1. 1 60-265 COMPUTER ARCHITECTURE I: Digital Design Akshai Aggarwal

2. 2 Course Outline Binary, Octal and Hexadecimal number system Digital logic and Boolean Algebra Combinational and sequential circuit design Digital components: decoders, multiplexers, registers, counters, memory etc.; Digital Integrated Circuits Register transfer and micro-operations Basic computer organization and design CPU structure, control unit design, interrupt handling  HOW DOES A COMPUTER WORK? Elementary assembly language instruction set and the execution process

3. 3 Grading Scheme Quizzes/ Individual Assignments 10% Quizzes: to be conducted at the beginning of some of the labs; For dates, please see the web-site. Midterm 115%Saturday May 30, 12.30 PM, Venue: Erie 1118 Midterm 215%Saturday June 13, 12.30 PM, Venue: Erie 1118 Final40%As scheduled by the university Labs10%Attendance is mandatory Group Assignment 10% Submission of Assignment: in the Lab on Monday 1 ST June

4. Quizzes 4 Day and DateTime of Quiz for Section 51 Time of Quiz for Section 52 Wednesday, 13 th May4 PM5.30 PM Wednesday, 20 th May4 PM5.30 PM Wednesday, 27 th May4 PM5.30 PM Wednesday, 3 rd June4 PM5.30 PM Wednesday,10 th June4 PM5.30 PM

5. 5 Digital Computers DIGITAL : information represented by variables that take a limited number of values BINARY : - reliability of hardware -binary nature of human logic DIGITAL COMPUTER: A discrete information processing system A SYSTEM: An organized collection of components, that interact through communication links among themselves and with their environment to provide a predefined functionality.

6. 6 ARRAYS OF BITS: 1001011 may represent 75 in decimal, or K in ASCII code or some control code.

7. 7 History: ENIAC Electrical Numerical Integrator And Computer (ENIAC): developed by Professor John Mauchly and his graduate student John Presper Eckert For army’s Ballistic Research Lab for calculating range and trajectory tables for new weapons Start 1943; development completed in 1946; used for Nuclear bomb computations 18000 vacuum tubes, 140 kW power, 30 ton, 1500 sq ft of floor space Decimal machine; programming manually by setting switches and plugging and unplugging cables

8. 8 1945: A New Project: Electronic Discrete Variable Computer (EDVAC) EDVAC: planned by John Von Neumann as a Stored Program machine Developed at Princeton Institute for Advanced Studies as the IAS computer from 1946 -1952 consisted of main memory Arithmetic Logic Unit and Control Unit  CPU  Registers in CPU: Accumulator, Address Register, Program Counter, Data Register and other Registers Input/Output Used binary numbers

9. 9 FUNCTIONAL PARTS: - hardware: CPU, memory, I/O units -software System software Application program Logical view vs. Physical components

10. 10 Architecture Architecture of a computer: those properties, which directly affect the logical working of a program; the attributes, which are apparent to a programmer Examples: instruction set, number of bits used to represent data

11. 11 Von Neumann Architecture of a digital computer: Von Neumann Architecture of a Digital Computer Memory Central processing unit I/O Processor Input Output Devices

12. 12 Structure and behavior of the computer as seen by the user. It includes: 1.Instruction set. 2.Information formats. 3.Techniques for addressing memory. The course is an introduction to the three aspects. Computer Architecture:

13. Organization Organization: operational units and their interconnection for realizing the architectural specifications Determination of which hardware should be used and how the parts should be connected together 13

14. 14 Logic Gates and Boolean Algebra: 1832: 17 years old Boole: inspired to put logical statements in a mathematical form 1854: ‘The Laws of thought’ - George Boole. “An investigation of the laws of thought, on which are founded the Mathematical Theories of Logic and Probabilities” Binary Variables: Logic Operations – Truth Table, logic diagram, Boolean Expressions.

15. 15 Objectives of Boole to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolic language of a Calculus to deduce, from these studies, some probable imitations concerning the nature and constitution of the human mind.

16. 16 The Values of Boole’s variables "the Universe" and "Nothing" “True” and “False” 1 and 0 Boole thought: Boole’s Algebra: represented mathematical systemization of human thought.  Not true. But thinking about machines, which think like a human being  powerful machines.

17. 17 Hollerith  IBM 1881:Mechanical Engineer at American Census Bureau, Washington: devised a punched card system for processing the census data of 1890 – the idea from his boss, John Shaw Billings 1882-83: Instructor at MIT 1896: Hollerith’s Tabulating Machines Company  International Business Machines

18. Logic gates: 18 Buffer gate Inverter or NOT gate Logic Symbol Truth Table Algebraic Expression AOut 0101 1010 A, A’ – A Out NOT gate

19. 19 Logic Symbol Truth Table Algebraic Expression Gates (continued): AND and OR GATES A BOut 0 0 1 10 1 00010001 A B Out 0 0 1 10 1 01110111 A.B, AB, A*B, A^B A+B, Av B ABAB Out AND ABAB Out OR

20. 20 Gates (continued): XOR and NAND gates Logic symbol Truth Table Algebraic Expression A BOut 0 0 1 1 0 1 01100110 A BOut 0 0 1 1 0 1 11101110 A  B A.B, (A  B)’ ABAB Out XOR Out ABAB NAND

21. 21 Gates (continued): NOR and XNOR gates Logical Symbol Truth Table Algebraic Expression A BOut 0 0 1 10 1 10001000 A B Out 0 0 1 1 0 1 10011001 A + B A  B ABAB Out NOR ABAB Out XNOR

22. 22 Boolean Gates Gates: AND, OR, NOT XOR, XNOR NAND, NOR Buffer NOT and Buffer are single-input gates. All the others are multi-input gates. The number of inputs to a gate is known as its Fan-in.

