# Computer Organization and Levels of Abstraction

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1 Computer Organization and Levels of Abstraction

2 Announcements Today: PS 7 Lab 8: Sound Lab tonight bring machines and headphones! PA 7 Tomorrow: Lab 9 Friday: PS8

3 Today (Short) Floating point review Boolean logic Combinational Circuits Levels of Abstraction

4 Floating point +/- Exponent Mantissa 1 bit 8 bits 23 bits Sign is a 0 or 1 Exponent is an binary integer Mantissa is a binary fraction

5 Floating point x /- Exponent Mantissa 1 bit 8 bits 23 bits Sign is a 0 or 1 Exponent is an binary integer Mantissa is a binary fraction

6 Floating point Sign x /- Exponent Mantissa 1 bit 8 bits 23 bits Sign is a 0 or 1

7 Exponent x Exponent Is an unsigned integer But exponent can be negative how to distinguish? IEEE-754 specifies a bias: 127 This gives us a range of -126 to +127 Makes comparison easier (for large and small values)

8 Floating point Mantissa x _ /- Exponent Mantissa 1 bit 8 bits 23 bits Pad the mantissa

9 Floating point Mantissa x _ /- Exponent Mantissa 1 bit 8 bits 23 bits Pad the mantissa

10 Floating point Mantissa x

11 11 You should be able to Identify basic gates Describe the behavior of a gate or circuit using Boolean expressions, truth tables, and logic diagrams Transform one Boolean expression into another given the laws of Boolean algebra

12 12 Conceptualizing bits and circuits ON or 1: true OFF or 0: false circuit behavior: expressed in Boolean logic or Boolean algebra

13 13 Boolean Logic (Algebra) Computer circuitry works based on Boolean Logic (Boolean Algebra) : operations on True (1) and False (0) values. A B A Λ B (A AND B) (conjunction) A ν B (A OR B) (disjunction) A A (NOT A) (negation) A and B in the table are Boolean variables, AND and OR are operations (also called functions).

14 14 Boolean Logic & Truth Tables Example: You can think of A Ù B below as is fun and is useful where A stands for the statement is fun, B stands for the statement is useful. A B A Λ B (A AND B) (conjunction) A ν B (A OR B) (disjunction) A A (NOT A) (negation)

15 15 Logic Gates A gate is a physical device that implements a Boolean operator by performing basic operations on electrical signals. Nowadays, gates are built from transistors. Physical behavior of circuits is beyond the scope of our course. A B A Ù B AND A B A Ú B OR A NOT

16 16 Combinational Circuits The logic states of inputs at any given time determine the state of the outputs. AND OR AND A Ù B B Ú C AND C Ù B OR (B Ú C) Ù (C Ù B) What is Q? (A Ù B) Ú ((B Ú C) Ù (C Ù B))

17 17 Truth Table of a Circuit AND OR OR AND AND Q = (A Ù B) Ú ((B Ú C) Ù (C Ù B)) A B C Q How do I know that there should be 8 rows in the truth table? Describes the relationship between inputs and outputs of a device

18 18 Truth Table of a Circuit AND OR OR AND AND Q = (A Ù B) Ú ((B Ú C) Ù (C Ù B)) A B C Q Describes the relationship between inputs and outputs of a device

19 19 Describing Behavior of Circuits Boolean expressions Circuit diagrams Equivalent notations Truth tables

20 20 Why manipulate circuits? The design process simplify a complex design for easier manufacturing, faster or cooler operation, Boolean algebra helps us find another design guaranteed to have same behavior

21 Logical Equivalence Q = (A Ù B) Ú ((B Ú C) Ù (C Ù B)) OR AND OR AND AND AND OR Q = B Ù (A Ú C) A B C Q This smaller circuit is logically equivalent to the one above: they have the same truth table. By using laws of Boolean Algebra we convert a circuit to another equivalent circuit. 21

22 22 Laws for the Logical Operators Ù and Ú (Similar to and +) Commutative: A Ù B = B Ù A A Ú B = B Ú A Associative: A Ù B Ù C = (A Ù B) Ù C = A Ù (B Ù C) A Ú B Ú C = (A Ú B) Ú C = A Ú (B Ú C) Distributive: A Ù (B Ú B) = (A Ù B) Ú (A Ù C) A Ú (B Ù C) = (A Ú B) Ù (A Ú C) Identity: A Ù 1 = A A Ú 0 = A Dominance: A Ù 0 = 0 A Ú 1 = 1 Idempotence: A Ù A = A A Ú A = A Complementation: A Ù A = 0 A Ú A = 1 Double Negation: A = A

