Add solutions to exercises 2.53-57

ex-2.53.txt

@@ -0,0 +1,7 @@
+(list 'a 'b 'c) -> (a b c)
+(list (list 'george)) -> ((george))
+(cdr '((x1 x2) (y1 y2))) -> ((y1 y2))
+(cadr '((x1 x2) (y1 y2))) -> (y1 y2)
+(pair? (car '(a short list))) -> #f
+(memq 'red '((red shoes) (blue socks))) -> #f
+(memq 'red '(red shoes blue socks)) -> (red shoes blue socks)

ex-2.54.scm

@@ -0,0 +1,8 @@
+#lang sicp
+
+(define (equal? a b)
+  (cond ((and (pair? a) (pair? b))
+         (and (equal? (car a) (car b)) (equal? (cdr a) (cdr b))))
+        ((and (not (pair? a)) (not (pair? b)))
+         (and (eq? a b)))
+        (else #f)))

ex-2.55.txt

@@ -0,0 +1,6 @@
+When typing (car ''abracadabra), the interpreter will transform the expression
+into: (car (quote (quote abracadabra))), then it will evaluate.
+First the symbol car becomes the car procedure, then quote gets evaluated as a
+special form, which means the surrounded expression should be left
+unevaluated, so we get something like: (#<car:procedure> (quote abracadabra)),
+now car gets applied, and we get the expected result.

ex-2.56.scm

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+#lang sicp
+
+(define (variable? x) (symbol? x))
+(define (same-variable? v1 v2)
+  (and (variable? v1) (variable? v2) (eq? v1 v2)))
+
+(define (=number? exp num) (and (number? exp) (= exp num)))
+
+(define (make-sum a1 a2)
+  (cond ((=number? a1 0) a2)
+        ((=number? a2 0) a1)
+        ((and (number? a1) (number? a2))
+         (+ a1 a2))
+        (else (list '+ a1 a2))))
+
+(define (make-product m1 m2)
+  (cond ((or (=number? m1 0) (=number? m2 0)) 0)
+        ((=number? m1 1) m2)
+        ((=number? m2 1) m1)
+        ((and (number? m1) (number? m2)) (* m1 m2))
+        (else (list '* m1 m2))))
+
+(define (sum? x) (and (pair? x) (eq? (car x) '+)))
+(define (addend s) (cadr s))
+(define (augend s) (caddr s))
+(define (product? x) (and (pair? x) (eq? (car x) '*)))
+(define (multiplier p) (cadr p))
+(define (multiplicand p) (caddr p))
+
+; for the purposes of our exponentiation exercise, we need to determine whether
+; the exponent is constant so that we can apply the rule correctly
+(define (constant? x var)
+  (or (number? x)
+      (and (variable? x) (not (same-variable? x var)))
+      (and (sum? x)
+           (constant? (addend x) var)
+           (constant? (augend x) var))
+      (and (product? x)
+           (constant? (multiplier x) var)
+           (constant? (multiplicand x) var))
+      (and (exponentiation? x)
+           (constant? (base x) var)
+           (constant? (exponent x) var))))
+
+(define (make-exponentiation b e)
+  (cond ((=number? e 0) 1)
+        ((=number? e 1) b)
+        ((=number? b 0) 0)
+        ((and (number? b) (number? e)) (expt b e))
+        (else (list '** b e))))
+        
+
+(define (exponentiation? x) (and (pair? x) (eq? (car x) '**)))
+(define (base e) (cadr e))
+(define (exponent e) (caddr e))
+
+(define (deriv exp var)
+  (cond ((number? exp) 0)
+        ((variable? exp) (if (same-variable? exp var) 1 0))
+        ((sum? exp) (make-sum (deriv (addend exp) var)
+                              (deriv (augend exp) var)))
+        ((product? exp)
+         (make-sum
+          (make-product (multiplier exp)
+                        (deriv (multiplicand exp) var))
+          (make-product (deriv (multiplier exp) var)
+                        (multiplicand exp))))
+        ((exponentiation? exp)
+         (if (constant? (exponent exp) var)
+             (make-product (make-product (exponent exp)
+                                         (make-exponentiation (base exp)
+                                                              (make-sum
+                                                               (exponent exp)
+                                                               -1)))
+                           (deriv (base exp) var))
+             (error "non-constant exponents not yet supported: DERIV" exp)))
+        (else
+         (error "unknown expression type: DERIV" exp))))

