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;; ====================================================================
;; SICP - 1.2: Procedures and the Processes They Generate
;; ====================================================================
;; ====================================================================
;; 1.2.1: Linear Recursion and Iteration
;; ====================================================================
(define (inc n) (+ n 1))
(define (dec n) (- n 1))
;; ====================================================================
;; *Exercise 1.9:* Each of the following two procedures defines a
;; method for adding two positive integers in terms of the procedures
;; `inc', which increments its argument by 1, and `dec', which
;; decrements its argument by 1.
;;
;; (define (+ a b)
;; (if (= a 0)
;; b
;; (inc (+ (dec a) b))))
;;
;; (define (+ a b)
;; (if (= a 0)
;; b
;; (+ (dec a) (inc b))))
;;
;; Using the substitution model, illustrate the process generated by
;; each procedure in evaluating `(+ 4 5)'. Are these processes
;; iterative or recursive?
;; --------------------------------------------------------------------
;; First procedure: Recursive
(+ 4 5)
(inc (+ 3 5))
(inc (inc (+ 2 5)))
(inc (inc (inc (+ 1 5))))
(inc (inc (inc (inc (+ 0 5)))))
(inc (inc (inc (inc 5))))
(inc (inc (inc 6)))
(inc (inc 7))
(inc 8)
9
;; Second procedure: Iterative
(+ 4 5)
(+ 3 6)
(+ 2 7)
(+ 1 8)
(+ 0 9)
9
;; ====================================================================
;; *Exercise 1.10:* The following procedure computes a mathematical
;; function called Ackermann's function.
;;
(define (A x y)
(cond ((= y 0) 0)
((= x 0) (* 2 y))
((= y 1) 2)
(else (A (- x 1)
(A x (- y 1))))))
;;
;; What are the values of the following expressions?
;;
;; (A 1 10)
1024
;; (A 2 4)
65536
;;
;; (A 3 3)
65536
;;
;; Consider the following procedures, where `A' is the procedure
;; defined above:
;;
;; (define (f n) (A 0 n))
;;
;; (define (g n) (A 1 n))
;;
;; (define (h n) (A 2 n))
;;
;; (define (k n) (* 5 n n))
;;
;; Give concise mathematical definitions for the functions computed
;; by the procedures `f', `g', and `h' for positive integer values of
;; n. For example, `(k n)' computes 5n^2.
;; --------------------------------------------------------------------
;; `(f n)' computes 2n
;; `(g n)' computes ...
;; ====================================================================
;; 1.2.2: Tree Recursion
;; ====================================================================
(define (fib n)
(fib-iter 1 0 n))
(define (fib-iter a b count)
(if (= count 0)
b
(fib-iter (+ a b) a (- count 1))))
;; ====================================================================
;; *Exercise 1.11:* A function f is defined by the rule that f(n) = n
;; if n<3 and f(n) = f(n - 1) + 2f(n - 2) + 3f(n - 3) if n>= 3.
;; Write a procedure that computes f by means of a recursive process.
;; Write a procedure that computes f by means of an iterative
;; process.
;; --------------------------------------------------------------------
(define (f-recursive n)
(if (< n 3)
n
(+ (f-recursive (- n 1))
(* 2 (f-recursive (- n 2)))
(* 3 (f-recursive (- n 3))))))
(define (f-iterative n)
(define (do-iter a b c n)
(if (< n 3)
a
(do-iter (+ a (* 2 b) (* 3 c)) a b (- n 1))))
(if (< n 3)
n
(do-iter 2 1 0 n)))
;; ====================================================================
;; *Exercise 1.12:* The following pattern of numbers is called "Pascal's
;; triangle".
;;
;; 1
;; 1 1
;; 1 2 1
;; 1 3 3 1
;; 1 4 6 4 1
;;
;; The numbers at the edge of the triangle are all 1, and each number
;; inside the triangle is the sum of the two numbers above it.(4)
;; Write a procedure that computes elements of Pascal's triangle by
;; means of a recursive process.
