Computer Systems 1 Fundamentals of Computing Computer Logic

CS1: Week 18 What’s Logic? Truth Tables Simple Logic Gates Simple Logic Circuits Other Logic Gates Other Logic Circuits Computer Systems 1 (2004-2005)

What’s Logic? “The grand aim of all science is to cover the greatest number of empirical facts by logical deduction from the smallest number of hypotheses or axioms.” Albert Einstein (1879 - 1955) “Against logic there is no armor like ignorance.” Laurence J. Peter (1919 - 1988) “A page of history is worth a pound of logic.” Oliver Wendell Holmes Jr. (1841 - 1935) “Logic is like the sword--those who appeal to it shall perish by it.” Samuel Butler (1835 - 1902) “Somebody who thinks logically is a nice contrast to the real world.” The Law of Thumb “Insanity is often the logic of an accurate mind overtaxed.” Oliver Wendell Holmes (1809 - 1894) Computer Systems 1 (2004-2005)

It’s Logical, Captain... Logic is concerned with conditions Conditions are either achieved or not achieved There are two states used within logic: True False This could also be thought on as: ON and OFF 1 and 0 YES and NO Computer Systems 1 (2004-2005)

It’s Logical, Captain... Logic is used in a variety of situations, most importantly: Real life situations Inside the computer To perform logical and arithmetic functions Real life logical situations: There’s a buzzer in your car that sounds when the headlights are on and the door is open The fire alarm installed in your home will go off if it senses heat or smoke If I’m not tired then I will go to the pub tonight Computer Systems 1 (2004-2005)

Logic Gates Each logic gate has it’s own features: Symbol Boolean Algebraic Expression Truth Table Gates can be used to build logic circuits Nothing to do with Bill GATES Even more thankfully, nothing to do with Gareth GATES Each gate has a number of inputs and outputs Usually multiple inputs and ONE output Circuits can be created to perform situation testing and produce and output Computer Systems 1 (2004-2005)

Truth Tables Truth tables are used to represent the functionality of a logic gate or circuit Truth tables are constructed by analysing all possible combinations of values that can be sent to a logic gate or circuit All possible outputs are then calculated Truth tables allow us to show the functionality of a logic gate or circuit We can also derive expressions and simplify complex circuits by analysing the truth tables More on that next week Computer Systems 1 (2004-2005)

Boolean Algebra Named after George Boole English Mathematician Provides a method to express functions and transforms using logical variables Commonly letters of the alphabet A, B, D, X, Y, Z, etc. E.g.- X + Y = Z Logic gates and circuits work on the principles of Boolean logic In the computer TRUE or FALSE is represented by a high or low voltage 1 or 0 Computer Systems 1 (2004-2005)

AND Gate Two or more inputs All inputs must be true to produce a true output E.g.- A AND B must be true All other combinations between inputs result in a false output Boolean expression for AND gate with 2 inputs (X AND Y): X•Y Computer Systems 1 (2004-2005)

OR Gate Two or more inputs At least one input must be true to produce a true output E.g.- A OR B must be true Both inputs being true result in a true output Boolean expression for OR gate with 2 inputs (X OR Y): X+Y Computer Systems 1 (2004-2005)

NOT Gate Usually only one input The value of the input is inverted TRUE becomes FALSE FALSE becomes TRUE E.g.- A is NOT true/false Boolean expression for OR gate with 1 input (X): X Sometimes ~X Computer Systems 1 (2004-2005)

Simple Logic Circuits Circuits comprise of one or more logic gates Gates are joined together Usually the process flow moves left to right Truth tables can be constructed for circuits Helped by deriving Boolean algebra expressions for gates and the output of the circuit Circuits are used to construct useful logical processes The computer (CPU) is a complex logic circuit Computer Systems 1 (2004-2005)

Simple Logic Circuits What’s the truth table? E.g.- Computer Systems 1 (2004-2005)

Simple Logic Circuits What’s the truth table? E.g.- OUTPUT= A+B + A•B Computer Systems 1 (2004-2005)

NAND Gate NAND Two or more inputs NOT AND Two or more inputs If all inputs are true then output is false (0) All other combinations between inputs result in a true output Boolean expression for NAND gate with 2 inputs (X NAND Y): X•Y Computer Systems 1 (2004-2005)

NOR Gate NOR Two or more inputs NOT OR Two or more inputs At least one input must be true to produce a false (0) output If both inputs are false then output becomes true Boolean expression for NOR gate with 2 inputs (X NOR Y): X+Y Computer Systems 1 (2004-2005)

XOR Gate XOR Two or more inputs Exclusive OR Sometimes EOR Two or more inputs Inputs must be different to produce a true output Both inputs being true or false result in a false output Boolean expression for XOR gate with 2 inputs (X XOR Y): X•Y + X•Y Computer Systems 1 (2004-2005)

XNOR Gate XNOR Two or more inputs Exclusive NOR Two or more inputs One input must be true to produce a false output and both inputs must be different Both inputs being true or false result in a true output Boolean expression for XNOR gate with 2 inputs (X XNOR Y): X•Y + X•Y Computer Systems 1 (2004-2005)

Other Logic Circuits Simplify last circuit using NOR gate Becomes: Computer Systems 1 (2004-2005)

Other Logic Circuits E.g.- OUTPUT= X•Y + (Y•Z + Y•Z) X Y Z Computer Systems 1 (2004-2005)

Other Logic Circuits X Y Z E.g.- Computer Systems 1 (2004-2005)

Do you know anything now? What’s Logic? Conditions Truth Tables Simple Logic Gates AND OR NOT Simple Logic Circuits Other Logic Gates NAND NOR XOR XNOR Other Logic Circuits Computer Systems 1 (2004-2005)