1. Boolean Functions 1111 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 0 0 0 1 1 1 x 2 x 3 x f mapping truth table

2. Representation of Boolean Functions Disjunctive Normal Form (canonical): 1111 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 0 1 0 1 0 Example: OR of AND terms (not unique): 321321321321 ),,(xxxxxxxxxxxxf  3231321 ),,(xxxxxxxf  1 x 2 x 3 x f

3. Representation of Boolean Functions Conjunctive Normal Form (canonical): Example: AND of OR terms (not unique): )( )()( )()(),,( 321 321321 321321321 xxx xxxxxx xxxxxxxxxf    )()(),,( 213321 xxxxxxf  1111 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 x 2 x 3 x f

4. Representation of Boolean Functions Example: XOR of AND terms: unique? 1111 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 0 0 0 1 0 1 0 1 x 2 x 3 x f )1 as represent literals; positive(only  xx 3213231321 ),,(xxxxxxxxxxf 

5. representation is unique. Dependence on variables is explicit. XOR of AND terms: Representation of Boolean Functions For m variables, express a boolean function as the sum of some combination of the product terms: m 2 3211312121 ||,,,|,,|1xxxxxxxxxxxx nnn   sums,distinct2 m 2

6. Logic Circuits Logic GateBuilding Block:

7. Logic Circuits Logic GateBuilding Block: feed-forward device

8. Logic Circuits “AND” gate 0 0 0 1 Common Gate: 0 0 1 1 0 1 0 1

9. Logic Circuits “OR” gate 0 0 1 1 0 1 0 1 0 1 1 1 Common Gate:

10. Logic Circuits “XOR” gate 0 0 1 1 0 1 0 1 0 1 1 0 Common Gate:

11. Logic Circuits “NOT” gate 1 0 Common Gate: 0 1

12. inputsoutputs Logic Circuits network of logic gates

13. Logic Circuits inputsoutputs network of logic gates gate

14. Example Logic Circuits x y x y z z c s

15. 0 1 0 1 1 1 0 1 1 0 1 Example

16. Logic Circuits Size: number of gates. Depth: longest path from an input to an output. Measures x y x y z z c s size = 5, depth = 3

17. Logic Circuits Why study logic circuits? It all seems simple enough.... 1 0 1 1 0 1 0 0 Construct XOR from AND/OR/NOT gates. 12 3 111 0 0 0 0 1 1 1 0 0 1 1 0 0 1 0 1 0 1 0 1 0 xx x 3 variables (draw circuit)

18. Logic Circuits Construct XOR from AND/OR/NOT gates. 4 variables (draw circuit) 1 1 1 1 1 1 1 1 12 3 011 1 0 0 0 1 1 1 0 1 0 0 1 0 1 0 0 1 0 1 1 0 xx x 4 1 0 0 0 1 0 1 1 x...

19. Example: Logic Gates Models of Computation

20. Example: Linear Threshold Gates

21. Models of Computation Example: Comparators and Balancers x y min(x, y) max(x, y) x y

22. Models of Computation Example: Switching Circuits a b c ed SD dcbacedeabf 

23. Switching Circuits (Shannon, 1938)

24. Example: Encoding FSMs Systems with states analyzed. Finite State Machine input current state next state A B D F

25. Logic Circuits Construct XOR from AND/OR/NOT gates. construction: lower bound: XOR of n variables gates For n > 4, optimal size is unknown. ≤ optimal size ≤

26. Linear Threshold Gates 1 x 2 x n x 1 w 2 w n w 0 w...

27. Linear Threshold Gates Useful Model?