23 Laws for the Logical Operators Ù and Ú (Similar to and +) Commutative: A Ù B = B Ù A A Ú B = B Ú A Associative: A Ù B Ù C = (A Ù B) Ù C = A Ù (B Ù C) A Ú B Ú C = (A Ú B) Ú C = A Ú (B Ú C) Distributive: A Ù (B Ú C) = (A Ù B) Ú (A Ù C) A Ú (B Ù C) = (A Ú B) Ù (A Ú C) Not true for + and Identity: A Ù 1 = A A Ú 0 = A The A s and B s here are schematic variables! You can instantiate them with any expression that has a Boolean value: (x Ú y) Ù z = z Ù (x Ú y) (by commutativity) A Ù B = B Ù A

24 Applying Properties for Ù and Ú Showing (x Ù y) Ú ((y Ú z) Ù (z Ù y)) = y Ù (x or z) Commutativity A Ù B = B Ù A (x Ù y) Ú ((z Ù y) Ù (y Ú z)) Distributivity A Ù (B Ú C) = (A Ù B) Ú (A Ù C) (x Ù y) Ú (z Ù y Ù y) Ú(z Ù y Ù z) Associativity, Commutativity, Idempotence (x Ù y) Ú ((z Ù y) Ú (y Ù z)) Commutativity, idempotence A Ù A = A ( y Ù x) Ú (y Ù z) Distributivity (backwards) (A Ù B) Ú (A Ù C) = A Ù (B Ú C) y Ù (x Ú z) Conclusion: (x Ù y) Ú ((y Ú z) Ù (z Ù y)) = y Ù (x Ú z)

25 25 Extending the system more gates and DeMorgan s laws

26 More gates (NAND, NOR, XOR) A B A nand B A nor B nand ( not and ): A nand B = not (A and B) A xor B A B (A Ù B) nor ( not or ): A nor B = not (A or B) xor ( exclusive or ): A xor B = (A and not B) or (B and not A) A B A B (A Ú B) A Å B 26

27 A curious fact Functional Completeness of NAND and NOR Any logical circuit can be implemented using NAND gates only Same applies to NOR

28 28 DeMorgan s Law Nand: (A Ù B) = A Ú B Nor: (A Ú B) = A Ù B

29 29 DeMorgan s Law Nand: (A Ù B) = A Ú B if not (x > 15 and x < 110):... is logically equivalent to if (not x > 15) or (not x < 110):... Nor: (A Ú B) = A Ù B if not (x < 15 or x > 110):... is logically equivalent to if (not x < 15) and (not x > 110):...

30 30 A circuit for parity checking Boolean expressions and circuits

31 The circuit 3-bit odd parity checker P = ( A Ù BÙ C) Ú ( A Ù B Ù C) Ú (A Ù BÙ C) Ú (A Ù B Ù C) A B C P

32 The circuit 3-bit odd parity checker P = ( A Ù BÙ C) Ú ( A Ù B Ù C) Ú (A Ù BÙ C) Ú (A Ù B Ù C) A B C P There are specific methods for obtaining canonical Boolean expressions from a truth table, such as writing it as a disjunction of conjunctions or as a conjunction of disjunctions. Note we have four subexpressions above each of them corresponding to exactly one row of the truth table where P is 1.

33 33 The circuit 3-bit odd parity checker P = ( A Ù BÙ C) Ú ( A Ù B Ù C) Ú (A Ù BÙ C) Ú (A Ù B Ù C) A B C P A B P = (A Å B) Å C C P

34 Circuits for arithmetic

35 36 Adding Binary Numbers A: B: Adding two 1-bit numbers without taking the carry into account A B Sum Sum = A Å B How can we handle the carry?

36 37 Adding Binary Numbers A: B: A B Sum Carry Half Adder: adds two single digits

37 A Full Adder C out C in A B C in C out S A B S

38 A Full Adder C out C in A B S A B C in C out S S = A Å B Å C in C out = ((A Å B) Ù C in ) Ú (A Ù B)

39 A Full Adder S: 1 when there is an odd number of bits that are 1 C out C in A B C in C out S A B S C out : 1 if both A and B are 1 or, one of the bits and the carry in are 1. S = A Å B Å C in C out = ((A Å B) Ù C in ) Ú (A Ù B)

40 Full Adder (FA) S = A Å B Å C in C out = ((A Å B) Ù C in ) Ú (A Ù B) A B C out 1-bit Full Adder S C in More abstract representation of the above circuit. Hides details of the circuit above.

41 8-bit Full Adder A 7 B 7 A 2 B 2 A 1 B 1 A 0 B 0 C out 1-bit Full Adder... 1-bit Full Adder 1-bit Full Adder 1-bit Full Adder C in S 7 S 2 A B S 1 S C out 8-bit FA 8 S C in More abstract representation of the above circuit. Hides details of the circuit above.

