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CS/COE0447

CS/COE0447. Computer Organization & Assembly Language. Logic Design Appendix C. 1. Outline. • Example to begin: let’s implement a MUX • Gates, Truth Tables, and Logic Equations • Combinatorial Logic • Constructing an ALU • Memory Elements: Flip-flops, Latches, and Registers. 2.

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CS/COE0447

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  1. CS/COE0447 Computer Organization & Assembly Language Logic Design Appendix C 1

  2. Outline • Example to begin: let’s implement a MUX • Gates, Truth Tables, and Logic Equations • Combinatorial Logic • Constructing an ALU • Memory Elements: Flip-flops, Latches, and Registers 2

  3. Logic Gates A 2-input AND Y=A&B Y B A Y=A|B 2-input OR Y B A 2-input NAND Y=~(A&B) Y B A 2-input NOR Y=~(A|B) Y B 3

  4. Multiplexor If S then C=B else C = A A 0 C 1 B How many bits is S? S 1, since it is choosing between 2 values Let’s see how to implement a 2-input MUX using gates. Hint: the answer uses AND gates, an OR gate, and one INVERTER Answer in lecture 4

  5. Computers and Logic • Digital electronics operate with only two voltage levels of interest: high and low voltage. – All other voltage levels are temporary and occur while transitioning between values • We’ll talk about them as signals that are – Logically true; 1; asserted – Logically false; 0, deasserted • 0 and 1 are complements and inverses of each other 5

  6. Combinational vs. Sequential Logic • Combinational logic – A function whose outputs depend only on the current input • Sequential logic – Memory elements, i.e., state elements – Outputs are dependent on current input and current state – Next state is also dependent on current input and current state 6

  7. Combinational Logic … … inputs outputs 7

  8. Sequential Logic … … inputs outputs next state current state clock 8

  9. • The next set of topics [until the sequential logic picture we just saw pops up again] will only be about combinatorial logic 9

  10. Functions Implemented Using Gates ? … … inputs outputs Combinatorial logic blocks implement logical functions, mapping inputs to outputs 10

  11. Describing a Function • OutputA= F(Input0, Input1, …, InputN-1) • OutputB= F’(Input0, Input1, …, InputN-1) • OutputC= F’’(Input0, Input1, …, InputN-1) • Methods – Truth table (since combinatorial logic has no memory, it can be completely specified by a truth table) – …[in a moment] 11

  12. Truth Table Input Output A B Cin 0 S Cout 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 12

  13. Truth Tables • In a truth table, there is one row for every possible combination of values of the inputs • Specifically, if there are N inputs, the possible combinations are the binary numbers 0 through 2EN - 1. For example: – 3 bits (0-7): 000 through 111 – 4 bits (0-15): 0000 through 1111 – 5 bits (0-31): 00000 through 11111 • While we could always use a truth table, they quickly grow in size and become hard to understand and work with • Boolean logic equations are more succinct 13

  14. Describing a Function • OutputA= F(Input0, Input1, …, InputN-1) • OutputB= F’(Input0, Input1, …, InputN-1) • OutputC= F’’(Input0, Input1, …, InputN-1) • Methods – Truth table – Boolean logic equations • Sum of products • Products of sums 14

  15. Truth Table and Equations Input Output A B Cin 0 S Cout 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 • S = A’B’Cin+A’BCin’+AB’Cin’+ABCin • Cout= A’BCin+AB’Cin+ABCin’+ABCin Column in the truth table Each output has its own…? 15 And its own Boolean equation

  16. Truth Tables and Equations • All functions specified by truth tables can also be specified by Boolean formulas [and vice versa] • So, let’s look more closely at Boolean algebra 16

  17. Boolean Algebra • Boole, George (1815~1864): mathematician and philosopher; inventor of Boolean Algebra, the basis of all computer arithmetic • Binary values: 0, 1 • Two binary operations: AND (/), OR () – AND is also called the logical product since its result is 1 only if both operands are 1 – OR is also called the logical sum since its result is 1 if either operand is 1 • One unary operation: NOT (~) 17

