# 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 Digital signals Binary (or logical ) values: or, on or off, high or low voltage Wires. Propagate logical values from place to place. Signals "flow" from left to right. Signals and Wires A drawing convention, sometimes violated Actually: flow from producer to consumer(s) of signal Input Output 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 8

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 9

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 Some Functions of 3 Variables x y z AND OR MAJ ODD

10 Universality of AND, OR, NOT Any Boolean function can be expressed using AND, OR, NOT. "Universal." XOR(x,y) = xy' + x'y x Expressing XOR Using AND, OR, NOT 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 ) Notation x' x y x + y Meaning NOT x x AND y x OR 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 2

12 Translate Boolean Formula to Boolean Circuit Use sum-of-products form. XOR(x, y) = xy' + x'y. 3

13 Translate Boolean Formula to Boolean Circuit Use sum-of-products form. MAJ(x, y, z) = x'yz + xy'z + xyz' + xyz. 4

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 5

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. 6

16 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 8

17 ODD Parity Circuit ODD(x, y, z). if odd number of inputs are. otherwise. 9

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 + z 3 z 2 z z + c 3 c 2 c c x 3 x 2 x x y 3 y 2 y y c + y 3 y 2 y y z 3 z 2 z z 2

19 Let's Make an Adder Circuit Goal: x + y = z for 4-bit integers. Step 2. (first attempt) Build truth table. Why is this a bad idea? + c x 3 x 2 x x y 3 y 2 y y z 3 z 2 z z 28-bit adder: rows > # electrons in universe 4-Bit Adder Truth Table c x 2 x y 2 y z 2 z x 3 x y 3 y z 3 z 2 8+ = 52 rows

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 y i c i z i x i 22

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 y i c i c i+ MAJ y i c i z i ODD x i x i 23

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. 24

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. 25

24 Subtractor Subtractor circuit: z = x - y. One approach: new 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 y - x - y y 3 y 2 y y carry z 4-Bit Subtractor Interface 4-Bit Subtractor Implementation 26

25 TOY 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 27

26 n = 8 for main memory 2 n -to- Multiplexer 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 y x x 4 x x 2 x 3 x 4 x 5 x 6 8 to MUX select y x 5 x 6 x 7 x 7 s s s 2 s 2 s s 8-to- Mux Interface 8-to- Mux Implementation 28

27 TOY Arithmetic Logic Unit: Implementation Input 6 Input 2 6 ~ + carry in & MUX 6 ^^ << >> op 2 +, - & 3 ^ <<, >> input 2 subtract shift direction ALU select 29

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

29 Combinational circuits. Sequential vs. 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 3

30 Flip-Flop Flip-flop. A small and useful sequential circuit. Abstraction that "remembers" one bit. Basis of important computer components: memory counter We will consider several flavors. 32

31 SR Flip-Flop What is the value of Q if: S = and R =? Q is surely S Q R 33

32 SR Flip-Flop What is the value of Q if: S = and R =? Q is surely S = and R =? Q is surely S Q R 34

33 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 S Q R 35

34 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 Q R 36

35 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. 37

36 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 38

37 Clock Clock. Fundamental abstraction. regular on-off pulse From some oscillating device, possibly external. Synchronizes operations of different circuit elements. GHz clock means billion pulses per second. cycle time Clock 39

38 How much does it Hert? Frequency is inverse of cycle time. Expressed in hertz. Frequency of Hz means that there is cycle per second. Hence: kilohertz (khz) means cycles/sec. megahertz (MHz) means million cycles/sec. gigahertz (GHz) means billion cycles/sec. terahertz (THz) means trillion cycles/sec. By the way, no such thing as hert Heinrich Rudolf Hertz ( ) 4

39 Clocked SR Flip-Flop Clocked SR Flip-Flop. Same as SR flip-flop except S and R only active when clock is. S clk R Implementation SR flip flop S Q R Clocked SR flip flop S clk R Q Interface Q clk R S 4

40 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 clk R Implementation Q Clocked D flip flop D clk Q Interface Q clk D 42

41 Summary Combinational circuits implement Boolean functions Gates and wires Fundamental building blocks. Truth tables. Describe Boolean functions. Sum-of-products. Systematic method to implement functions. Sequential circuits add "state" to digital hardware. Flip-flop. Represents bit. TOY register. 6 D flip-flops. TOY main memory. 256 registers. Next time: we build a complete TOY computer (oh yes). 43

42 George Boole (85 864) Claude Shannon (96 2) 44

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