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C00004 00003	.if false then begin "answers"
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.cb	CS206	Computing with Symbolic Expressions	Spring 1977
.cb	Midterm Examination
.cb	May 2 1977
.skip 2

.FILL

Write the following routines using either Maclisp or Blackboard notation.

You may use any books or notes that you wish on this test.

.SKIP 2
1.a. %2Least[u]%1 is the least integer in the list %2u%1 of integers.

b. %2Least1[x]%1 is the least integer in the s-expression %2x%1, all of whose
atoms are assumed to be integers.
.SKIP 1
2. Let %2e%1 be a Lisp term in internal notation. %2Frees[e]%1 returns a list of
the free variables in %2e%1. Remember that labels in PROGs are not free
variables.  Forget about the problems that are caused by considering FEXPRS;
i.e. assume that all arguments to functions WILL be evaluated. If you have extra
time, explain for extra credit why you are glad to make this assumption.
.SKIP 1
3. The polynomial %2a0 + a1*x + ... +an*x↑n%1 may be represented by the list:
%2(a0 a1 ... an)%1 of its coefficients.

a. %2Quot[p1,p2]%1 is the quotient of the polynomials %2p1%1 and %2p2%1.

b. %2Rem[p1,p2]%1 is the remainder polynomial.

; Answers to CS 206 Midterm
; May 3, 1977

(DEFUN LEAST (U) (LEEST (CAR U) (CDR U))) ; Compares first to rest

(DEFUN LEEST (WIN REST) ; Compares winner so far to the rest
(COND ((NULL REST) WIN)
(T (LEEST (COND ((LESSP WIN (CAR REST)) WIN)
(T (CAR REST)))
(CDR REST)))))

(DEFUN LEAST1 (X) ; Keeps winner of least of car vs. least of cdr
(COND ((NUMBERP X) X)
(T ((LAMBDA (A D) (COND ((LESSP A D) A) (T D)))
(LEAST1 (CAR X))
(LEAST1 (CDR X))))))

(defun min-l (l); least written without an auxiliary function
((lambda (ans)
(mapc (function (lambda (q)
(cond ((lessp q ans)
(setq ans q))))) (cdr l))
ans) (car l)))

; least1 written to compare first to rest
(defun min-x (x) (min-ex x (first-number x)))

(defun min-ex(x ans)
(cond ((numberp x) x)
(t
(mapc (function (lambda (q)
((lambda(a)
(cond ((lessp a ans)(setq ans a))))
(min-x q))))
x) ans)))

(defun first-number (l)
(cond ((numberp l) l)
(t (first-number (car l)))))

;Quotient written to do convential polynomial division, just like long division of numbers

(DEFUN QUOT (P1 P2) (REVERSE (QUOT1 (REVERSE P1) (REVERSE P2))))

(DEFUN QUOT1 (P1 P2) ; Does the division using lists with highest power first
(COND ((LESSP (LENGTH P1) (LENGTH P2)) NIL)
(T ((LAMBDA (Q)
(CONS Q
(QUOT1 (VDIFF (CDR P1)
(SCALPROD Q (CDR P2)))
P2)))
(DIV (CAR P1) (CAR P2))))))

(DEFUN VDIFF (U V) ;Takes the difference element by element of two lists
(COND ((NULL U) (MAPCAR (FUNCTION MINUS) V)); invert sign and return
((NULL V) U)
(T (CONS (DIFFERENCE (CAR U) (CAR V))
(VDIFF (CDR U) (CDR V))))))

; Forces floating point arithmetic ... otherwise (quotient 4 3)=1
(DEFUN DIV (Q1 Q2) (QUOTIENT (PLUS Q1 0.0) (PLUS Q2 0.0)))

(DEFUN SCALPROD (S V) ; Multiplies S times each element of V
(MAPCAR (FUNCTION (LAMBDA (V1) (TIMES S V1))) V))

(DEFUN REM (P1 P2) (REVERSE (REM1 (REVERSE P1) (REVERSE P2))))

; Just like quot1 except throws away the quotient and keeps remainder
(DEFUN REM1 (P1 P2)
(COND ((LESSP (LENGTH P1) (LENGTH P2)) P1)
(T (REM1 (VDIFF (CDR P1)
(SCALPROD (DIV (CAR P1) (CAR P2))
(CDR P2)))
P2))))

