EXPLORATIONMATHEMATICS HIGHER LEVEL

Title: Exploring the usage of Boolean algebra in Minecraft by creation of a ripple carry adder.

Exam Session: May 2015School Name: UWC Mahindra College of IndiaName: Aditya KaranCandidate Number: 000969 0039

IntroductionVideo games have been an interest of mine since I was a very young child. I have grown up playing video games, and have throughout my childhood played a large number of games across numerous genres which were intrinsic to my development as an individual. I noticed that over the years, the assimilation of information and experiences that I later associated with video games were primarily subtle and hidden. This changed entirely when I was first introduced Minecraft in 2014. Minecraft is a sandbox game that hands the power to create to the player. By manipulating the placement of blocks with various properties in a three dimensional grid, the player is capable of creating a seemingly infinite variety of constructions. The game is based around survival in the wilderness and the player can obtain resources by interfacing with the environment using certain tools, and manipulating these resources to create relatively complex structures. The most interesting notion in Minecraft is the idea that only basic parameters such as the properties of blocks are defined by the game while the possibility for creation is only otherwise limited by the creativity of the player and processing power of the system running the game. The player-base constantly pushes the limits of the game and promotes innovation while pushing constructions into the game that beyond their ability to innovate, also impart conceptual understanding through hands on, practical means.My interest in computing grew from my attachment to video games but I had, for the most part, no access to any materials through which I may have created circuits to understand the basics of computing. My introduction to Minecraft changed this, as with a little bit of innovation, the game allowed me to explore circuitry for no additional cost, using only the game itself.Boolean algebra is the tool I will be using for this exploration. Boolean algebra is defined as the studyof mathematicaloperationsperformed on certainvariables(calledbinary variables) that can have only twovalues: true (represented by 1) orfalse(represented by 0). Itprovidesa set ofrules(calledBoolean logic) that are indispensable indigitalcomputer-circuit and switching-circuitdesign. Booleanoperationare carried out with algebraicoperators(calledBoolean operators), the most basic of which are NOT, AND, and OR. Named after itsinventor, the UK mathematician andcomputerpioneer George Boole (1815-65).1

This exploration is an attempt to utilize concepts from Boolean algebra within the scope of Minecraft to create a device that is capable of carrying out the addition of two numbers entered as binary inputs. This will be done by exploring some fundamental aspects of Boolean algebra and using these concepts to create a ripple carry adder, an instrument capable of adding two 4-bit binary numbers.

Minecraft:Minecraft is at its core a game about breaking, crafting and placing blocks with various properties in a three dimensional grid. Each block, for the most part, may only occupy one spot in the grid, which includes objects in the game that are not cubes, but are still defined as blocks within the game such as torches, levers etc. Minecraft has over the years garnered a community of players that push the limits of what may be possible using objects and mechanics within the game. Minecraft contains a certain class of objects within the game, referred to Redstone and Redstone devices that may be utilized to simulate the flow of current and create circuits that are analogous to electrical devices that we may encounter in our everyday life. The following are Redstone related blocks essential to the construction of the circuits described further in the exploration:Redstone: Redstone is an ore that can be mined within the game. It is analogous to wires in real life, and may be used to carry a current in the game world by transmitting their powered state to adjacent blocks in the direction of flow of the current.. Redstone blocks are closely similar to current conducting wires, except for the fact that they do not need to be part of a closed and complete circuit to carry current. They can be in two states, a powered, or an unpowered state, which is dependent on the blocks that they are adjacent to, and whether they are capable of supplying power to the wire.Redstone and Redstone wire in schematics:

Redstone wireRedstone

Lever: A lever is a Redstone device that can be used to power adjacent blocks, and therefore induce a current in adjacent Redstone wire. A lever simply has an on and off state. In a circuit, an input lever in the on state passes an input of (1) and a lever in the off state passes an input (0). Lever in schematics:

OFF positionON positionTorch: A Redstone torch is a device that may be used to power a circuit. It also has the property of being capable of inverting any input when used in conjunction with a regular powerable block.

