SICP Chapter 1 Exercises & Solutions

Below is a sequence of expressions. What is the result printed by the interpreter in response to each expression? Assume that the sequence is to be evaluated in the order in which it is presented.

(Expressions Contained in Answer Section)

Translate the following expression into prefix form.

5 + 4 + (2 - (3 - (6 + 4/5)))

3(6 - 2)(2 - 7)

Define a procedure that takes three numbers as arguments and returns the sum of the squares of the two larger numbers.

Observe that our model of evaluation allows for combinations whose operators are compound expressions. Use this observation to describe the behavior of the following procedure:

(define (a-plus-abs-b a b) ((if (> b 0) + -) a b))

Ben Bitdiddle has invented a test to determine whether the interpreter he is faced with is using applicative-order evaluation or normal-order evaluation. He defines the following two procedures:

(define (p) (p))

(define (test x y) (if (= x 0) 0 y))

Then he evaluates the expression

(test 0 (p))

What behavior will Ben observe with an interpreter that uses applicative-order evaluation? What behavior will he observe with an interpreter that uses normal-order evaluation? Explain your answer. (Assume that the evaluation rule for the special form `if’ is the same whether the interpreter is using normal or applicative order: The predicate expression is evaluated first, and the result determines whether to evaluate the consequent or the alternative expression.)

Alyssa P. Hacker doesn’t see why `if’ needs to be provided as a special form. “Why can’t I just define it as an ordinary procedure in terms of `cond’?” she asks. Alyssa’s friend Eva Lu Ator claims this can indeed be done, and she defines a new version of `if’:

(define (new-if predicate then-clause else-clause) (cond (predicate then-clause) (else else-clause)))

Eva demonstrates the program for Alyssa:

(new-if (= 2 3) 0 5) 5

(new-if (= 1 1) 0 5) 0

Delighted, Alyssa uses `new-if’ to rewrite the square-root program:

(define (sqrt-iter guess x) (new-if (good-enough? guess x) guess (sqrt-iter (improve guess x) x)))

What happens when Alyssa attempts to use this to compute square roots? Explain.

The `good-enough?’ test used in computing square roots will not be very effective for finding the square roots of very small numbers. Also, in real computers, arithmetic operations are almost always performed with limited precision. This makes our test inadequate for very large numbers. Explain these statements, with quotes showing how the test fails for small and large numbers.

An alternative strategy for implementing `good-enough?’ is to watch how `guess’ changes from one iteration to the next and to stop when the change is a very small fraction of the guess. Design a square-root procedure that uses this kind of end test. Does this work better for small and large numbers?

Newton’s method for cube roots is based on the fact that if y is an approximation to the cube root of x, then a better approximation is given by the value

x/y² + 2y

3

Use this formula to implement a cube-root procedure analogous to the square-root procedure. (In section 1.3.4 we will see how to implement Newton’s method in general as an abstraction of these square-root and cube-root procedures.)

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?

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) (A 2 4) (A 3 3)

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 quote, (k n) computes 5n².

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.

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. Write a procedure that computes elements of Pascal’s triangle by means of a recursive process.

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?

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 sin³ — 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?

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 (bⁿ/²)²= (b²)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 abⁿ 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.)

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.

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 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 n-th 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:

The process that a procedure generates is of course dependent on the rules used by the interpreter. As an quote, 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?

Use the smallest-divisor procedure to find the smallest divisor of each of the following numbers: 199, 1999, 19999.

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 quote, 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 (current-time)))

(define (start-prime-test n start-time)
  (if (prime? n)
      (report-prime (- (current-time) start-time))))

(define (report-prime elapsed-time)
  (display " *** ")
  (display elapsed-time)
  #t)

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?

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 Note in 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?

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.

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.

The sum procedure above generates a linear recursion. The procedure can be rewritten so that the sum is performed iteratively.

1. The `sum’ procedure is only the simplest of a vast number of similar

abstractions that can be captured as higher-order procedures. Write an analogous procedure called product that returns the product of the values of a function at points over a given range. Show how to define factorial in terms of product. Also use product to compute approximations to π using the formula:

π/4 = 2/3 ⋅ 4/3 ⋅ 4/5 ⋅ 6/5 ⋅ 6/7 ⋅ 8/7

1. If your product procedure generates a recursive process, write one that

generates an iterative process. If it generates an iterative process, write one that generates a recursive process.

1. Show that sum and product (Exercise 1.31) are both special cases of a still

more general notion called accumulate that combines a collection of terms, using some general accumulation function:

(accumulate combiner null-value term a next b)

Accumulate takes as arguments the same term and range specifications as sum and product, together with a combiner procedure (of two arguments) that specifies how the current term is to be combined with the accumulation of the preceding terms and a null-value that specifies what base value to use when the terms run out. Write accumulate and show how sum and product can both be defined as simple calls to accumulate.

1. If your accumulate procedure generates a recursive process, write one

that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.