23. Boolean Algebra Boolean Algebra uses Boolean variables. A Boolean variable can have only two possible values: 0 or 1. Boolean operations: any of the operations defined by logic gates Property of CLOSURE: If A and B are Boolean variables, A + B and A.B are also Boolean variables.

24. Identity Element identity element (or neutral element) is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them. This is used for groups and related concepts. (Ref: http://en.wikipedia.org/wiki/Identity_element)http://en.wikipedia.org/wiki/Identity_element OR operation: 0 is the Identity Element for OR: 1 + 0 = 1; 0 + 0 = 0 AND operation: 1 is the Identity Element for AND: 1.1 = 1; 0.1 = 0 24

25. 25 1 x + 0 = x x. 0 = 0 2 x + 1 = 1 x. 1 = x 3 x + x = x (Idempotency) x. x = x (Idempotency) 4 x + x’ = 1 x. x’ = 0 5 x + y = y + x (Commutativity) x. y = y. x (Commutativity) 6x + (y + z) = (x + y) + z (Associativity) x. (y. z) = (x. y). z (Associativity) Boolean identities:

26. 26 Continuation of Boolean Identities: 7 x.(y + z) = x.y + x.z (Distributive) x + (y.z)=(x + y).(x + z) (Distributive) 8 (x + y)’ = x’.y’ (De Morgan’s Theorem) (x.y)’ = x’ + y’ (De Morgan’s Theorem) 9 (x’)’ = x 10 x + xy = xx (x + y) = x 11 x + x’y = x + yx (x’ + y) = xy 12 xy + xy’ = x(x + y).(x + y’) = x 13xy + x’z + yz = xy + x’z (Consensus Theorem) (x+y)(x’+z)(y+z)=(x+y)(x’+z) (Consensus Theorem)

27. 27 A few comments: 7(b) x + yz = (x + y)(x + Z) RHS = x + xy + xz + yz = x(1 + y + z) + yz = x + yz = LHS Example 7(b):

28. 28 De Morgan’s Theorems 8(a) (x + y)’ = x’.y’ De Morgan’s Theorem: XyXy F NOR XYXY F

29. 29 x yx + y (x+y)’ x’ y’x’. y’ 0 0 0 1 1 1 1 0 1 1 0 1 0 0 1 0 1 0 0 1 0 1 1 1 0 0 0 0 Proof: Continuation of the proof: De’ Morgan’s theorem

30. Another Method for proving De Morgan’s theorem …1 Identity 4(a): A + A’ = 1 Identity 4(b): A.A’ = 0 To prove: A is the complement of B  prove that (i) A + B = 1 and (ii) A.B = 0  To prove: (x + y)’ = x’.y’, consider A = x’.y’ and B = x + y; Prove that (i) x’.y’ + (x + y) = 1 and (ii) x’.y’.(x + y) = 0  A is the complement of B. 30

31. Another Method for proving De Morgan’s theorem …2 Proof: (i) LHS = x’.y’ + (x + y) = (x’.y’ + x) + y = x + y’ + y by using Th. 11(a) = x + 1 by using Th. 4(a) = 1 by using Th. 2(a) (ii) LHS = x’.y’.(x + y) = x.x’.y’ + x’.y’.y = 0.y’ + x’.0 by using Th. 4(b) = 0 Hence (x + y)’ = x’.y’. 31

32. 32 8(b) (x y)’ = x’ + y’; The proof is similar to that for 8(a). 10(b) LHS = x (x + y) = x + xy = x (1 + y) = x = RHS Examples 8(b) and 10 (b):

33. 33 Examples 11(a): 11 (a) LHS = x + x’y Use x = x.(1 + y) by using 1 + y = 1 and x.1 = x = x + x.y = x.x + x.y by using x = x.x = x.x + x.y + x.x’ by using x.x’ = 0 and A + 0 = A LHS = (x.x + x.y + x.x’) + x’.y = x.(x + y) + x’.(x + y) = (x + x’).(x + y) = x + y by using x + x’ = 1 and 1.B = B

34. Examples 11(b) and 12: 11(b) x.(x’ + y) = x.x’ + x.y = x.y by using x.x’ = 0 and 0 + A = A 12 (b) (x + y).(x + y’) = x + x.y’ + x.y + y.y’ = x.(1 + y’ + x) by using x.x’ = 0 and 0 + A = A = x by using 1 + y’ + x = 1 and x.1 = x 34

35. 35 13 (a) LHS = x.y + x’.z + y.z = x.y + x’.z + (x + x’).y.z = x.y.(1 + z) + x’.z.(1 + y) = xy + x’z = RHS 13 (b) LHS = (x + y).(x’ + z).(y + z) = (x.y + x.z + y + y.z).(x’ + z) = (x.z + y.(1 + x + z)). (x’ + z) = (y + x.z).(x’ + z) = x’.y + y.z + x.x’.z + x.z = xx’ + x’y + yz + xz = x’.(x + y) + z.(x + y) = (x + y).(x’ + z) = RHS Example 13:

36. 36 Boolean Algebra Closure: X + Y, X.Y Identity Element: 0, 1 Commutation: X + Y = Y + X, X.Y = Y.X Distribution: X.(Y + Z) = X.Y + X.Z, X + (Y.Z) = (X + Y).(X + Z) Complements: X + X’ = 1, X.X’ = 0 Idempotency X + X = X, X.X = X

37. Boolean cont… 37 Decimal A B C D 0 0 000 1 0 010 2 0 100 3 0 110 4 1 000 5 1 010 6 1 100 7 1 111