: (algebraic-structures stream foldable)
• Reducible: (algebraic-structures reducible)
  • List: (algebraic-structures list reducible)
  • Vector: (algebraic-structures vector reducible)
  • Stream: (algebraic-structures stream reducible)
• Functor: (algebraic-structures functor)
  • List: (algebraic-structures list functor)
  • Vector: (algebraic-structures vector functor)
  • Stream: (algebraic-structures stream functor)
• Applicative: (algebraic-structures applicative)
  • List: (algebraic-structures list applicative)
  • List (zip): (algebraic-structures list zip applicative)
  • Vector (zip): (algebraic-structures vector zip applicative)
  • Stream: (algebraic-structures stream applicative)
  • Stream (zip): (algebraic-structures stream zip applicative)
• Monad: (algebraic-structures monad)
  • List: (algebraic-structures list monad)
  • Stream: (algebraic-structures stream monad)
• Alternative: (algebraic-structures alternative)
  • List: (algebraic-structures list alternative)
  • Stream: (algebraic-structures stream alternative)

Examples

• Group
  • mod7
• Monad
  • optional
  • state
• Monad (do syntax)
  • pythagorean-triples