  18. Laws of Boolean Algebra Identity, Zero, and One laws – aa = a+a = a – a1 =a; a+0 = a [“copy” operations] – a0 =0; a+1 = 1 [deassert by ANDing with 0; assert by ORing with 1] Inverse – aa = 0; a+a = 1 Commutative – ab = ba – a+b = b+a Associative – a(bc) = (ab)c – a+(b+c) = (a+b)+c Distributive – a(b+c) = ab + ac – a+(bc) = (a+b)(a+c) • • • • • 18

  19. Laws of Boolean Algebra • De Morgan’s laws – ~(a+b) = ~a~b – ~(ab) = ~a+~b • More… – a+(ab) = a – a(a+b) = a – ~~a=a • You’ll see this again in CS441 and CS1502 19

  20. Examples To get used to Boolean equations To see the relationships among Truth Tables, Boolean Equations, and hardware implementations in gates To see that a “sum of products” formula can always be derived from a truth table To see that equations can often be simplified • • • • 20

  21. Example equation • E = (A’ B C) + (A B’ C) + (A B C’) • What is the value of the equation if A = 1, B = 0 and C = 0? • E = (1’ 0 0) + (1 0’ 0) + (1 0 0’) • E = (0 0 0) + (1 1 0) + (1 0 1) = 0 • What is the value of the equation if A = 0, B = 1, and C = 1? • E = (0’ 1 1) + (0 1’ 1) + (0 1 1’) • E = (1 1 1) + (0 0 1) + (0 1 0) = 1 21

  22. Truth Table for E A B C D E F 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 1 1 1 1 0 1 0 0 1 0 0 1 0 1 1 1 0 1 1 0 1 1 0 1 1 1 1 0 1 •You can read our equation for E right from the truth table: • E = (A’ B C) + (A B’ C) + (A B C’) •These are the three cases when E is 1. •Now, give a Boolean equation for F: F = A B C 22

  23. Give a Boolean Equation for D A B C D E F 0 0 0 0 0 0 0 0 1 1 0 0 0 1 0 1 0 0 0 1 1 1 1 0 1 0 0 1 0 0 1 0 1 1 1 0 1 1 0 1 1 0 1 1 1 1 0 1 D = (A’ B’ C) + (A’ B C’) + (A’ B C) + (A B’ C’) + (A B’ C) + (A B C’) + (A B C) There are many logically equivalent equations (in general) D = (A’ B’ C’)’ [D is true in all cases except A=0 B=0 C=0.] Apply DeMorgan’s law: D = A’’ + B’’ + C’’ = A + B + C 23

  24. Example: boolean equation of a circuit First add the boolean equations at the output for each AND gate A•B A B Y C B•C 24

  25. Example: Next add the Boolean equations at the output for the OR gate A B A•B (A•B) + (B•C) Y C B•C The Boolean equation for the complete logic circuit is: Y = (A•B)+(B•C) 25

  26. Example: Truth Table Y = (A•B)+(B•C) A B C Y 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 0 1 0 1 0 1 1 0 1 1 1 1 1 Reading an equation from the Table:Y = (A’ B C) + (A B C’) + (A B C) The equations are logically equivalent: one way to see this is to consider each row in the truth table. If the two equations have the same outputs for each input, then they are logically equivalent. 26

  27. Example: MUX (A S’) A (A S’) + (B S) C B If the equation below were implemented directly: If the equation below were implemented directly: four (3 four (3- -input) AND gates and one (4 input) AND gates and one (4- -input) OR gate would be needed gate would be needed (B S) input) OR S A B S C Again, the two formulas are equivalent [next slide] 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 1 1 1 1 C = (A’ B S) + (A B’ S’) + (A B S’) + (A B S) Equation read from the Table: 27

