Random Bricks

Level 3 - Reading https://bricklayerdotorg.wordpress.com/generating-random-bricks/ Random Bricks:

Random Numbers

Random Numbers In this discussion we will ignore the idiosyncrasies surrounding how computers represent real numbers and issues surrounding equality comparisons. Virtually all programming languages provide a variety of ways to generate a random (real) number that, mathematically speaking, is in the range 0.0 ≤ n < 1.0 Note that, in this range, n can never equal 1.0, but can equal 0.0.

A Random Number Function To get a random number, one typically calls a special function. The basic random number function is a nullary function. Its name can vary from language to language, but the function is typically called random or rand. Let random denote a function that when called generates a random number in the range 0.0 ≤ n < 1.0 Let us assume that random is a nullary function which can therefore be called as follows: random ()

Simulating a Coin Flip

An Algorithm The following algorithm shows how the random function can be used to simulate a coin flip. In the algorithm below, H stands for “heads” and T stands for “tails”. Let n = random () if n < 0.5 then print (“H”) else print (“T”) This algorithm can be easily implemented in SML. An important question is: What kind of output could one expect from a program that flipped a coin, say, 10 times? Would you expect 5 heads and 5 tails? Would you expect 10 heads and 0 tails?

An Experiment I conducted an experiment where I flipped an actual coin 10 times and got the following sequence of heads (H) and tails (T). H T T T T T H H H H One important thing to note about the sequence is that random does not mean “evenly distributed”. For example, after seeing a sequence of 3 tails, it is incorrect to assume that the next coin flip must yield heads. What would happen if I ran the experiment again? What would be the chances (i.e., the odds) that same sequence of heads and tails would occur?

Properties of a Random Function Let us consider writing a program that conducts the coin- flipping experiment on our behalf. What would you want to have happen if you ran such a program twice? Would you want the program to produce the same output every time it is executed? If not, how might this impact your ability to test the program? Would you want the program to produce the same output if it is executed on a different computer?

Simulating the Roll of a Die

An Algorithm The following algorithm shows how the random function can be used to simulate the roll of a six- sided die. The basic idea here is to partition the  range 0.0 ≤ n < 1.0 into six equal sections and associate each section with (aka, map each section to) a side of the die. Let n = random ()        if 0.0/6.0 ≤ n < 1.0/6.0 then print (“1”) else if 1.0/6.0 ≤ n < 2.0/6.0 then print (“2”) else if 2.0/6.0 ≤ n < 3.0/6.0 then print (“3”) else if 3.0/6.0 ≤ n < 4.0/6.0 then print (“4”) else if 4.0/6.0 ≤ n < 5.0/6.0 then print (“5”) else if 5.0/6.0 ≤ n < 6.0/6.0 then print (“6”) This algorithm can also be easily implemented in SML.

An Experiment I conducted an experiment where I rolled an actual die 10 times and got the following sequence of numbers. 3 5 1 2 1 5 2 2 5 2 Note that the numbers 4 and 6 never appeared. Does this imply that the die is “loaded”? No. If I roll the die five more times, will I be guaranteed that 4 and/or 6 will appear? No. It is important to appreciate that random number sequences have such properties.

Bricklayer A Random Brick Function

The generateRandomBrickFn function Bricklayer provides a function, called generateRandomBrickFn, that can be used to generate random bricks. This function implements an algorithm similar to the coin-flipping and die-rolling algorithms described previously. Let brickList denote a list of bricks. The evaluation of the expression generateRandomBrickFn brickList will produce a nullary function value as its result. A val-declaration can be used to bind a variable (i.e., our desired function name) to this nullary function value as follows. val myName = generateRandomBrickFn brickList; The evaluation of the function call “myName ()” will return a brick, randomly selected from brickList.

Predefined Brick Lists Bricklayer provides a set of predefined brick lists that can be used to generate random brick functions. This set includes the following brick lists. grayscale greenScale blueScale purpleScale redScale warmScale brownScale clearScale allOneBitBricks

Example 1a A one dimensional sequence of RED and BLACK bricks.

Question: In what order are these bricks “put”? open Level_3; val brickList = [BLACK,RED]; val randomBrick = generateRandomBrickFn brickList; fun sequence0 (x,z) = let val delta = 1; val brick1 = randomBrick (); val brick2 = randomBrick (); in put2D (1,1) brick1 (x + 0 * delta, z); put2D (1,1) brick2 (x + 1 * delta, z) end; fun sequence1 (x,z) = val delta = 2*1; sequence0 (x + 0 * delta, z ); sequence0 (x + 1 * delta, z ) fun sequence2 (x,z) = val delta = 2*2*1; sequence1 (x + 0 * delta, z ); sequence1 (x + 1 * delta, z ) fun sequence3 (x,z) = let val delta = 2*2*2*1; in sequence2 (x + 0 * delta, z ); sequence2 (x + 1 * delta, z ) end; fun sequence4 (x,z) = val delta = 2*2*2*2*1; sequence3 (x + 0 * delta, z ); sequence3 (x + 1 * delta, z ) build2D (32,32); sequence4 (0,0); show2D "random in 1D"; Question: In what order are these bricks “put”? Question: How can we confirm this?

Example 1b Tracing a one dimensional sequence of RED and BLACK bricks.

fun showPoint (x,z) = let val pointStr = "(" ^ Int.toString x ^ "," ^ Int.toString z ^ ")"; in print("\n(x,z) = " ^ pointStr) end; fun myPut dim brick p = ( showPoint p; put2D dim brick p ); fun sequence0 (x,z) = val delta = 1; val brick1 = randomBrick (); val brick2 = randomBrick (); myPut (1,1) brick1 (x + 0 * delta, z); myPut (1,1) brick2 (x + 1 * delta, z) fun random1D p = ( print "\n\n"; sequence4 (0,0); print "\n\n" ); build2D (32,32); random1D (0,0); show2D "random in 1D";

Example 2 A four-colored two dimensional random brick sequence.

open Level_3; val brickList = [RED, GREEN, YELLOW, BLUE]; val randomBrick = generateRandomBrickFn brickList; fun board0 (x,z) = let val delta = 1; val brick1 = randomBrick (); val brick2 = randomBrick (); val brick3 = randomBrick (); val brick4 = randomBrick (); in put2D (1,1) brick1 (x , z ); put2D (1,1) brick2 (x + delta, z ); put2D (1,1) brick3 (x + delta, z + delta); put2D (1,1) brick4 (x , z + delta) end; fun board1 (x,z) = val delta = 2*1; board0 (x , z ); board0 (x + delta, z ); board0 (x + delta, z + delta); board0 (x , z + delta) fun board2 (x,z) = let val delta = 2*2*1; in board1 (x , z ); board1 (x + delta, z ); board1 (x + delta, z + delta); board1 (x , z + delta) end; fun board3 (x,z) = val delta = 2*2*2*1; board2 (x , z ); board2 (x + delta, z ); board2 (x + delta, z + delta); board2 (x , z + delta) build2D (32,32); board3 (0,0); show2D "random 2D";

Question: In what order are these bricks “put”? Question: How can we confirm this?

The End