6.1 Combinational Circuits. George Boole ( ) Claude Shannon ( )

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1 6. Combinational Circuits George Boole (85 864) Claude Shannon (96 2)

2 Signals and Wires Digital signals Binary (or logical ) values: or, on or off, high or low voltage Wires. Propagate digital signals from place to place. Signals "flow" from left to right. A drawing convention, sometimes violated Actually: flow from producer to consumer(s) of signal 2

3 Logic Gates Logical gates. Fundamental building blocks. x x' x y xy x y x + y NOT AND OR 3

4 Multiway AND Gates AND(x, x, x 2, x 3, x 4, x 5, x 6, x 7 ). if all inputs are. otherwise. 4

5 Multiway OR Gates OR(x, x, x 2, x 3, x 4, x 5, x 6, x 7 ). if at least one input is. otherwise. 5

6 Boolean Algebra History. Developed by Boole to solve mathematical logic problems (847). Shannon master's thesis applied it to digital circuits (937). Basics. Boolean variable: value is or. Boolean function: function whose inputs and outputs are,. Relationship to circuits. Boolean variables: signals. Boolean functions: circuits. "possibly the most important, and also the most famous, master's thesis of the [2th] century" --Howard Gardner 6

7 Truth Table Truth table. Systematic method to describe Boolean function. One row for each possible input combination. N inputs 2 N rows. AND Truth Table x y AND(x, y) AND 7

8 Truth Table for Functions of 2 Variables Truth table. 6 Boolean functions of 2 variables. every 4-bit value represents one Truth Table for All Boolean Functions of 2 Variables x y ZERO AND x y XOR OR Truth Table for All Boolean Functions of 2 Variables x y NOR EQ y' x' NAND ONE 8

9 Truth Table for Functions of 3 Variables Truth table. 6 Boolean functions of 2 variables. every 4-bit value represents one 256 Boolean functions of 3 variables. every 8-bit value represents one 2^(2^N) Boolean functions of N variables! x Some Functions of 3 Variables y z AND OR MAJ ODD 9

10 Universality of AND, OR, NOT Any Boolean function can be expressed using AND, OR, NOT. "Universal" Example: XOR(x,y) = xy' + x'y Expressing XOR Using AND, OR, NOT Notation x' x y x + y Meaning NOT x x AND y x OR y x y x' y' x'y xy' x'y + xy' XOR Exercise. Show {AND, NOT}, {OR, NOT}, {NAND}, {AND, XOR} are universal. Hint. Use DeMorgan s Law: (xy) = (x + y ) and (x + y) = (x y )

11 Sum-of-Products Any Boolean function can be expressed using AND, OR, NOT. Sum-of-products is systematic procedure. form AND term for each in truth table of Boolean function OR terms together x'yz Expressing MAJ Using Sum-of-Products z xyz' xyz xy'z MAJ y x x'yz + xy'z + xyz' + xyz

12 Translate Boolean Formula to Boolean Circuit Use sum-of-products form. XOR(x, y) = xy' + x'y. 2

13 Translate Boolean Formula to Boolean Circuit Use sum-of-products form. MAJ(x, y, z) = x'yz + xy'z + xyz' + xyz. 3

14 Simplification Using Boolean Algebra Many possible circuits for each Boolean function. Sum-of-products not necessarily optimal in: number of gates (space) depth of circuit (time) MAJ(x, y, z) = x'yz + xy'z + xyz' + xyz = xy + yz + xz. size = 8, depth = 3 size = 4, depth = 2 4

15 Expressing a Boolean Function Using AND, OR, NOT Ingredients. AND gates. OR gates. NOT gates. Wire. Instructions. Step : represent input and output signals with Boolean variables. Step 2: construct truth table to carry out computation. Step 3: derive (simplified) Boolean expression using sum-of products. Step 4: transform Boolean expression into circuit. 5

16 6 ODD Parity Circuit ODD(x, y, z). if odd number of inputs are. otherwise. x'y'z Expressing ODD Using Sum-of-Products z xy'z' xyz x'yz' ODD y x x'y'z + x'yz' + xy'z' + xyz