Powered Redstone torch placed adjacent to a generic block.Powered and unpowered Redstone torches placed on a generic block.

Blocks: There are a large variety of generic blocks in Minecraft with simulated properties of wood, stone, steel etc. Within the framework of the game, some blocks may interact in specific ways with Redstone, but for the purpose of this exploration, all blocks not described above will be considered to be generic blocks (light gray color in schematics), with a property common to most blocks in Minecraft. Most blocks in Minecraft may be activated if placed next to powered Redstone or another device capable of inducing a current, and will transmit power to all blocks adjacent to them. This property allows for the creation of the NOT gate, fundamental to most circuit designs in Minecraft.

Basic operations: The following are basic operations in Boolean algebra, their respective logic gates (the manner in which the operation is graphically represented in a circuit) and the logic gates recreated in Minecraft.There are three fundamental operations in Boolean algebra; each one defined over the set {1,0} by use of a truth table, with a corresponding logic gate.ANDThe AND () operator is defined such that, for A B = Y, only if both A and B are equal to one, Y equals 1. In every other case, Y is 0.The operator may be expressed as,A B = 1 if A=B=1. For all other pairs of A and B, A B = Y = 0.Table 1.0: Truth table for the AND operator.ABA B =Y

000

010

100

111

The AND logic gate, which symbolizes the operator in a circuit diagram is drawn as follows;

AAB

B

The figure is read from left to right. The two lines entering the gate symbolize the inputs A and B while the line exiting the gate symbolizes the output, Y.The AND gate may be created using the following design in Minecraft using the properties of a Redstone torch. In the following diagram, if either one of the levers is switched off, either torch T1 or T2 powers Redstone wire R1, resulting in no output. Therefore, the following gate only gives an output of 1 if both levers are switched on.

T2R1T1ORThe OR () operator is defined such that, for A B = Y, if either A or B equals 1, then Y will equal 1. Only if both A and B are equal 0, will Y equal 0.Therefore, the operator may be expressed as,A B = 0 if A=B=0. For all other pairs of A and B, AB = Y = 1.Table 1.1: Truth table for the OR operator.ABA B =Y

000

011

101

111

The OR logic gate, which symbolizes the operator in a circuit diagram is drawn as follows;

ABAB

The OR gate in Minecraft can be created from the following design. It is simple as if either or both of the levers are in the ON position, the gate will output a signal of one, since the Redstone will light up if adjacent blocks are powered on.

NOTThe final fundamental operator, the NOT () operator, is defined over a single input, and simply inverts the input value. Therefore if A=1, A = 0 and if A=0 A = 1.Table 1.2: Truth table for the NOT operator.A A =Y

10

01

The NOT logic gate, which symbolizes the operator in a circuit diagram is drawn as follows;

AA

The NOT gate in Minecraft is essentially a Redstone torch connected to block, due to which, as defined by the mechanics of the game, the torch inverts a signal passed into the block. Thus any signal A input into the gate will result in output A.

Complex operatorsUsing the fundamental operators described above, we can set out to create more complicated circuit components, i.e. logic gates that are defined as operators. A vast number of these operators exist, but for the purpose of this exploration only the XOR operator is essential.XORThe XOR () operator, sometimes referred to as EXOR is abbreviated to Exclusive OR. The XOR operator is defined such that, for A B = Y, if A=1 or B=1 and A = B, then Y = 1. For all other pairs of A and B, Y=0.Table 1.3: Truth table for the XOR operator.ABA B =Y

000

011

101

110

Most importantly, the XOR operator is not a fundamental operator, it is in fact a composition of previously described fundamental operators, and can be described as follows:A B = (A B) (B A)Proof of statementTo be capable of proving the above statement, first, a Karnough map will have to be defined.A Karnough map is a diagram consisting of a rectangular array of squares, each representing a different combination of the variables of a Boolean function.2 it is a manner of organizing a truth table such that relationships between variables result in easily observable patterns.The above expression is derived from the Karnough map of the XOR truth table (Table 1.4), which is the following.