;Frees uses auxiliaries to get rid of duplicates and to carry the list of bound variables
(DEFUN FREES (E) (REMOVDUPS (FREE1 E NIL)))

(DEFUN FREE1 (E BOUND) ; Looks for any variable not in Bound
(COND ((OR (NULL E) (EQ E T) (NUMBERP E)) NIL); Ignores constants
((ATOM E) (COND ((MEMBER E BOUND) NIL) (T (LIST E)))); collects unbound vars
((EQ 'GO (CAR E)) NIL); Ignores labels in GO statements
((EQ 'LAMBDA (CAR E))  ; Collects bound vars whenever possible
(FREE1 (CDDR E) (APPEND (CADR E) BOUND)))
((EQ 'DEFUN (CAR E))
(FREE1 (CDDDR E) (APPEND (CADDR E) BOUND)))
((EQ 'PROG (CAR E)) ;Calls stripatoms to avoid labels in the PROG
(FREE1 (STRIPATOMS (CDDR E)) (APPEND (CADR E) BOUND)))
((ATOM (CAR E)); Takes all the arguments of function but not fun name
(MAPCARAPP (FUNCTION FREE1) (CDR E) BOUND))
(T (MAPCARAPP (FUNCTION FREE1) E BOUND)))) ;Takes all elements of the list

; This is Map Car Append, like system function MAPCAN, but with extra arguments allowed
(DEFUN MAPCARAPP (FUN LISTARG ARG2)
(COND ((NULL LISTARG) NIL)
(T (APPEND (APPLY FUN (LIST (CAR LISTARG) ARG2))
(MAPCARAPP FUN (CDR LISTARG) ARG2)))))

(DEFUN REMOVDUPS (U) ; Gets rid of any element that occurs later in list
(COND ((NULL U) NIL)
((MEMBER (CAR U) (CDR U)) (REMOVDUPS (CDR U)))
(T (CONS (CAR U) (REMOVDUPS (CDR U))))))

; Returns all of list except atoms at top level (labels in this case)
(DEFUN STRIPATOMS (U)
(COND ((NULL U) NIL)
((ATOM (CAR U)) (CDR U))
(T (CONS (CAR U) (STRIPATOMS (CDR U))))))

; You can also use a lambda in writing frees

(defun frees (l) (rem-dup (frees1 l nil)))

(defun frees1 (l bound)
(cond 	((or (null l)(eq l t)(numberp l)) nil)
((and (atom l) (not (memq l bound))) (ncons l))
((atom l) nil)
((eq (car l) 'defun)((lambda (bound)
(mapcan (function
(lambda (q) (frees1 q bound))) (cdddr l)))
((or (eq (car l) 'lambda)
(eq (car l) 'prog))
((lambda (bound)
(mapcan (function
(lambda (q) (frees1 q bound)))(cddr l)))
((atom (car l))(mapcan (function
(lambda (q)(frees1 q bound))) (cdr l)))
(t (mapcan (function
(lambda (q) (frees1 q bound))) l))))

; Rem-dup uses memq, a version of member that tests with EQ rather than EQUAL
(defun rem-dup (l)
(cond 	((null l) nil)
((memq (car l)(cdr l))(rem-dup (cdr l)))
(t (cons (car l)(rem-dup (cdr l))))))

;The reason FEXPRS should be ignored when writing Frees is illustrated by QUOTE.
;If the expression "(QUOTE X)" appeared, the simple-minded version of Frees will
;report x as a free variable (assuming it hasn't been bound). To "fix" Frees so
;it wouldn't make this mistake, you could rule out the args of QUOTE. But then
;two other problems arise.

;1. Suppose you encounter "(MaybeQuote X)" where maybequote is a fexpr that under
;certain conditions defined by free variables quotes its argument and other times
;evaluates it. What then ?
;
;2. Suppose the previously mentioned reference to "(QUOTE X)" occurred in the
;following context: "(APPLY FOO (QUOTE X))".  In this case we see that X is in
;fact a candidate for being a free variable, and must be considered.
;
;The general tendency displayed by these examples hints of the opening of a whole
;can of worms, and the impression is correct. The general question of whether an
;arbitrary variable will be evaluated is as hard as the question of whether an
;arbitrary program will halt; i.e. it is undecidable.