Invitation to Computer Science 6th Edition Chapter 4 The Building Blocks: Binary Numbers, Boolean Logic, and Gates

Introduction Chapter 4 focuses on hardware design (also called logic design) How to represent and store information inside a computer--The binary numbering system How to use the principles of symbolic logic to design gates--Boolean logic and gates How to use gates to construct circuits that perform operations such as adding and comparing numbers, and fetching instructions--Building computer circuits Control circuits Invitation to Computer Science, 6th Edition

Introduction Computing agent Abstract concept representing any object capable of understanding and executing our instructions Fundamental building blocks of all computer systems Binary representation Boolean logic Gates Circuits Invitation to Computer Science, 6th Edition 3

The Binary Numbering System Binary representation of numeric and textual information Two types of information representation External representation Internal representation Binary is a base-2 positional numbering system Invitation to Computer Science, 6th Edition 4

Figure 4.1 Distinction Between External Memory and Internal Representation of Information Invitation to Computer Science, 6th Edition

Binary Representation of Numeric and Textual Information Invitation to Computer Science, 6th Edition

Binary Representation of Numeric and Textual Information Binary-to-decimal algorithm Whenever there is a 1 in a column, add the positional value of that column to a running sum Whenever there is a 0 in a column, add nothing The final sum is the decimal value of this binary number Invitation to Computer Science, 6th Edition

Figure 4.2 Binary-to-Decimal Conversion Table Invitation to Computer Science, 6th Edition

Binary Representation of Numeric and Textual Information (continued) To convert a decimal value into its binary equivalent Use the decimal-to-binary algorithm Maximum number of binary digits that can be used to store an integer: 16, 32, or 64 bits Given k bits, the largest unsigned integer is 2k-1 Given 16 bits the largest is 216-1=65535 Arithmetic overflow Operation that produces an unsigned value greater than 65,535 Invitation to Computer Science, 6th Edition

Binary Representation of Numeric and Textual Information (continued) Invitation to Computer Science, 6th Edition

Binary Representation of Numeric and Textual Information (continued) How to express signed integer? Sign/Magnitude notation (old computer) Two's complement notation (current computer) Invitation to Computer Science, 6th Edition

Sign/Magnitude notation 1(sign bit) 110001 (total: 7 bits, computer A) When it is a signed value, it is -49 When it is a unsigned value, it is 113 1(sign bit) 0110001 (total: 8 bits, computer B) When it is a unsigned value, it is 177 You must tell the computer if it is signed or unsigned integer. The number of bit depends on computer (8bit, 16bit, 32bit, 64bit) Invitation to Computer Science, 6th Edition

Invitation to Computer Science, 6th Edition

Two's complement notation Why we have this solution? Because of two zeros problem in Sign/Magnitude notation : 10000, 00000 if (a = b) do operation 1 else do operation 2 Invitation to Computer Science, 6th Edition

Two's complement notation If A > 0 then do nothing else get complement value of each bit a <- a+1 Eg: -3 (3bit): 3 -> 011 -> 100 -> 101 -3 (4bit): 3 -> 0011 -> 1100 -> 1101 The number of bit depends on computer (8bit, 16bit, 32bit, 64bit) Easier method: Get complement value of each bit until first 1 Eg: -3 (3bit): 3-> 011 -> 101 -3 (4bit) : 3 -> 0011 -> 1101 No substraction in two’s complement notation. Convert to complement value. Eg 5 – 3 = 5 + (-3) Invitation to Computer Science, 6th Edition

Two's complement notation Bit pattern Decimal Value 001 +1 010 +2 011 +3 100 -4 101 -3 110 -2 111 -1 Invitation to Computer Science, 6th Edition

