Reading: Randomness and simulation

Read By
Wednesday, Oct 31, 2018
Summary
Up to this point, all of the programs we have written are, in some sense, predictable. That is, the output depends only on the input. However, we want some programs, such as games and simulations, to be less predictable. We may also want to have make drawings using unpredictable procedures, so that we can derive inspiration from unexpected results. In this reading, we consider Scheme’s tools for supporting such unpredictability, particularly the random procedure.

Introduction: Simulation

Many computing applications involve the simulation of games or events, with the hope of gaining insights and identifying underlying principles. In some cases, simulations can apply definite, well-known formulae. For example, in studying the effect of a pollution source in a lake or stream, one can keep track of pollutant concentrations in various places. Then, since the flow of water and the interactions of pollutants is reasonably well understood, one can follow the flow of the pollutants over a period of time, according to known equations.

In other cases, specific outcomes involve some chance. For example, when an automobile begins a trip and encounters a traffic light, it may be a matter of chance whether the light is green, yellow, or red. Similar uncertainties arise when considering genetic mutations or when tabulating outcomes involving flipping a coin, tossing a die, or dealing cards. In these cases, one may know about the probability of an event occurring (a head occurs about half the time), but the outcome of any one event depends on chance.

In studying events that involve some chance, one approach is to model the event or game, using a random-number generator as the basis for decisions. If such a model is simulated many times on a computer, the results may give some statistical information about what outcomes are likely and how often each type of outcome might be expected to occur. This approach to problem solving is called the Monte Carlo Method.

The random procedure

A random number generator for a typical computer language is a procedure that produces an unpredictable value each time it is called. Such procedures simulate a random selection process. Scheme provides the procedure random for this purpose. This procedure returns integer values that depend on its parameter. In particular, random returns an unpredictable integer value between 0 and one less than its parameter, inclusive. By “unpredictable” we mean that we are unlikely to be able to predict the number that random will return.

> (random 10)
1
> (random 10)
9
> (random 10)
7
> (random 10)
0
> (random 10)
5
> (random 10)
1
> (random 10)
0

Simulating a die

We can use random to write a program to simulate the rolling of a die. The simulation generates integers from 1 to 6, to correspond to the faces on the die cube. The details of this simulation are shown in the following procedure:

;;; Procedure:
;;;   roll-a-die
;;; Parameters:
;;;   None
;;; Purpose:
;;;   To simulate the rolling of one six-sided die.
;;; Produces:
;;;   An integer between 1 and 6, inclusive.
;;; Preconditions:
;;;   [None]
;;; Postconditions:
;;;   Returns an integer between 1 and 6, inclusive.
;;;   It should be difficult (or impossible) to predict which
;;;     number is produced.
(define roll-a-die
  (lambda ()
    (+
      (random 6)    ; a value in the range [0 .. 5]
      1)))          ; now in the range [1 .. 6]

We can use that procedure to simulate the roll of multiple dice.

;;; Procedure:
;;;   roll-dice
;;; Parameters:
;;;   n, an integer
;;; Purpose:
;;;   Roll n six-sided dice and make a list of their values.
;;; Produces:
;;;   rolls, a list of integers
;;; Preconditions:
;;;   [No additional]
;;; Postconditions:
;;;   (length rolls) = n
;;;   Each element of rolls is a value between 1 and 6.
;;;   The values in rolls are reasonably evenly distributed.
;;;   The values in rolls are difficult to predict.
(define roll-dice
  (lambda (n)
    (if (zero? n)
        null
        (cons (roll-a-die) (roll-dice (- n 1))))))

Generating text

We can also use random to select “unpredictable” elements of a list. Let’s start with a simple procedure.

;;; Procedure:
;;;   random-elt
;;; Parameters:
;;;   lst, a non-empty list 
;;; Purpose:
;;;   Unpredictably pick an element of lst.
;;; Produces:
;;;   val, a value
;;; Preconditions:
;;;   [No additional]
;;; Postconditions:
;;;   * val is an element of lst.
;;;   * If lst contains more than one element, it is difficult to predict 
;;;     which element val is.
(define random-elt
  (lambda (lst)
    (list-ref lst (random (length lst)))))

There are many ways to apply random-elt. For example, here’s a collection of procedures that make an unpredictable sentence.

(define sentence
  (lambda ()
     (string-append
       (random-person) " "
       (random-transitive-verb) " "
       (random-object) ".")))

(define random-person
  (let ([people (list "Anh Thu" "Charlie" "Fahmida" "Halle"
                      "John" "Maisy" "Yesheng" "Zoe")])
    (lambda ()
      (random-elt people))))

(define random-transitive-verb
  (let ([verbs (list "saw" "watched" "threw" "ate" "borrowed")])
    (lambda ()
      (random-elt verbs))))

(define random-object
  (let ([articles (list "the" "a")]
        [adjectives (list "heavy" "blue" "green" "hot" "cold" "disgusting")]
        [nouns (list "cup of coffee" "computer" "classroom"
                     "PBJ algorithm" "homework assignment")])
    (lambda ()
      (string-append
        (random-elt articles) " "
        (random-elt adjectives) " "
        (random-elt nouns)))))

Here’s one example of the primary procedure in action.

> (sentence)
"Anh Thu borrowed a cold cup of coffee."

Self checks

Check 1: Testing random

a. When you give the procedure random the parameter n, it will produce one of how many unique values? What is the smallest value? What is the largest?

b. Evaluate the expression (random 10) several times. What values do you get?

c. What values do you expect to get if you call random with 1 as a parameter?

d. Check your hypothesis experimentally.

e. What do you expect to happen if you call random with 0 or -1 as a parameter?

f. Check your hypothesis experimentally.

g. What do you expect to happen if you call random with non-integer parameters.

h. Check your hypothesis experimentally.

i. Try calling random with no parameters. What happens?

Check 2: Rolling dice

a. Add the following two procedures to the definitions pane.

;;; Procedure:
;;;   roll-a-die
;;; Parameters:
;;;   None
;;; Purpose:
;;;   To simulate the rolling of one six-sided die.
;;; Produces:
;;;   An integer between 1 and 6, inclusive.
;;; Preconditions:
;;;   [None]
;;; Postconditions:
;;;   Returns an integer between 1 and 6, inclusive.
;;;   It should be difficult (or impossible) to predict which
;;;     number is produced.
(define roll-a-die
  (lambda ()
    (+
      (random 6)    ; a value in the range [0 .. 5]
      1)))          ; now in the range [1 .. 6]

;;; Procedure:
;;;   roll-dice
;;; Parameters:
;;;   n, an integer
;;; Purpose:
;;;   Roll n six-sided dice and make a list of their values.
;;; Produces:
;;;   rolls, a list of integers
;;; Preconditions:
;;;   [No additional]
;;; Postconditions:
;;;   (length rolls) = n
;;;   Each element of rolls is a value between 1 and 6.
;;;   The values in rolls are reasonably evenly distributed.
;;;   The values in rolls are difficult to predict.
(define roll-dice
  (lambda (n)
    (if (zero? n)
        null
        (cons (roll-a-die) (roll-dice (- n 1))))))

b. Using roll-dice, roll ten dice.

c. Using roll-dice, roll ten dice. (Yes, this instruction is the same as the previous instruction. You should do it twice.)

d. Did you get the same list of values each time? Why or why not?

e. What other procedures have you encountered that may return different values each time you call them with the same parameters?