42 Control Circuits In addition to circuits for basic logical and arithmetic operations, there are also circuits that determine the order in which operations are carried out and to select the correct data values to be processed.

43 44 Multiplexer (MUX) A multiplexer chooses one of its inputs. 2 n input lines, n selector lines, and 1 output line A B F D 1 D 2 D 3 D 4 MUX F 0 0 D D D D 4 A B hides details of the circuit on the left

44 Arithmetic Logic Unit (ALU) OP 1 OP 0 Carry In & OP OP 0 OP 1 F 0 0 A Ù B 0 1 A Ú B 1 0 A 1 1 A + B Depending on the OP code Mux chooses the result of one of the functions (and, or, identity, addition) 45

45 46 Building A Complete Computer from Parts

46 Computing Machines An instruction is a single arithmetic or logical operation. A program is a sequence of instructions that causes the desired function to be calculated. A computing system is a combination of program and machine (computer). How can we build a computing system that calculates the desired function specified by a program?

47 Stored Program Computer A stored program computer is electronic hardware that implements an instruction set. 48

48 49 Von Neumann Architecture Big idea: Data and instructions to manipulate the data are both bit sequences Modern computers built according to the Von Neumann Architecture includes separate units To process information (CPU): reads and executes instructions of a program in the order prescribed by the program To store information (memory)

49 50 Stored Program Computer adder, multiplier, multiplexor, etc. small amount of memory in the CPU

50 Central Processing Unit (CPU) A CPU contains: Arithmetic Logic Unit to perform computation Registers to hold information Instruction register (current instruction being executed) Program counter (to hold location of next instruction in memory) Accumulator (to hold computation result from ALU) Data register(s) (to hold other important data for future use) Control unit to regulate flow of information and operations that are performed at each instruction step 51

51 Memory The simplest unit of storage is a bit (1 or 0). Bits are grouped into bytes (8 bits). Memory is a collection of cells each with a unique physical address. We use the generic term cell rather than byte or word because the number of bits in each addressable location varies from machine one machine to another. A machine that can generate, for example, 32-bit addresses, can utilize a memory that contains up to 2 32 memory cells. 52

52 Memory Layout Address 100: 104: 108: 112: 116: Content Address : : : : : Content We saw this picture in Unit 6. It hid the bit representation for readability. Assumes that memory is byte addressable and each integer occupies 4 bytes. In this picture and in reality, addresses and memory contents are sequences of bits. 53

53 Stored Program Computer instruction fetch, decode, execute adder, multiplier, multiplexor, Etc. program counter, instruction register, Etc. Two specialized registers: the instruction register holds the current instruction to be executed and the program counter contains the address of the next instruction to be executed.

54 Processing Instructions Both data and instructions are stored in memory as bit patterns Instructions stored in contiguous memory locations Data stored in a different part of memory The address of the first instruction is loaded into the program counter and and the processing cycle starts. 55

55 Fetch-Decode-Execute Cycle Modern computers include control logic that implements the fetch-decode-execute cycle introduced by John von Neumann: Fetch next instruction from memory into the instruction register. Decode instruction to a control signal and get any data it needs (possibly from memory). Execute instruction with data in ALU and store results (possibly into memory). Repeat. Note that all of these steps are implemented with circuits of the kind we have seen in this unit. 56

56 Power of abstraction

57 58 Using Abstraction in Computer Design We can use layers of abstraction to hide details of the computer design. We can work in any layer, not needing to know how the lower layers work or how the current layer fits into the larger system. Low High transistors gates circuits (adders, multiplexors ) central processing units (ALU, registers ) computer A component at a higher abstraction layer uses components from a lower abstraction layer without having to know the details of how it is built. It only needs to know what it does.

58 59 Abstraction in Programming The set of all operations that can be executed by a processor is called its instruction set. Instructions are built into hardware: electronics of the CPU recognize binary representations of the specific instructions. That means each CPU has its own machine language that it understands. But we can write programs without thinking about on what machine our program will run. This is because we can write programs in high-level languages that are abstractions of machine level instructions.

59 A High-Level Program # This programs displays Hello, World! print( Hello world! )

60 A Low-Level Program

61 62 Obtaining Machine Language Instructions Programs are typically written in higher-level languages and then translated into machine language (executable code). A compiler is a program that translates code written in one language into another language. An interpreter translates the instructions one line at a time into something that can be executed by the computer s hardware.

62 Summary A computing system is a combination of program and machine (computer). In this lecture, we focused on how a machine can be designed using levels of abstraction: gates à circuits for elementary operations à basic processing units à computer

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