Compare value range Suppose we have k bit. Sign/Magnitude notation: -(2k-1 – 1) to (2k-1 – 1) Eg 3bit: -3 ~ +3 Two's complement notation: -(2k-1) to (2k-1 – 1) Eg 3bit: -4 ~ +3 Why different? Because Sign/Magnitude notation has two zeros while two’s complement notation has only one zero. Though two’s complement notation is difficult to human, it much clearer to computer. Invitation to Computer Science, 6th Edition

Invitation to Computer Science, 6th Edition

Binary <-> Decimal (Fractional number) Decimal -> Binary 0.8125 0.8125 * 2 = 1.625 ------ Get 1 0.625 * 2 = 1.25 ------ Get 1 0.25 * 2 = 0.5 ------ Get 0 0.5 * 2 = 1 ------ Get 1 Final: 0.1101 (Attention: compare to integer conversion)

Fractional Numbers Fractional numbers (12.34 and –0.001275) Can be represented in binary by using signed-integer techniques Scientific notation ±M x B±E M is the mantissa, B is the exponent base (usually 2), and E is the exponent Normalize the number First significant digit is immediately to the right of the binary point Invitation to Computer Science, 6th Edition

Binary Representation of Numeric and Textual Information (continued) Representing real numbers Real numbers should be put into binary scientific notation: a x 2b Example: 101.11 x 20 Number then normalized so that first significant digit is immediately to the right of the binary point Example: .10111 x 23 Invitation to Computer Science, 6th Edition

Normalization Example Binary Normalization 5.75 = 101.11 = 0.10111 * 23 Mantissa and exponent then stored ±M * B ±E 00010111000011 (8 bits for mantissa, 6 bits for exponent) 5.75 = 101.11 = 0.010111 * 24 = 1.0111 * 22 0.010111 * 24 and 1.0111 * 22 are not normalized number

Invitation to Computer Science, 6th Edition

Textual Information Code mapping ASCII UNICODE Assigning each printable letter or symbol in our alphabet a unique number ASCII International standard for representing textual information in the majority of computers Uses 8 bits to represent each character UNICODE Uses a 16-bit representation for characters rather than the 8-bit format of ASCII Invitation to Computer Science, 6th Edition

Figure 4.3 ASCII Conversion Table Invitation to Computer Science, 6th Edition

Binary Representation of Sound and Images Digital representation Values for a given object are drawn from a finite set Analog representation Objects can take on any value Figure 4.4 Amplitude of the wave: measure of its loudness Period of the wave (T): time it takes for the wave to make one complete cycle (pitch—highness or lowness) Frequency f: total number of cycles per unit time (cycles/seconds or hertz) Invitation to Computer Science, 6th Edition

Figure 4.4 Example of Sound Represented as a Waveform Invitation to Computer Science, 6th Edition

Binary Representation of Sound and Images (continued) Digitizing means periodic sampling of amplitude values Sampling rate Measures how many times per second we sample the amplitude of the sound wave Bit depth Number of bits used to encode each sample MP3 Most popular and widely used digital audio format Mp3 44,100 16 bits Invitation to Computer Science, 6th Edition

Figure 4.5 Digitization of an Analog Signal (a) Sampling the Original Signal (b) Re-creating the Signal from the Sampled Values Invitation to Computer Science, 6th Edition

Binary Representation of Sound and Images (continued) From samples, original sound can be approximated To improve the approximation Sample more frequently Use more bits for each sample value Invitation to Computer Science, 6th Edition

Binary Representation of Sound and Images (continued) Representing image data Images are sampled by reading color and intensity values Each sampled point is a pixel Image quality depends on number of bits at each pixel 0.05~0.1mm Scanning Measuring the intensity values of distinct points located at regular intervals across the image’s surface Invitation to Computer Science, 6th Edition

Binary Representation of Sound and Images (continued) Invitation to Computer Science, 6th Edition

Invitation to Computer Science, 6th Edition

Data Compression Why we need data compression? Because the original data need too much space. Eg. 3,000,000 pixels/photograph * 24 bits/pixel = 72million bits. Data compression algorithms Attempt to represent information in ways that preserve accuracy while using significantly less space Invitation to Computer Science, 6th Edition