  28. Example: MUX BS C = (A S’) + (B S) C = (A’ B S) + (A B’ S’) + (A B S’) + (A B S) AS’ Methods perform such simplifications automatically If B ==0: (AS’) + 0 If B == 1: 0 + (AS’) So, this is the same as AS’ A B S C You can see they are equivalent by comparing values for each row 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 1 0 1 28 1 1 1 1

  29. Expressive Power • Any Boolean algebra function can be constructed using AND gates, OR gates, and Inverters – [For your interest: NAND and NOR are both universal: any logic function can be built with just that one gate type] • There are “canonical forms” for Boolean functions: all equations can be expressed in these forms • This made it possible to create translation programs that, given a logic equation or truth table as input, can automatically design a circuit that implements it 29

  30. Outline • Example to begin: let’s implement a MUX! • Gates, Truth Tables, and Logic Equations • Combinatorial Logic • Constructing an ALU • Memory Elements: Flip-flops, Latches, and Registers 30

  31. Since we were talking about MUXs… • How are larger MUXs implemented – Wider inputs than 1 bit – More choices 31

  32. A 32-bit wide 2-to-1 Multiplexor 1-bit input to to all 32 MUXs Each MUX is the same; just like the one we saw earlier Choosing between 2 32-bit wide buses Bus: collection of data lines treated as a single value. E.g., MUX controlled by MemtoReg. 32

  33. Use a Decoder to build a MUX with more choices Decoder n bit input value and 2^n outputs I1 I2 O3 O2 O1 O0 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 1 1 0 0 0 This is a 2-to-4 decoder Appendix C shows the truth table for a 3-to-8 decoder 33

  34. Decoder: implementation with gates Decoder n bit input value and 2^n outputs A = X • Y B = X • Y C = X • Y D = X • Y X Y A B C D 0 0 0 0 0 1 0 1 0 0 1 0 1 0 0 1 0 0 1 1 1 0 0 0 X Y A B C C D 34

  35. N input MUX using a decoder • Example in lecture 35

  36. Implementing Combinatorial Logic • PLA (Programming Logic Array) – A direct implementation of sum of products form pla.html (thanks to: www.cs.umd.edu/class/spring2003/cmsc311/Notes/C omb/pla.html) • ROM (Read Only Memory) – Interpret the truth table as fixed values stored in memory • Using logic gate chips (74LS…) 36

  37. 74LS Series • Chips contain several logic gates 32 08 04 1 2 1 2 3 3 13 12 11 10 4 5 4 5 6 6 9 8 5 6 9 9 8 8 10 10 3 4 1 2 12 13 12 13 11 11 SN 74LS32 Quad 2-input OR gate SN 74LS04 Hex inverter gate SN 74LS08 Quad 2-input AND gate 37

  38. ALU Symbol • Note that it’s combinational logic 38

  39. Building a 1-bit ALU • ALU = Arithmetic Logic Unit 39

  40. Building a 1-bit Adder Input Output A B Cin 0 S Cout 0 0 0 0 0 0 1 1 0 0 1 0 1 0 0 1 1 0 1 1 0 0 1 0 1 0 1 0 1 1 1 0 0 1 1 1 1 1 1 • S = A’B’Cin+A’BCin’+AB’Cin’+ABCin • Cout = AB+BCin+ACin (after simplification) • E.g., build a pla 40

  41. Building a 32-bit ALU 41

  42. Implementing “SUB” 42

  43. Implementing “NAND”/”NOR” 43

  44. Implementing “SLT” 44

  45. Implementing “SLT”, cont’d 45

  46. 4-bit datapath “Operation” same for all “Binvert” same for all “Ainvert” same for all Bit 0 Bit 1 0 0 Bit 3 Bit 2 0 0 0 0 46

  47. Supporting “BEQ”/”BNE” • Need a “zero-detector” 47

  48. ALU Symbol • Note that it’s a combinational logic 48

  49. Sequential Logic … … inputs outputs next state current state clock 49

  50. RS Latch • Note that there are feedbacks! 50

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