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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)))