Data Compression Lossless compression schemes No information is lost in the compression It is possible to exactly reproduce the original data Lossy compression schemes Do not guarantee that all of the information in the original data can be fully and completely recreated Compression ratio Measures how much compression schemes reduce storage requirements of data Compression rate = size of the uncompressed data / size of the compressed data Invitation to Computer Science, 6th Edition

Binary Representation of Sound and Images (continued) Run-length encoding Replaces a sequence of identical values v1, v2, . . ., vn by a pair of values (v, n) for image compression Eg. (255 255 255), (255, 0, 0), (255, 255, 255), (255, 0, 0), (255, 255, 255) → (255, 4), (0, 2), (255, 4), (0, 2), (255, 3) Variable length code sets Often used to compress text Can also be used with other forms of data Invitation to Computer Science, 6th Edition

Data Compression Simple compression method - “variable length code sets” - for text compress Letter 4 bit encoding Variable length encoding A 0000 00 I 0001 10 H 0010 010 W 0100 110 H A W A I I → 0010, 0000, 0100, 0000, 0001, 0001 → 001, 00, 110, 00, 10, 10 Invitation to Computer Science, 6th Edition

Figure 4.8 Using Variable Length Code Sets (a) Fixed Length (b) Variable Length Invitation to Computer Science, 6th Edition

Invitation to Computer Science, 6th Edition

The Reliability of Binary Representation Computers use binary representation for reasons of reliability Building a base-10 “decimal computer” Requires finding a device with 10 distinct and stable energy states that can be used to represent the 10 unique digits (0, 1, . . . , 9) of the decimal system Bistable environment Only two (rather than 10) stable states separated by a huge energy barrier Invitation to Computer Science, 6th Edition

Binary Storage Devices Magnetic core ‰Historic device for computer memory ‰Tiny magnetized rings; flow of current sets the direction of magnetic field direction of magnetic field Binary values 0 and 1are represented using the direction of the magnetic field Invitation to Computer Science, 6th Edition

Figure 4.9 Using Magnetic Cores to Represent Binary Values Invitation to Computer Science, 6th Edition

Binary Storage Devices (continued) Transistors ‰Solid-state switches; either permit or block current flow ‰A control input causes state change ‰Constructed from semiconductors ‰Can be printed photographically on a wafer of silicon to produce a device known as an integrated circuit to produce a device known as an integrated circuit „Circuit board ‰Interconnects all the different chips needed to run a computer system Invitation to Computer Science, 6th Edition

Figure 4.10 Relationships Among Transistors, Chips, and Circuit Boards Invitation to Computer Science, 6th Edition

Binary Storage Devices (continued) Figure 4.11 Control (base): used to open or close the switch inside the transistor ON state: current coming from the In line (Collector) can flow directly to the Out line (Emitter), and the associated voltage can be detected by a measuring device Invitation to Computer Science, 6th Edition

Figure 4.11 Simplified Model of a Transistor Invitation to Computer Science, 6th Edition

Boolean Logic Boolean logic Construction of computer circuits is based on this Boolean expression Any expression that evaluates to either true or false Constructed by combining together Boolean operations Example: (a AND b) OR ((NOT b) AND (NOT a)) Truth table A column for each input plus the output A row for each combination of input values Invitation to Computer Science, 6th Edition

Figure 4.12 Truth Table for the AND Operation Truth Table: Can express the idea that the AND operation produces the value true if and only if both of its components are true Invitation to Computer Science, 6th Edition

Boolean Logic (continued) Boolean operations a AND b True only when a is true and b is true a OR b True when a is true, b is true, or both are true NOT a True when a is false and vice versa Invitation to Computer Science, 6th Edition

Boolean Logic (continued) Boolean operations AND, OR, NOT Binary operators Require two operands Unary operator Requires only one operand NOT operation Reverses, or complements, the value of a Boolean expression Invitation to Computer Science, 6th Edition

