Boolean Functions x 2 x 3 x f mapping truth table

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

Representation of Boolean Functions Conjunctive Normal Form (canonical): Example: AND of OR terms (not unique): )( )()( )()(),,( xxx xxxxxx xxxxxxxxxf    )()(),,( xxxxxxf  x 2 x 3 x f

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

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 ||,,,|,,|1xxxxxxxxxxxx nnn   sums,distinct2 m 2

Logic Circuits Logic GateBuilding Block:

Logic Circuits Logic GateBuilding Block: feed-forward device

Logic Circuits “AND” gate Common Gate:

Logic Circuits “OR” gate Common Gate:

Logic Circuits “XOR” gate Common Gate:

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

inputsoutputs Logic Circuits network of logic gates

Logic Circuits inputsoutputs network of logic gates gate

Example Logic Circuits x y x y z z c s

Example

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

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

Logic Circuits Construct XOR from AND/OR/NOT gates. 4 variables (draw circuit) xx x x...

Example: Logic Gates Models of Computation

Example: Linear Threshold Gates

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

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

Switching Circuits (Shannon, 1938)

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

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 ≤

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

Linear Threshold Gates Useful Model?