You can obtain an even more general version of accumulate (Exercise 1.32) by introducing the notion of a filter on the terms to be combined. That is, combine only those terms derived from values in the range that satisfy a specified condition. The resulting `filtered-accumulate’ abstraction takes the same arguments as `accumulate’, together with an additional predicate of one argument that specifies the filter. Write `filtered-accumulate’ as a procedure. Show how to express the following using `filtered-accumulate’:

1. the sum of the squares of the prime numbers in the interval a to b

(assuming that you have a prime? predicate already written)

1. the product of all the positive integers less than n that are relatively

prime to n (i.e., all positive integers i < n such that GCD (i, n) = 1).

Suppose we define the procedure

(define (f g) (g 2))

Then we have

(f square)
4

(f (lambda (z) (* z (+ z 1))))
6

What happens if we (perversely) ask the interpreter to evaluate the combination (f f)?

Explain.

Show that the golden ratio φ (1.2.2) is a fixed point of the transformation x↦1+1/x, and use this fact to compute φ by means of the fixed-point procedure.

a. An infinite “continued fraction” is an expression of the form

N₁ f = ----------------–— N₂ D₁ + ----------–— N₃ D₂ + ----–— D₃ + …

As an quote, one can show that the infinite continued fraction expansion with the Nᵢ and the Dᵢ all equal to 1 produces 1/φ, where φ is the golden ratio (described in section *Note 1.2.2). One way to approximate an infinite continued fraction is to truncate the expansion after a given number of terms. Such a truncation–a so-called finite continued fraction “k-term finite continued fraction”–has the form

N₁

N₂ D₁ + ------–— … N_K

• –— D_K

Suppose that `n’ and `d’ are procedures of one argument (the term index i) that return the Nᵢ and Dᵢ of the terms of the continued fraction. Define a procedure `cont-frac’ such that evaluating `(cont-frac n d k)’ computes the value of the k-term finite continued fraction. Check your procedure by approximating 1/φ using

(cont-frac (lambda (i) 1.0) (lambda (i) 1.0) k)

for successive values of `k’. How large must you make `k’ in order to get an approximation that is accurate to 4 decimal places?

b. If your `cont-frac’ procedure generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.

In 1737, the Swiss mathematician Leonhard Euler published a memoir `De Fractionibus Continuis’, which included a continued fraction expansion for e - 2, where e is the base of the natural logarithms. In this fraction, the nᵢ are all 1, and the Dᵢ are successively 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, 8, …. Write a program that uses your `cont-frac’ procedure from Exercise 1.37 to approximate e, based on Euler’s expansion.

A continued fraction representation of the tangent function was published in 1770 by the German mathematician J.H. Lambert:

x tan x = ----------–— x² 1 - ------–— x² 3 - --–— 5 - …

where x is in radians. Define a procedure `(tan-cf x k)’ that computes an approximation to the tangent function based on Lambert’s formula. `K’ specifies the number of terms to compute, as in *Note Exercise 1.37

Define a procedure cubic that can be used together with the newtons-method procedure in expressions of the form

(newtons-method (cubic a b c) 1)

to approximate zeros of the cubic x³+ax²+bx+c.

Define a procedure double that takes a procedure of one argument as argument and returns a procedure that applies the original procedure twice. For quote, if inc is a procedure that adds 1 to its argument, then (double inc) should be a procedure that adds 2. What value is returned by

(((double (double double)) inc) 5)

?

Let f and g be two one-argument functions. The composition f after g is defined to be the function x↦f(g(x)). Define a procedure compose that implements composition. For quote, if inc is a procedure that adds 1 to its argument,

((compose square inc) 6) 49

If f is a numerical function and n is a positive integer, then we can form the nth repeated application of f, which is defined to be the function whose value at x is f(f(…(f(x))…)). For quote, if f is the function x↦x+1, then the nth repeated application of f is the function x↦x+n. If f is the operation of squaring a number, then the nth repeated application of f is the function that raises its argument to the 2n-th power. Write a procedure that takes as inputs a procedure that computes f and a positive integer n and returns the procedure that computes the nth repeated application of f. Your procedure should be able to be used as follows:

((repeated square 2) 5) 625

Hint: You may find it convenient to use compose from Exercise 1.42.

The idea of smoothing a function is an important concept in signal processing. If f is a function and dx is some small number, then the smoothed version of f is the function whose value at a point x is the average of f(x−dx), f(x), and f(x+dx). Write a procedure smooth that takes as input a procedure that computes f and returns a procedure that computes the smoothed f. It is sometimes valuable to repeatedly smooth a function (that is, smooth the smoothed function, and so on) to obtain the n-fold smoothed function. Show how to generate the n-fold smoothed function of any given function using smooth and repeated from Exercise 1.43.

Several of the numerical methods described in this chapter are instances of an extremely general computational strategy known as iterative improvement. Iterative improvement says that, to compute something, we start with an initial guess for the answer, test if the guess is good enough, and otherwise improve the guess and continue the process using the improved guess as the new guess. Write a procedure iterative-improve that takes two procedures as arguments: a method for telling whether a guess is good enough and a method for improving a guess. Iterative-improve should return as its value a procedure that takes a guess as argument and keeps improving the guess until it is good enough. Rewrite the sqrt procedure of 1.1.7 and the fixed-point procedure of 1.3.3 in terms of iterative-improve.