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Computer Architecture D75P 34

Why Computers Use Boolean Logic

By now you will have grasped that computers can only understand, store and manipulate two values – binary equivalents of 1 and 0. In real life, these values are held as combinations of high/ground voltage; charged/non-charged capacitors; magnetized/non-magnetized sectors on hard drive; and so on. For programming and modelling purposes, however, we think of these states as being 1 v 0 or true v false.

Counting from 0 to 15 with only four lightbulbs!

Today, all our computers employ a system we call Boolean Logic. Boole’s system works by taking combinations of true/false and analysing them to produce a single result, which will either be true or false. (You may already have some experience of this through searching on the Internet – “Python AND snake NOT Monty” is what is known as a Boolean Expression).

In reality, the inputs are processed by the ALU, which performs the analysis and provides the result. The ALU consists of a number of complex switching mechanisms, called Logic Gates – four basic types and three composite ones, which we will look at in turn.

The basic gates are AND, OR, XOR and NOT.

AND gates will only yield a TRUE result (that is, a binary 1) if all input is TRUE. Therefore, any gate with a false input (even if only one) will produce a FALSE (binary 0) result.

AND gate

OR gates are less fussy. An OR gate will send a TRUE result if any of its input is TRUE. Only if every input is FALSE will it produce a 0 result.

OR gate

XOR (Exclusive OR - sometimes called EOR) looks for differences in inputs – it is used for comparing things. If the inputs are all identical (no matter whether 1 or 0) it produces a 0 result. If any of them are different from all the others, it generates a 1.

XOR gate

NOT gates, on the other hand, do just one thing. A NOT gate accepts only a single input, either TRUE or FALSE, which it promptly reverses.

NOT gate

The composite gates are AND, OR or XOR combined with NOT to make NAND (not and), NOR (not or) and XNOR (not XOR) gates. These new gates process input in the usual manner and then reverse the result.

NAND gate

NOR gate

XNOR gate

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Try this for yourself using the applet below. Choose a value for both inputs, then click on the choice box to select a gate. The output will then be displayed. (It is convention to call the inputs A, B, C etc. and the output Q. If we called the output “O” it might get confused with 0 (zero)).

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Venn Diagrams.

Boolean logic can also be depicted as a Venn diagram. Consider the following. A “Mr Whippy” machine can dispense chocolate, vanilla or strawberry ice-cream. How many combinations can you have, and what do you ask for?

Strawberry OR vanilla. Actually there are three possibilities here – a pink cone, a white cone or a mixture of the two. You can have one, or the other, or both.

Strawberry OR Vanilla

Strawberry AND vanilla. Only a strawberry/vanilla swirl is acceptable – strawberry on it’s own, or just plain vanilla, do not count. You need to have both.

Strawberry AND Vanilla

NOT strawberry. You could have chocolate, vanilla, or choc/vanilla combo.

NOT strawberry

Strawberry XOR chocolate. This is a tough one! You must have one, or the other, but not nothing, and not both. This translates to strawberry, OR chocolate, but you can’t have a combo, or no ice cream at all.

Strawberry XOR Chocolate

Chocolate AND strawberry AND vanilla. You must have a neapolitan!

You must have EVERYTHING!

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Boolean logic appears in many programming languages as both “boolean expressions” and a variable type in it’s own right. You declare and initialise it like this :

cone: boolean; cone := false; {pascal}
boolean cone = false; //java
dim cone as boolean; cone = false; ‘visual basic

The following examples will all use Pascal, but you’ll get the idea.

IF ((flavour = “strawberry”) OR (flavour = “chocolate”))THEN cone := true ELSE cone := false;

What the computer does here is to evaluate the expression in brackets, and decide whether or not it is true (you chose strawberry OR chocolate – it only needs one condition to be fulfilled). Then it knows what to do with “cone”.

IF ((flavour1 = “strawberry”) AND (flavour2 = “chocolate”)) THEN cone := true ELSE cone := false;

In this case, BOTH the conditions have to be true in order for “cone” to become true as well.

Mr. Whippy at Hazlehead Park, 1999

Later on, I will also add a page about logic circuits.

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Last Updated - 06/10/03

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