17 ODD Parity Circuit ODD(x, y, z). if odd number of inputs are. otherwise. 7

18 Let's Make an Adder Circuit Goal: x + y = z for 4-bit integers. We build 4-bit adder: 9 inputs, 4 outputs. Same idea scales to 28-bit adder. Key computer component. Step. Represent input and output in binary x 3 x 2 x x y 3 y 2 y y + z 3 z 2 z z + + x 3 x 2 x x y 3 y 2 y y z 3 z 2 z z 8

19 Let's Make an Adder Circuit Goal: x + y = z for 4-bit integers. c Step 2. (first attempt) Build truth table. Why is this a bad idea? + x 3 y 3 z 3 x 2 y 2 z 2 x y z x y z 28-bit adder: rows > # electrons in universe! 4-Bit Adder Truth Table c x 3 x 2 x x y 3 y 2 y y z 3 z 2 z z = 52 rows! 9

20 Let's Make an Adder Circuit Goal: x + y = z for 4-bit integers. Step 2. (do one bit at a time) Build truth table for carry bit. Build truth table for summand bit. + c 3 c = c 2 c x 3 x 2 x x y 3 y 2 y y z 3 z 2 z z Carry Bit x i y i c i c i+ Summand Bit x i y i c i z i 2

21 Let's Make an Adder Circuit Goal: x + y = z for 4-bit integers. Step 3. Derive (simplified) Boolean expression. + c 3 c = c 2 c x 3 x 2 x x y 3 y 2 y y z 3 z 2 z z Carry Bit Summand Bit x i y i c i c i+ MAJ x i y i c i z i ODD 2

22 Let's Make an Adder Circuit Goal: x + y = z for 4-bit integers. Step 4. Transform Boolean expression into circuit. Chain together -bit adders. 22

23 Let's Make an Adder Circuit Goal: x + y = z for 4-bit integers. Step 4. Transform Boolean expression into circuit. Chain together -bit adders. 23

24 Subtractor Subtractor circuit: z = x - y. One approach: design like adder circuit. Better idea: reuse adder circuit. 2's complement: to negate an integer, flip bits, then add x 3 x 2 x x z 3 x + z 2 z - x - y y 3 z y 2 y y y 4-Bit Subtractor Interface 4-Bit Subtractor Implementation 24

25 Arithmetic Logic Unit: Interface ALU Interface. Add, subtract, bitwise and, bitwise xor, shift left, shift right, copy. Associate 3-bit integer with 5 primary ALU operations. ALU performs operations in parallel control wires select which result ALU outputs op 2 Input 6 +, - & ALU 6 ^ <<, >> input 2 Input ALU select shift direction subtract 25

26 Arithmetic Logic Unit: Implementation Input Input ~ + carry in & 6 MUX ^ << >> op 2 +, - & 3 ^ <<, >> input 2 subtract shift direction ALU control 26

27 2 n -to- Multiplexer n = 8 for main memory 2 n -to- multiplexer. n select inputs, 2 n data inputs, output. Copies "selected" data input bit to output. x x x 2 x 3 x x 4 y x x 2 x 3 x 4 8 to MUX y x 5 x 6 x 5 x 7 x 6 x 7 s s s 2 s 2 s s 8-to- Mux Interface 8-to- Mux Implementation 27

28 6.2: Sequential Circuits S Q R COS 26: General Computer Science

29 Sequential vs. Combinational Circuits Combinational circuits. Output determined solely by inputs. Can draw solely with left-to-right signal paths. Sequential circuits. Output determined by inputs AND previous outputs. Feedback loop. S R Q 29

30 SR Flip-Flop What is the value of Q if: S = and R =? Q is surely. S = and R =? Q is surely. S = and R =? Q is possibly... or possibly. S old Q Q R While S = R =, Q remembers what it was the last time S or R was. 3

31 SR Flip-Flop SR Flip-Flop. S =, R = (set) Flips bit on. S =, R = (reset) Flops bit off. S = R = Status quo. S = R = Not allowed. S Q SR flip flop R S R Q Implementation Interface 3