Figure 4.13 Truth Table for the OR Operation Invitation to Computer Science, 6th Edition

Figure 4.14 Truth Table for the NOT Operation Invitation to Computer Science, 6th Edition

Boolean Logic (continued) Example: (a AND b) OR ((NOT b) and (NOT a)) a b Value 1 Invitation to Computer Science, 6th Edition

Invitation to Computer Science, 6th Edition

Invitation to Computer Science, 6th Edition

Gates Gates Hardware devices built from transistors to mimic Boolean logic Transforms a set of (0 1) input values into a single ‰Transforms a set of (0,1) input values into a single (0,1) output value AND gate Two input lines, one output line Outputs a 1 when both inputs are 1 Invitation to Computer Science, 6th Edition

Gates (continued) OR gate NOT gate Two input lines, one output line Outputs a 1 when either input is 1 NOT gate One input line, one output line Outputs a 1 when input is 0 and vice versa Invitation to Computer Science, 6th Edition

Figure 4.15 The Three Basic Gates and Their Symbols Invitation to Computer Science, 6th Edition

Figure 4.16 Construction of a NOT Gate Invitation to Computer Science, 6th Edition

Figure 4.17 Construction of NAND and AND Gates (a) A Two-transistor NAND Gate (b) A Three-transistor AND Gate Invitation to Computer Science, 6th Edition

Gates (continued) NAND (acronym for NOT AND) To construct an OR gate Produces the complement of the AND operation To construct an OR gate Start with two transistors Transistors are connected in parallel Invitation to Computer Science, 6th Edition

Figure 4.18 Construction of NOR and OR Gates (a) A Two-transistor NOR Gate (b) A Three-transistor OR Gate Invitation to Computer Science, 6th Edition

Gates (continued) „Abstraction in hardware design ‰Map hardware devices to Boolean logic ‰Design more complex devices in terms of logic, not electronics ‰Conversion from logic to hardware design can be automated Invitation to Computer Science, 6th Edition

Building Computer Circuits Invitation to Computer Science, 6th Edition

Building Computer Circuits Every Boolean expression: Can be represented pictorially as a circuit diagram Every output value in a circuit diagram: Can be written as a Boolean expression Invitation to Computer Science, 6th Edition

Figure 4.19 Diagram of a Typical Computer Circuit Invitation to Computer Science, 6th Edition

A Circuit Construction Algorithm Sum-of-products algorithm is one way to design circuits Truth table to Boolean expression to gate layout Step 1: Truth Table Construction Determine how the circuit should behave under all possible circumstances If a circuit has N input lines and if each input line can be either a 0 or a 1, then: There are 2N combinations of input values, and the truth table has 2N rows Invitation to Computer Science, 6th Edition

A Truth Table for a Circuit with 8 Input Combinations Invitation to Computer Science, 6th Edition

A Circuit Construction Algorithm (continued) Step 2: Subexpression Construction Using AND and NOT Gates Choose any one output column of the truth table built in step 1, and scan down that column Every place that you find a 1 in that output column, you build a Boolean subexpression that produces the value 1 for exactly that combination of input values and no other Invitation to Computer Science, 6th Edition

Output Column Labeled Output-1 from the Previous Truth Table Invitation to Computer Science, 6th Edition

Taking Snapshots Step 3: Subexpression Combination Using OR Gates Take each of the subexpressions produced in step 2 and combine them, two at a time, using OR gates Step 4: Circuit Diagram Production Construct the final circuit diagram Algorithms for circuit optimization Reduce the number of gates needed to implement a circuit Invitation to Computer Science, 6th Edition

Figure 4.20 Circuit Diagram for the Output Labeled Output-1 Invitation to Computer Science, 6th Edition

Figure 4.21 The Sum-of-Products Circuit Construction Algorithm Invitation to Computer Science, 6th Edition

