6: Combinational Circuits

Transcription

1 Computer Architecture 6: Combinational Circuits Previous two lectures. von Neumann machine. This lectures. Boolean circuits. Later in the course. Putting it all together and building a TOY machine. George Boole (85 864) Claude Shannon (96 2) Introduction to Computer Science Robert Sedgewick and Kevin Wane 2 Digital Circuits Abstract Switch What is a digital sstem?! Digital: signals are or.! Analog: signals var continuousl. Wh digital sstems?! Accurac and reliabilit.! Staggeringl fast and cheap. Basic abstractions.! On, off.! Switch that can turn something on or off. Digital circuits and ou.! Computer microprocessors.! Antilock brakes.! Cell phones. Abstract switch.! Electrical (transistor).! Electro-mechanical (rela).! Hdraulic (water valve). Hdraulic valve.! 3 connections: input, output, control.! Pressure on control pushes on a piston that turns on water flow from input to output. valve is alwas entirel open or closed amplification, restoring logic design Control! Control pipe affects output pipe, but output does not affect control; establishes forward flow of information over time. In Out 3 4

2 Computer Computer Hdraulic computer. Connect hdraulic valves and pipe. Electrical computer. Same fundamental abstractions. Abstraction Hdraulic Computer Electrical Computer open Connector Pipe Wire closed Signal Water pressure Voltage Switch Hdraulic valve Metal-oide transistor Hdraulic OR Gate Reference: W. D. Hill, The Pattern on the Stone. 5 6 Wires Logic Gates Wires.! Propagate logical values from place to place.! Signals "flow" from producers to consumers.! Tpical drawing convention: "flow" from left to right. Logical gates.! Fundamental building blocks. ' Input Output NOT AND OR 7 8

3 Multiwa AND Gates Multiwa OR Gates AND(,, 2, 3, 4, 5, 6, 7 ).! if all inputs are.! otherwise. OR(,, 2, 3, 4, 5, 6, 7 ).! if at least one input is.! otherwise. 9 Boolean Algebra Truth Table Histor.! Developed b Boole to solve mathematical logic problems (847).! Shannon first applied to digital circuits (937). Basics.! Boolean variable: value is or.! Boolean function: function whose inputs and outputs are,. Truth table.! Sstematic method to describe Boolean function.! One row for each possible input combination.! N inputs! 2 N rows. AND(, ) Relationship to circuits.! Boolean variables: signals.! Boolean functions: circuits. AND Truth Table AND 3

4 Truth Table for Functions of 2 Variables Truth Table for Functions of 3 Variables Truth table.! 6 Boolean functions of 2 variables. ever 4-bit value represents one Truth Table for All Boolean Functions of 2 Variables ZERO NOR AND XOR Truth Table for All Boolean Functions of 2 Variables EQ ' ' NAND OR ONE Truth table.! 6 Boolean functions of 2 variables. ever 4-bit value represents one! 256 Boolean functions of 3 variables. ever 8-bit value represents one! 2^(2^N) Boolean functions of N variables! Some Functions of 3 Variables z AND OR MAJ ODD 4 5 Universalit of AND, OR, NOT Sum-of-Products An Boolean function can be epressed using AND, OR, NOT.! {AND, OR, NOT} are "universal."! E: XOR(,) = ' '. Notation ' Epressing XOR Using AND, OR, NOT ' ' ' ' ' ' XOR Eercise: {AND, NOT}, {OR, NOT}, {NAND}, {AND, XOR} are universal. Hint. DeMorgan's law: ('')' =. Meaning NOT AND OR An Boolean function can be epressed using AND, OR, NOT.! Sum-of-products is sstematic procedure. form AND term for each in Boolean function OR terms together Epressing MAJ Using Sum-of-Products z MAJ 'z 'z z' z 'z 'z z' z 6 7

5 Translate Boolean Formula to Boolean Circuit Translate Boolean Formula to Boolean Circuit Use sum-of-products form.! XOR(, ) = ' '. Use sum-of-products form.! MAJ(,, z) = 'z 'z z' z. 8 9 Simplification Using Boolean Algebra Epressing a Boolean Function Using AND, OR, NOT Man possible circuits for each Boolean function.! Sum-of-products not necessaril optimal in: number of gates (space) depth of circuit (time)! MAJ(,, z) = 'z 'z z' z = z z. Ingredients.! AND gates.! OR gates.! NOT gates.! Wire. Instructions.! Step : represent input and output signals with Boolean variables.! Step 2: construct truth table to carr out computation.! Step 3: derive (simplified) Boolean epression using sum-of products.! Step 4: transform Boolean epression into circuit. size = 8, depth = 4 size = 4, depth = 3 2 2

6 ODD Parit Circuit ODD Parit Circuit ODD(,, z).! if odd number of inputs are.! otherwise. ODD(,, z).! if odd number of inputs are.! otherwise. Epressing ODD Using Sum-of-Products z ODD ''z 'z' 'z' z ''z 'z' 'z' z Goal: = z for 4-bit integers.! We build 4-bit adder: 9 inputs, 4 outputs.! Same idea scales to 28-bit adder.! Ke computer component Goal: = z for 4-bit integers. Step 2. (first attempt)! Build truth table.! Wh is this a bad idea? c z 3 z 2 z z Step. Represent input and output in binar. 28-bit adder: rows > # electrons in universe! c z 3 z 2 z z z 3 z 2 z z c Bit Adder Truth Table 3 2 z 3 z 2 z z 2 8 = 52 rows! 24 25

7 Goal: = z for 4-bit integers. c 3 c = c 2 c Goal: = z for 4-bit integers. c 3 c = c 2 c Step 2. (do one bit at a time)! Build truth table for carr bit. 3 2 Step 3.! Derive (simplified) Boolean epression. 3 2! Build truth table for summand bit. z 3 z 2 z z z 3 z 2 z z Carr Bit Summand Bit Carr Bit Summand Bit i i c i c i i i c i z i i i c i c i MAJ i i c i z i ODD Goal: = z for 4-bit integers. Goal: = z for 4-bit integers. Step 4.! Transform Boolean epression into circuit.! Chain together -bit adders. Step 4.! Transform Boolean epression into circuit.! Chain together -bit adders

8 Subtractor Summar Subtractor circuit: z = -.! One approach: design like adder circuit.! Better idea: reuse adder circuit.! Recall 2's complement: to negate an integer, flip bits, then add carr z 3 z 2 z z Lessons for software design appl to hardware design!! Interface describes behavior of circuit.! Implementation gives details of how to build it. Laers of abstraction appl with a vengeance!! On/off.! Controlled switch (transistor).! Gates (AND, OR, NOT).! Boolean circuit (MAJ, ODD).! Adder.! Arithmetic logic unit.!.! TOY machine. 4-Bit Subtractor Interface 4-Bit Subtractor Implementation 3 3