32 Clock Clock. Fundamental abstraction. regular on-off pulse External analog device. Synchronizes operations of different circuit elements. GHz clock means billion pulses per second. cycle time Clock 32

33 Clocked SR Flip-Flop Clocked SR Flip-Flop. Same as SR flip-flop except S and R only active when clock is. S Cl R SR flip flop S Q R Implementation Clocked SR flip flop S Cl R Q Interface 33

34 Clocked D Flip-Flop Clocked D Flip-Flop. Output follows D input while clock is. Output is remembered while clock is. D Cl Clocked SR flip flop S Cl R Implementation Q Clocked D flip flop D Cl Q Interface 34

35 Stand-Alone Register k-bit register. Stores k bits. Register contents always available on output. If write enable is asserted, k input bits get copied into register. Ex: Program Counter, 6 TOY registers, 256 TOY memory locations. x D Cl Q y x D Cl Q y 6 reg 6 write data write enable read data x 5 Write D Cl Q y 5 6-bit Register Interface 6-bit Register Implementation 35

36 Registers TOY registers: fancy 6 x 6-bit register file. Want to be able to read two registers, and write to a third in the same instructions: R R2 + R3. 3 address inputs, data input, 2 data outputs. Add decoders and muxes for additional ports. Write Data Write Address A Address R R R2 R3 R4 R5 R8 R9 RA RB RC RD 6 6 A Data B Data B Address 4 R6 R7 RE RF Cl Write 36

37 37 Main Memory TOY main memory: 256 x 6-bit register file. 6 Cl Write Address 8 Read Data 6 Write Data F E D B A 9 C F E D B A 9 C F 2E 2D 2B 2A 29 2C F 3E 3D 3B 3A 39 3C F 4E 4D 4B 4A 49 4C F 5E 5D 5B 5A 59 5C F 6E 6D 6B 6A 69 6C F 7E 7D 7B 7A 79 7C F 8E 8D 8B 8A 89 8C F 9E 9D 9B 9A 99 9C 98 A7 A6 A5 A3 A2 A A4 A AF AE AD AB AA A9 AC A8 B7 B6 B5 B3 B2 B B4 B BF BE BD BB BA B9 BC B8 C7 C6 C5 C3 C2 C C4 C CF CE CD CB CA C9 CC C8 D7 D6 D5 D3 D2 D D4 D DF DE DD DB DA D9 DC D8 E7 E6 E5 E3 E2 E E4 E EF EE ED EB EA E9 EC E8 F7 F6 F5 F3 F2 F F4 F FF FE FD FB FA F9 FC F8

38 6.3: Machine Architecture

39 The TOY Datapath 8 6 pc for branch, jump PC + addr for loads, stores pc for jal Memory Addr W Data R Data W 8 load IR op d s t result of arithmetic, logic, or addr for load addr 2 addr Registers W Data B Data W Addr A Addr B Addr A Data W Cond Eval A L U = > pc + 8 store data 39

40 Control Control: controls components, enables connections. Input: opcode, clock, conditional evaluation. (green) Output: control wires. (orange) Registers Cond Eval = > W Data PC Memory Addr R Data IR op d s 4 2 A Data B Data W Addr A Addr A L U 5 + W Data W t B Addr W -bit counter Opcode Execute Fetch Clock Control 4

41 Summary Combinational circuits: how to compute things And, Or, Not primitives sufficient for any Boolean function Systematic method: truth tables and sum-of-products Examples Majority Binary adder Multiplexor Sequential circuits: where to put things Flip flop primitive holds one bit Many flip flops make a register (6 for TOY) Many registers make a register file Lots and lot of registers make a memory (256 for TOY) A whole computer Uses combinational circuits to perform computations Uses sequential circuits to store results Uses a little of each for control 4

42 The final secret A" B" OUT" A" B" OUT" A" OUT" All three of our logic primitives can be made using a single* type of electronic primitive: the transistor! *not counting the passive resistors 42

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