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5.15 Commutative Rings

Scheme provides a consistent and capable set of numeric functions. Inexacts implement a field; integers a commutative ring (and Euclidean domain). This package allows one to use basic Scheme numeric functions with symbols and non-numeric elements of commutative rings.

(require 'commutative-ring)

The commutative-ring package makes the procedures +, -, *, /, and ^ careful in the sense that any non-numeric arguments they do not reduce appear in the expression output. In order to see what working with this package is like, self-set all the single letter identifiers (to their corresponding symbols).

(define a 'a)
…
(define z 'z)

Or just (require 'self-set). Now try some sample expressions:

(+ (+ a b) (- a b)) ⇒ (* a 2)
(* (+ a b) (+ a b)) ⇒ (^ (+ a b) 2)
(* (+ a b) (- a b)) ⇒ (* (+ a b) (- a b))
(* (- a b) (- a b)) ⇒ (^ (- a b) 2)
(* (- a b) (+ a b)) ⇒ (* (+ a b) (- a b))
(/ (+ a b) (+ c d)) ⇒ (/ (+ a b) (+ c d))
(^ (+ a b) 3) ⇒ (^ (+ a b) 3)
(^ (+ a 2) 3) ⇒ (^ (+ 2 a) 3)

Associative rules have been applied and repeated addition and multiplication converted to multiplication and exponentiation.

We can enable distributive rules, thus expanding to sum of products form:

(set! *ruleset* (combined-rulesets distribute* distribute/))

(* (+ a b) (+ a b)) ⇒ (+ (* 2 a b) (^ a 2) (^ b 2))
(* (+ a b) (- a b)) ⇒ (- (^ a 2) (^ b 2))
(* (- a b) (- a b)) ⇒ (- (+ (^ a 2) (^ b 2)) (* 2 a b))
(* (- a b) (+ a b)) ⇒ (- (^ a 2) (^ b 2))
(/ (+ a b) (+ c d)) ⇒ (+ (/ a (+ c d)) (/ b (+ c d)))
(/ (+ a b) (- c d)) ⇒ (+ (/ a (- c d)) (/ b (- c d)))
(/ (- a b) (- c d)) ⇒ (- (/ a (- c d)) (/ b (- c d)))
(/ (- a b) (+ c d)) ⇒ (- (/ a (+ c d)) (/ b (+ c d)))
(^ (+ a b) 3) ⇒ (+ (* 3 a (^ b 2)) (* 3 b (^ a 2)) (^ a 3) (^ b 3))
(^ (+ a 2) 3) ⇒ (+ 8 (* a 12) (* (^ a 2) 6) (^ a 3))

Use of this package is not restricted to simple arithmetic expressions:

(require 'determinant)

(determinant '((a b c) (d e f) (g h i))) ⇒
(- (+ (* a e i) (* b f g) (* c d h)) (* a f h) (* b d i) (* c e g))

Currently, only +, -, *, /, and ^ support non-numeric elements. Expressions with - are converted to equivalent expressions without -, so behavior for - is not defined separately. / expressions are handled similarly.

This list might be extended to include quotient, modulo, remainder, lcm, and gcd; but these work only for the more restrictive Euclidean (Unique Factorization) Domain.

5.16 Rules and Rulesets

The commutative-ring package allows control of ring properties through the use of rulesets.

Variable: *ruleset*

Contains the set of rules currently in effect. Rules defined by cring:define-rule are stored within the value of *ruleset* at the time cring:define-rule is called. If *ruleset* is #f, then no rules apply.

Function: make-ruleset rule1 …
Function: make-ruleset name rule1 …

Returns a new ruleset containing the rules formed by applying cring:define-rule to each 4-element list argument rule. If the first argument to make-ruleset is a symbol, then the database table created for the new ruleset will be named name. Calling make-ruleset with no rule arguments creates an empty ruleset.

Function: combined-rulesets ruleset1 …
Function: combined-rulesets name ruleset1 …

Returns a new ruleset containing the rules contained in each ruleset argument ruleset. If the first argument to combined-ruleset is a symbol, then the database table created for the new ruleset will be named name. Calling combined-ruleset with no ruleset arguments creates an empty ruleset.

Two rulesets are defined by this package.

Constant: distribute*

Contains the ruleset to distribute multiplication over addition and subtraction.

Constant: distribute/

Contains the ruleset to distribute division over addition and subtraction.

Take care when using both distribute* and distribute/ simultaneously. It is possible to put / into an infinite loop.

You can specify how sum and product expressions containing non-numeric elements simplify by specifying the rules for + or * for cases where expressions involving objects reduce to numbers or to expressions involving different non-numeric elements.

Function: cring:define-rule op sub-op1 sub-op2 reduction

Defines a rule for the case when the operation represented by symbol op is applied to lists whose cars are sub-op1 and sub-op2, respectively. The argument reduction is a procedure accepting 2 arguments which will be lists whose cars are sub-op1 and sub-op2.

Function: cring:define-rule op sub-op1 'identity reduction

Defines a rule for the case when the operation represented by symbol op is applied to a list whose car is sub-op1, and some other argument. Reduction will be called with the list whose car is sub-op1 and some other argument.

If reduction returns #f, the reduction has failed and other reductions will be tried. If reduction returns a non-false value, that value will replace the two arguments in arithmetic (+, -, and *) calculations involving non-numeric elements.

The operations + and * are assumed commutative; hence both orders of arguments to reduction will be tried if necessary.