;; --------------------------------------------------------------------
(define (pascal row column)
(cond ((= column 1) 1)
((= row column) 1)
(#t (+ (pascal (- row 1) (- column 1))
(pascal (- row 1) column)))))
;; ====================================================================
;; *Exercise 1.13:* Prove that _Fib_(n) is the closest integer to
;; [phi]^n/[sqrt](5), where [phi] = (1 + [sqrt](5))/2. Hint: Let
;; [illegiblesymbol] = (1 - [sqrt](5))/2. Use induction and the
;; definition of the Fibonacci numbers (see section *Note 1-2-2::) to
;; prove that _Fib_(n) = ([phi]^n - [illegiblesymbol]^n)/[sqrt](5).
;; --------------------------------------------------------------------
;; http://www.billthelizard.com/2009/12/sicp-exercise-113-fibonacci-and-golden.html
;; ====================================================================
;; 1.2.3: Orders of Growth
;; ====================================================================
;; *Exercise 1.14:* Draw the tree illustrating the process generated
;; by the `count-change' procedure of section *Note 1-2-2:: in making
;; change for 11 cents. What are the orders of growth of the space
;; and number of steps used by this process as the amount to be
;; changed increases?
;;
;; *Exercise 1.15:* The sine of an angle (specified in radians) can
;; be computed by making use of the approximation `sin' xapprox x if
;; x is sufficiently small, and the trigonometric identity
;;
;; x x
;; sin x = 3 sin --- - 4 sin^3 ---
;; 3 3
;;
;; to reduce the size of the argument of `sin'. (For purposes of this
;; exercise an angle is considered "sufficiently small" if its
;; magnitude is not greater than 0.1 radians.) These ideas are
;; incorporated in the following procedures:
;;
;; (define (cube x) (* x x x))
;;
;; (define (p x) (- (* 3 x) (* 4 (cube x))))
;;
;; (define (sine angle)
;; (if (not (> (abs angle) 0.1))
;; angle
;; (p (sine (/ angle 3.0)))))
;;
;; a. How many times is the procedure `p' applied when `(sine
;; 12.15)' is evaluated?
;;
;; b. What is the order of growth in space and number of steps (as
;; a function of a) used by the process generated by the `sine'
;; procedure when `(sine a)' is evaluated?
;; ====================================================================
;; 1.2.4: Exponentiation
;; --------------------------------------------------------------------
;; *Exercise 1.16:* Design a procedure that evolves an iterative
;; exponentiation process that uses successive squaring and uses a
;; logarithmic number of steps, as does `fast-expt'. (Hint: Using the
;; observation that (b^(n/2))^2 = (b^2)^(n/2), keep, along with the
;; exponent n and the base b, an additional state variable a, and
;; define the state transformation in such a way that the product a
;; b^n is unchanged from state to state. At the beginning of the
;; process a is taken to be 1, and the answer is given by the value
;; of a at the end of the process. In general, the technique of
;; defining an "invariant quantity" that remains unchanged from state
;; to state is a powerful way to think about the design of iterative
;; algorithms.)
;;
;; *Exercise 1.17:* The exponentiation algorithms in this section are
;; based on performing exponentiation by means of repeated
;; multiplication. In a similar way, one can perform integer
;; multiplication by means of repeated addition. The following
;; multiplication procedure (in which it is assumed that our language
;; can only add, not multiply) is analogous to the `expt' procedure:
;;
;; (define (* a b)
;; (if (= b 0)
;; 0
;; (+ a (* a (- b 1)))))
;;
;; This algorithm takes a number of steps that is linear in `b'. Now
;; suppose we include, together with addition, operations `double',
;; which doubles an integer, and `halve', which divides an (even)
;; integer by 2. Using these, design a multiplication procedure
;; analogous to `fast-expt' that uses a logarithmic number of steps.
