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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 `car`s are sub-op1 and sub-op2, respectively. The argument reduction is a procedure accepting 2 arguments which will be lists whose `car`s 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)))
```