Invitation to Computer Science, 6th Edition

Invitation to Computer Science, 6th Edition

Examples of Circuit Design and Construction A Compare-For-Equality Circuit Tests two unsigned binary numbers for exact equality Produces the value 1 (true) if the two numbers are equal and the value 0 ( false) if they are not Built by combining together 1-bit comparison circuits (1-CE) Integers are equal if corresponding bits are equal (AND together 1-CE circuits for each pair of bits) Invitation to Computer Science, 6th Edition

A Compare-for-Equality Circuit (continued) 1-CE circuit truth table a b Output 1 Invitation to Computer Science, Java Version, Third Edition

A Compare-for-Equality Circuit (continued) 1-CE Boolean expression First case: (NOT a) AND (NOT b) Second case: a AND b Combined: ((NOT a) AND (NOT b)) OR (a AND b) Invitation to Computer Science, Java Version, Third Edition

Figure 4.22 One-Bit Compare for Equality Circuit Invitation to Computer Science, 6th Edition

Figure 4.23 N-Bit Compare for Equality Circuit Invitation to Computer Science, 6th Edition

An Addition Circuit Full adder Figure 4.27 Addition circuits Performs binary addition on two unsigned N-bit integers Figure 4.27 Shows the complete full adder circuit called ADD Addition circuits Found in every computer, workstation, and handheld calculator in the marketplace Built from 1-bit adders (1-ADD) Invitation to Computer Science, 6th Edition

Figure 4.24 The 1-ADD Circuit and Truth Table Invitation to Computer Science, 6th Edition

Figure 4.25 Sum Output for the 1-ADD Circuit Invitation to Computer Science, 6th Edition

An Addition Circuit (continued) Building the full adder Put rightmost bits into 1-ADD, with zero for the input carry Send 1-ADD’s output value to output, and put its carry value as input to 1-ADD for next bits to left Repeat process for all bits Invitation to Computer Science, Java Version, Third Edition

Figure 4.26 Complete 1-ADD Circuit for 1-Bit Binary Addition Invitation to Computer Science, 6th Edition

Figure 4.27 The Complete Full Adder ADD Circuit Invitation to Computer Science, 6th Edition

Control Circuits Used to: Multiplexor Determine the order in which operations are carried out Select the correct data values to be processed Multiplexor Circuit that has 2N input lines and 1 output line Function: to select exactly one of its 2N input lines and copy the binary value on that input line onto its single output line Invitation to Computer Science, 6th Edition

Figure 4.28 A Two-Input Multiplexor Circuit Invitation to Computer Science, 6th Edition

Figure 4.31 Example of the Use of a Multiplexor Circuit Invitation to Computer Science, 6th Edition

Control Circuits (continued) Decoder Has N input lines numbered 0, 1, 2, . . . , N – 1 and 2N output lines numbered 0, 1, 2, 3, . . . , 2N – 1 Determines the value represented on its N input lines and then sends a signal (1) on the single output line that has that identification number Invitation to Computer Science, 6th Edition

Figure 4.29 A 2-to-4 Decoder Circuit Invitation to Computer Science, 6th Edition

Figure 4.30 Example of the Use of a Decoder Circuit Invitation to Computer Science, 6th Edition

Summary Digital computers Binary values Boolean logic Use binary representations of data: numbers, text, multimedia Binary values Create a bistable environment, making computers reliable Boolean logic Maps easily onto electronic hardware Invitation to Computer Science, 6th Edition

Summary (continued) Circuits Computational and control circuits Constructed using Boolean expressions as an abstraction Computational and control circuits Can be built from Boolean gates Invitation to Computer Science, 6th Edition

Test 2 Chapter 3 & 4 18 questions: Multiple choice, Essay questions Time efficiency analysis Binary search algorithm Binary, Decimal, Hexadecimal number conversion Unsigned number Boolean expression Circuit construction Multiple choice, Essay questions Invitation to Computer Science, 6th Edition