;;
;; *Exercise 1.18:* Using the results of *Note Exercise 1-16:: and
;; *Note Exercise 1-17::, devise a procedure that generates an
;; iterative process for multiplying two integers in terms of adding,
;; doubling, and halving and uses a logarithmic number of steps.(4)
;;
;; *Exercise 1.19:* There is a clever algorithm for computing the
;; Fibonacci numbers in a logarithmic number of steps. Recall the
;; transformation of the state variables a and b in the `fib-iter'
;; process of section *Note 1-2-2::: a <- a + b and b <- a. Call
;; this transformation T, and observe that applying T over and over
;; again n times, starting with 1 and 0, produces the pair _Fib_(n +
;; 1) and _Fib_(n). In other words, the Fibonacci numbers are
;; produced by applying T^n, the nth power of the transformation T,
;; starting with the pair (1,0). Now consider T to be the special
;; case of p = 0 and q = 1 in a family of transformations T_(pq),
;; where T_(pq) transforms the pair (a,b) according to a <- bq + aq +
;; ap and b <- bp + aq. Show that if we apply such a transformation
;; T_(pq) twice, the effect is the same as using a single
;; transformation T_(p'q') of the same form, and compute p' and q' in
;; terms of p and q. This gives us an explicit way to square these
;; transformations, and thus we can compute T^n using successive
;; squaring, as in the `fast-expt' procedure. Put this all together
;; to complete the following procedure, which runs in a logarithmic
;; number of steps:(5)
;;
;; (define (fib n)
;; (fib-iter 1 0 0 1 n))
;;
;; (define (fib-iter a b p q count)
;; (cond ((= count 0) b)
;; ((even? count)
;; (fib-iter a
;; b
;; <??> ; compute p'
;; <??> ; compute q'
;; (/ count 2)))
;; (else (fib-iter (+ (* b q) (* a q) (* a p))
;; (+ (* b p) (* a q))
;; p
;; q
;; (- count 1)))))
;; ====================================================================
;; 1.2.5: Greatest Common Divisors
;; ====================================================================
;; *Exercise 1.20:* The process that a procedure generates is of
;; course dependent on the rules used by the interpreter. As an
;; example, consider the iterative `gcd' procedure given above.
;; Suppose we were to interpret this procedure using normal-order
;; evaluation, as discussed in section *Note 1-1-5::. (The
;; normal-order-evaluation rule for `if' is described in *Note
;; Exercise 1-5::.) Using the substitution method (for normal
;; order), illustrate the process generated in evaluating `(gcd 206
;; 40)' and indicate the `remainder' operations that are actually
;; performed. How many `remainder' operations are actually performed
;; in the normal-order evaluation of `(gcd 206 40)'? In the
;; applicative-order evaluation?
;; ====================================================================
;; 1.2.6: Example: Testing for Primality
;; ====================================================================
;; *Exercise 1.21:* Use the `smallest-divisor' procedure to find the
;; smallest divisor of each of the following numbers: 199, 1999,
;; 19999.
;;
;; *Exercise 1.22:* Most Lisp implementations include a primitive
;; called `runtime' that returns an integer that specifies the amount
;; of time the system has been running (measured, for example, in
;; microseconds). The following `timed-prime-test' procedure, when
;; called with an integer n, prints n and checks to see if n is
;; prime. If n is prime, the procedure prints three asterisks
;; followed by the amount of time used in performing the test.
;;
;; (define (timed-prime-test n)
;; (newline)
;; (display n)
;; (start-prime-test n (runtime)))
;;
;; (define (start-prime-test n start-time)
;; (if (prime? n)
;; (report-prime (- (runtime) start-time))))
;;
;; (define (report-prime elapsed-time)
;; (display " *** ")
;; (display elapsed-time))
;;
;; Using this procedure, write a procedure `search-for-primes' that
;; checks the primality of consecutive odd integers in a specified
;; range. Use your procedure to find the three smallest primes
;; larger than 1000; larger than 10,000; larger than 100,000; larger
;; than 1,000,000. Note the time needed to test each prime. Since
;; the testing algorithm has order of growth of [theta](_[sqrt]_(n)),
;; you should expect that testing for primes around 10,000 should
;; take about _[sqrt]_(10) times as long as testing for primes around
;; 1000. Do your timing data bear this out? How well do the data
;; for 100,000 and 1,000,000 support the _[sqrt]_(n) prediction? Is
;; your result compatible with the notion that programs on your
;; machine run in time proportional to the number of steps required
;; for the computation?
;;
;; *Exercise 1.23:* The `smallest-divisor' procedure shown at the
;; start of this section does lots of needless testing: After it
;; checks to see if the number is divisible by 2 there is no point in
;; checking to see if it is divisible by any larger even numbers.