The following rule is the definition for distributing * over +.

(cring:define-rule
 '* '+ 'identity
 (lambda (exp1 exp2)
   (apply + (map (lambda (trm) (* trm exp2)) (cdr exp1))))))

5.17 How to Create a Commutative Ring

The first step in creating your commutative ring is to write procedures to create elements of the ring. A non-numeric element of the ring must be represented as a list whose first element is a symbol or string. This first element identifies the type of the object. A convenient and clear convention is to make the type-identifying element be the same symbol whose top-level value is the procedure to create it.

(define (n . list1)
  (cond ((and (= 2 (length list1))
              (eq? (car list1) (cadr list1)))
         0)
        ((not (term< (first list1) (last1 list1)))
         (apply n (reverse list1)))
        (else (cons 'n list1))))

(define (s x y) (n x y))

(define (m . list1)
  (cond ((neq? (first list1) (term_min list1))
         (apply m (cyclicrotate list1)))
        ((term< (last1 list1) (cadr list1))
         (apply m (reverse (cyclicrotate list1))))
        (else (cons 'm list1))))

Define a procedure to multiply 2 non-numeric elements of the ring. Other multiplicatons are handled automatically. Objects for which rules have not been defined are not changed.

(define (n*n ni nj)
  (let ((list1 (cdr ni)) (list2 (cdr nj)))
    (cond ((null? (intersection list1 list2)) #f)
          ((and (eq? (last1 list1) (first list2))
                (neq? (first list1) (last1 list2)))
           (apply n (splice list1 list2)))
          ((and (eq? (first list1) (first list2))
                (neq? (last1 list1) (last1 list2)))
           (apply n (splice (reverse list1) list2)))
          ((and (eq? (last1 list1) (last1 list2))
                (neq? (first list1) (first list2)))
           (apply n (splice list1 (reverse list2))))
          ((and (eq? (last1 list1) (first list2))
                (eq? (first list1) (last1 list2)))
           (apply m (cyclicsplice list1 list2)))
          ((and (eq? (first list1) (first list2))
                (eq? (last1 list1) (last1 list2)))
           (apply m (cyclicsplice (reverse list1) list2)))
          (else #f))))

Test the procedures to see if they work.

;;; where cyclicrotate(list) is cyclic rotation of the list one step
;;; by putting the first element at the end
(define (cyclicrotate list1)
  (append (rest list1) (list (first list1))))
;;; and where term_min(list) is the element of the list which is
;;; first in the term ordering.
(define (term_min list1)
  (car (sort list1 term<)))
(define (term< sym1 sym2)
  (string<? (symbol->string sym1) (symbol->string sym2)))
(define first car)
(define rest cdr)
(define (last1 list1) (car (last-pair list1)))
(define (neq? obj1 obj2) (not (eq? obj1 obj2)))
;;; where splice is the concatenation of list1 and list2 except that their
;;; common element is not repeated.
(define (splice list1 list2)
  (cond ((eq? (last1 list1) (first list2))
         (append list1 (cdr list2)))
        (else (slib:error 'splice list1 list2))))
;;; where cyclicsplice is the result of leaving off the last element of
;;; splice(list1,list2).
(define (cyclicsplice list1 list2)
  (cond ((and (eq? (last1 list1) (first list2))
              (eq? (first list1) (last1 list2)))
         (butlast (splice list1 list2) 1))
        (else (slib:error 'cyclicsplice list1 list2))))

(N*N (S a b) (S a b)) ⇒ (m a b)

Then register the rule for multiplying type N objects by type N objects.

(cring:define-rule '* 'N 'N N*N))

Now we are ready to compute!

(define (t)
  (define detM
    (+ (* (S g b)
          (+ (* (S f d)
                (- (* (S a f) (S d g)) (* (S a g) (S d f))))
             (* (S f f)
                (- (* (S a g) (S d d)) (* (S a d) (S d g))))
             (* (S f g)
                (- (* (S a d) (S d f)) (* (S a f) (S d d))))))
       (* (S g d)
          (+ (* (S f b)
                (- (* (S a g) (S d f)) (* (S a f) (S d g))))
             (* (S f f)
                (- (* (S a b) (S d g)) (* (S a g) (S d b))))
             (* (S f g)
                (- (* (S a f) (S d b)) (* (S a b) (S d f))))))
       (* (S g f)
          (+ (* (S f b)
                (- (* (S a d) (S d g)) (* (S a g) (S d d))))
             (* (S f d)
                (- (* (S a g) (S d b)) (* (S a b) (S d g))))
             (* (S f g)
                (- (* (S a b) (S d d)) (* (S a d) (S d b))))))
       (* (S g g)
          (+ (* (S f b)
                (- (* (S a f) (S d d)) (* (S a d) (S d f))))
             (* (S f d)
                (- (* (S a b) (S d f)) (* (S a f) (S d b))))
             (* (S f f)
                (- (* (S a d) (S d b)) (* (S a b) (S d d))))))))
  (* (S b e) (S c a) (S e c)
     detM
     ))
(pretty-print (t))
-|
(- (+ (m a c e b d f g)
      (m a c e b d g f)
      (m a c e b f d g)
      (m a c e b f g d)
      (m a c e b g d f)
      (m a c e b g f d))
   (* 2 (m a b e c) (m d f g))
   (* (m a c e b d) (m f g))
   (* (m a c e b f) (m d g))
   (* (m a c e b g) (m d f)))

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