Let the above map represent the function Y.The Karnough map indicates that there are two possible combinations of the inputs A and B that return a high value, i.e. 1. This occurs when A=1 and B=0 or A=0 and B=1.(The cases highlighted in blue)This means, When A=1 and B=0, Y=1. Which means when A=1 and B=1, Y=1. This may be written as (A B) = 1.Similarly, Y=1 when A=0 and B=1, which means when A=1 and B=1, Y=1. This may be written as (A B) = 1.The function returns 1 when either of the above statements is true, Therefore the Boolean expression for the function is Y = (A B) (B A)

The XOR logic gate, which symbolizes the operator in a circuit diagram is drawn as follows;

A

A B

B

The circuit diagram may also be draw using fundamental operators using the derived expression A B = (A B) (B A);

BAB

A B

The XOR gate may be recreated by substituting the gates in the above diagram with their Minecraft counterparts.

BA BA

The binary adder:The adder, also referred to as a summer, is a circuit that performs addition of given input values. The adder is therefore essentially a representation of the addition (+) operator, and therefore the circuit can be formulated as done previously as with the XOR operator, by use of a truth table. Half-adderThe half adder is a highly simplified adder that only accepts 2 bits of input, in this case A and B, giving A+B as the output. Before a truth table is designed, we can make a simple observation. By assigning A = 1 and B = 1 as input values, by simple arithmetic, we can observe that A+B= 10, where 10 is 2 bits of information. For the half-adder to be capable of displaying this information, it has to be capable of returning 2 bits as output, and thus, for the truth table, two output variables, Y and C are defined, where Y is the sum and C is the Carry. For the above example, Cout = 1 and Y = 0. The reason that the second bit is denoted as the carry is that when a full adder is constructed, the Cout value is passed on to be added by the subsequent half adder. The concept will be illustrated clearly when the full adder is discussed.Table 2.0: Truth table for a half adder:ABCoutY

0000

0101

1001

1110

From the above truth table, expressions for Cout and Y can be derived by comparison with previously described logic gates. The relationship between A and B that results in output Cout is exactly as in Table 1.0 for A B. Similarly the relationship between A and B that results in output Y is exactly as in Table 1.3 for A B.Therefore the expressions for Cout and Y are,Cout = A B and Y = A B.These expressions may be used to design a circuit using an AND gate and an XOR that, by inputting 2 bits of information, (A,B), two bits of information, (C,Y) are returned that are equal to A+B.Circuit design

A

YB

Cout

The following is the schematic for a half-adder created in Minecraft by substituting the logic gates in the above diagram with their Minecraft counterparts.

CoutXOR gate: It is important to make sure that the inputs line up during constructionYA full adder is an adder that is capable of receiving 3 bits of input and returning their sum in as 2 bits, and is essentially composed of two half-adders. The additional input bit is the Cin bit that represents a bit carried over from a previous operation. The Cin is essential when larger numbers are added, during which the Cout of one adder is the Cin for the successive adder. The full adder accepts A,B and Cin as inputs and returns A + B + C in terms of the bits Y and Cout.Table 2.1: Truth table for a full adder:ABCinCoutY

00000

00101

01001

01110

10001

10110

11010

11111

From the truth table, the Karnough maps for output Y is;

From the Karnough map, the following expression may be derived;Y = (A B Cin) (A B Cin) (A B Cin) (A B Cin)The Karnough map for output Cout is;

From the Karnough map, the following expression may be derived;Cout = (A B) (B C) (C A)

CinBA

Y

Cout

The full-adder is yet again simply created by substituting the components of the above diagram with their Minecraft counterparts.