;; This suggests that the values used for `test-divisor' should not
;; be 2, 3, 4, 5, 6, ..., but rather 2, 3, 5, 7, 9, .... To
;; implement this change, define a procedure `next' that returns 3 if
;; its input is equal to 2 and otherwise returns its input plus 2.
;; Modify the `smallest-divisor' procedure to use `(next
;; test-divisor)' instead of `(+ test-divisor 1)'. With
;; `timed-prime-test' incorporating this modified version of
;; `smallest-divisor', run the test for each of the 12 primes found in
;; *Note Exercise 1-22::. Since this modification halves the number
;; of test steps, you should expect it to run about twice as fast.
;; Is this expectation confirmed? If not, what is the observed ratio
;; of the speeds of the two algorithms, and how do you explain the
;; fact that it is different from 2?
;;
;; *Exercise 1.24:* Modify the `timed-prime-test' procedure of *Note
;; Exercise 1-22:: to use `fast-prime?' (the Fermat method), and test
;; each of the 12 primes you found in that exercise. Since the
;; Fermat test has [theta](`log' n) growth, how would you expect the
;; time to test primes near 1,000,000 to compare with the time needed
;; to test primes near 1000? Do your data bear this out? Can you
;; explain any discrepancy you find?
;;
;; *Exercise 1.25:* Alyssa P. Hacker complains that we went to a lot
;; of extra work in writing `expmod'. After all, she says, since we
;; already know how to compute exponentials, we could have simply
;; written
;;
;; (define (expmod base exp m)
;; (remainder (fast-expt base exp) m))
;;
;; Is she correct? Would this procedure serve as well for our fast
;; prime tester? Explain.
;;
;; *Exercise 1.26:* Louis Reasoner is having great difficulty doing
;; *Note Exercise 1-24::. His `fast-prime?' test seems to run more
;; slowly than his `prime?' test. Louis calls his friend Eva Lu Ator
;; over to help. When they examine Louis's code, they find that he
;; has rewritten the `expmod' procedure to use an explicit
;; multiplication, rather than calling `square':
;;
;; (define (expmod base exp m)
;; (cond ((= exp 0) 1)
;; ((even? exp)
;; (remainder (* (expmod base (/ exp 2) m)
;; (expmod base (/ exp 2) m))
;; m))
;; (else
;; (remainder (* base (expmod base (- exp 1) m))
;; m))))
;;
;; "I don't see what difference that could make," says Louis. "I
;; do." says Eva. "By writing the procedure like that, you have
;; transformed the [theta](`log' n) process into a [theta](n)
;; process." Explain.
;;
;; *Exercise 1.27:* Demonstrate that the Carmichael numbers listed in
;; *Note Footnote 1-47:: really do fool the Fermat test. That is,
;; write a procedure that takes an integer n and tests whether a^n is
;; congruent to a modulo n for every a<n, and try your procedure on
;; the given Carmichael numbers.
;;
;; *Exercise 1.28:* One variant of the Fermat test that cannot be
;; fooled is called the "Miller-Rabin test" (Miller 1976; Rabin
;; 1980). This starts from an alternate form of Fermat's Little
;; Theorem, which states that if n is a prime number and a is any
;; positive integer less than n, then a raised to the (n - 1)st power
;; is congruent to 1 modulo n. To test the primality of a number n
;; by the Miller-Rabin test, we pick a random number a<n and raise a
;; to the (n - 1)st power modulo n using the `expmod' procedure.
;; However, whenever we perform the squaring step in `expmod', we
;; check to see if we have discovered a "nontrivial square root of 1
;; modulo n," that is, a number not equal to 1 or n - 1 whose square
;; is equal to 1 modulo n. It is possible to prove that if such a
;; nontrivial square root of 1 exists, then n is not prime. It is
;; also possible to prove that if n is an odd number that is not
;; prime, then, for at least half the numbers a<n, computing a^(n-1)
;; in this way will reveal a nontrivial square root of 1 modulo n.
;; (This is why the Miller-Rabin test cannot be fooled.) Modify the
;; `expmod' procedure to signal if it discovers a nontrivial square
;; root of 1, and use this to implement the Miller-Rabin test with a
;; procedure analogous to `fermat-test'. Check your procedure by
;; testing various known primes and non-primes. Hint: One convenient
;; way to make `expmod' signal is to have it return 0.