ex-2.57.scm

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+#lang sicp
+
+; For these exercises, where I'm adding extra functionality to the deriv
+; system. I chose to build each new feature off of the results of the previous
+; exercises, rather than the unmodified code.
+; This might make the solutions a bit more complicated than they have to be
+
+(define (variable? x) (symbol? x))
+(define (same-variable? v1 v2)
+  (and (variable? v1) (variable? v2) (eq? v1 v2)))
+
+(define (=number? exp num) (and (number? exp) (= exp num)))
+
+(define (not-number? x) (not (number? x)))
+
+(define (filter predicate sequence)
+  (cond ((null? sequence) nil)
+        ((predicate (car sequence))
+         (cons (car sequence)
+               (filter predicate (cdr sequence))))
+        (else (filter predicate (cdr sequence)))))
+
+; only folds the beginning constants of the top-level expression
+; e.g. (+ 2 3 5 x) becomes (+ 10 x)
+;      (+ x y z) -> (+ x y z)
+;      (* 2 0 3 x) -> 0
+;      (+ 3 -3 x) -> x
+(define (fold exp)
+  (if (and (not (sum? exp)) (not (product? exp)))
+      exp ; if you don't know how to fold the exp, just leave it
+      (let* ((op (car exp))
+             (consts (filter number? (cdr exp)))
+             (rest (filter not-number? (cdr exp)))
+             (folded-const (if (eq? op '+)
+                               (apply + consts)
+                               (apply * consts)))
+             (args (cond ((and (eq? op '*) (= folded-const 0)) '(0))
+                         ((and (eq? op '*) (= folded-const 1)) rest)
+                         ((and (eq? op '+) (= folded-const 0)) rest)
+                         (else (append (list folded-const) rest)))))
+        (cond ((and (eq? op '+) (= (length args) 0)) 0)
+              ((and (eq? op '*) (= (length args) 0)) 1)
+              ((= (length args) 1) (car args))
+              (else (append (list op) args))))))
+
+; make-sum and make-product will be changed so that any arguments which are
+; sums themselves will be spliced into the list, and all numbers will be placed
+; in the front and folded
+(define (make-sum a1 a2)
+  ; l1 and l2 are lists to be spliced, if they're not sums, we make them lists
+  ; of a list, so effectively, they don't get spliced
+  (let ((l1 (if (sum? a1) (cdr a1) (list a1)))
+        (l2 (if (sum? a2) (cdr a2) (list a2))))
+    (fold (append '(+) l1 l2))))
+
+(define (make-product m1 m2)
+  (let ((l1 (if (product? m1) (cdr m1) (list m1)))
+        (l2 (if (product? m2) (cdr m2) (list m2))))
+    (fold (append '(*) l1 l2))))
+
+(define (sum? x) (and (pair? x) (eq? (car x) '+)))
+(define (addend s) (cadr s))
+(define (augend s) (if (= (length s) 3)
+                       (caddr s)
+                       (cons '+ (cddr s))))
+(define (product? x) (and (pair? x) (eq? (car x) '*)))
+(define (multiplier p) (cadr p))
+(define (multiplicand p) (if (= (length p) 3)
+                             (caddr p)
+                             (cons '* (cddr p))))
+
+; for the purposes of our exponentiation exercise, we need to determine whether
+; the exponent is constant so that we can apply the rule correctly
+(define (constant? x var)
+  (or (number? x)
+      (and (variable? x) (not (same-variable? x var)))
+      (and (sum? x)
+           (constant? (addend x) var)
+           (constant? (augend x) var))
+      (and (product? x)
+           (constant? (multiplier x) var)
+           (constant? (multiplicand x) var))
+      (and (exponentiation? x)
+           (constant? (base x) var)
+           (constant? (exponent x) var))))
+
+(define (make-exponentiation b e)
+  (cond ((=number? e 0) 1)
+        ((=number? e 1) b)
+        ((=number? b 0) 0)
+        ((and (number? b) (number? e)) (expt b e))
+        (else (list '** b e))))
+        
+
+(define (exponentiation? x) (and (pair? x) (eq? (car x) '**)))
+(define (base e) (cadr e))
+(define (exponent e) (caddr e))
+
+(define (deriv exp var)
+  (cond ((number? exp) 0)
+        ((variable? exp) (if (same-variable? exp var) 1 0))
+        ((sum? exp) (make-sum (deriv (addend exp) var)
+                              (deriv (augend exp) var)))
+        ((product? exp)
+         (make-sum
+          (make-product (multiplier exp)
+                        (deriv (multiplicand exp) var))
+          (make-product (deriv (multiplier exp) var)
+                        (multiplicand exp))))
+        ((exponentiation? exp)
+         (if (constant? (exponent exp) var)
+             (make-product (make-product (exponent exp)
+                                         (make-exponentiation (base exp)
+                                                              (make-sum
+                                                               (exponent exp)
+                                                               -1)))
+                           (deriv (base exp) var))
+             (error "non-constant exponents not yet supported: DERIV" exp)))
+        (else
+         (error "unknown expression type: DERIV" exp))))