CoutYABCinAddition of larger numbersGiven two numbers A and B, each of them can be deconstructed into single bits of information. For instance, if A = 1010, and B = 0011 then the numbers are deconstructed such that A0 = 0, A1= 1, A2 = 0, A3 = 1 and B0 = 1, B1= 1, B2 = 0, B3 = 0. The process of addition is analogous to conventional addition, and each bit of A is added to its corresponding bit of B, and the Cout from the addition is added to the next set of bits. Therefore, A0 is added to B0, resulting in two outputs, Y0 and Cout. Cout is then added to A1 + B1. The process continues in such a manner until the two numbers are added, and the output received is Y0 , Y1, Y2 and Y3. From the above example, Y0 =1 , Y1 =0 , Y2 = 1 ,Y3 = 1, which means Y = 1101.The following is a graphical representation of the concept described previously which illustrated the addition of each corresponding bit of the two numbers. The format is effective in displaying the similarity to addition in the decimal system.

The above process of addition translated into a circuit using a half-adder and multiple full adders. The half adder may be used to add the first two bits, i.e. A0 and B0 since there is no Cin. In every other instance a full adder is used. For this exploration I attempt to recreate a 4-bit ripple carry adder in Minecraft, which takes two numbers A and B as input, such that each number may have 4-bits of information, and outputs the sum of the two numbers, A+B in binary. The 4-bit adder uses one half-adder and three full adders connected such that the carry (Cout) from the half-adder is passed as the Cin for the first full adder. The resulting carry from the second full adder is passed on as input into the next adder and so forth. The following is a diagram for a 4-bit ripple carry adder.

Y3Y2Y1Y0A0B0A1B1A2B2A3B3

Full adderFull adderFull adderHalf-adder

C2C3C1

C4

Note: In the above diagram, C is used to represent both Cout and Cin.

Full adder (3)Full adder (2)Full-adder (1)Half-adderB3A3CoutY3B2A2B1A1B0A0Y2Y1Y0

Note: In the above diagram, for the sake of simplicity, the inputs of the ripple carry adder and the Carry out from the previous adder do not match the inputs of each full adder. This result in a simpler diagram, but it should be assumed that each input displayed in the ripple carry adder is in fact connected to the input of its corresponding full-adder.

The above schematic is the ultimate result of this exploration and represents a fully functional ripple carry adder. The ripple carry adder receives inputs from the levers in the schematic in the forms of 1s and 0s, which make up a number in the binary system and outputs the sum of the two input numbers in Binary in the form of powered Redstone, representing 1, and unpowered Redstone representing 0s.

ConclusionThe game of Minecraft itself is highly conducive to the creation of circuits evident from the simplicity involved in designing basic logic gates. Instead of involving complicated components, the game hands the player a NOT gate, and subsequently makes the creation of other logic gates very simple, which is not the same as in real life where components are relatively expensive and extremely difficult to create from scratch, due to which Minecraft is both very entertaining and educative in its capabilities. The major downside of the circuits described in previous sections is that they are not optimal, that is, they use more gates than absolutely necessary and result in a slower rate of information processing. The circuits may also not be optimal within the Minecraft context as the game renders each block individually as they change state when a signal is passed in a circuit, which means that they consume greater processing power to run within the game as opposed to more compact circuits which give a similar functionality. The lack of processing power efficiency may also be attributed to the fact that the above circuits were built without utilizing the ability for logic gates and circuits to be be constructed in three dimensions which further reduces the length of wiring required for the construction of circuits, reducing the number of blocks that need to be rendered. Over the course of this exploration I have learned considerably about Boolean algebra and Boolean operators with respect to logical gates and their implementation in computing, while also generally grasping the modeling of simple Boolean functions. I have also been able to apply these concepts, in the above exploration, using limitations defined by the world of Minecraft to design an instrument that is capable of conducting the addition of two 4-bit integers, and thus compute data at a simplistic level.

Bibliography1. "What Is Boolean Algebra? Definition and Meaning." BusinessDictionary.com. Web. 2 Apr. 2015.2. "Karnaugh Map." Academic Dictionaries and Encyclopedias. Web. 1 Apr. 2015.

A =1010B =0011+Y = 1101