eolas/Theory_of_Computation/Logic_gates.md

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Theory_of_Computation Logic Electronics binary

Logic gates

Logic gates are the basic building blocks of digital computing. A logic gate is an electrical circuit that has one or more than one input and only one output. The input controls the output and is isomorphic with logical conditions that can be expressed in the form of truth-tables.

Truth tables

I know from my study of logic that truth tables enable us to present the conditions under which logical propositions are true or false. To take the AND operator: AND evaluates to true if both of its constituent expressions are true and false in any other circumstances (e.g. if one proposition is true and the other false (or vice versa) and if both propositions are false ).

This is most clearly expressed in the following truth table:

Truth table for AND

p    q   p & q
_    _   _____
t    t     t
t    f     f
f    t     f
f    f     f   

Another example is the negation (NOT) operator in logic which is highly trivial. The negation operator (¬ or ~ ) switches the value of a proposition from true to false. When we put ~ before true it becomes false and when we put ~ before false it becomes true. We will see shortly that this corresponds to a basic on/off switch.

Truth table fo NOT

p    ~ p
_    __
t    f
f    t

NAND gates

A NAND gate is a logic gate that combines the truth conditions for AND and NOT . I

Let's first introduce the circuit:

The real-life circuit showing two switches corresponding to two transistors which control the LED light.

In this circuit, there are two transistors, each connected to a switch. The switches control the LED light. So the switches are the input and the LED is the output.

For clarity, we are not going to draw both transistors, we will simplify the diagram with a symbol for them which stands for the NAND gate:

Remember that a 'logic gate' is a logical abstraction of a physical process: the voltage passing through a transistor. The transistors register the charge and the switches control it's flow, the 'gate' is just the combination of transistors and how they are arranged. There is not a physical gate per se, there is only the transistor whose output we characterize in terms of logic.

The diagram below shows how the circuit models the truth conditions for AND

Diagram representing NAND gate:

• When both switches are off (corresponding to false false) the output is on (the bulb lights up).
• If either one of the switches are on, the output remains on (corresponding to true false or false true )
• It is only when both switches are on, that the output is off (corresponding to true true )

Remember that switch circuitry is counter intuitive: the switches being on corresponds to the output ceasing to execute because the switches break the circuit, they don't join it.

Transliterating the logic truth table to the switch behaviour

We can now present a truth table for NAND alongside the truth conditions for AND and NOT

// AND 
p    q   p & q
_    _   _____

t    t     t      (1)
t    f     f      (2)
f    t     f      (3)
f    f     f      (4)  

// NOT

p    ~ p
_    __
t    f
f    t

A    B   Output
_    _   _____

0    0     1       (1)
1    0     1       (2)
0    1     1       (3)
1    1     0       (4)

• So we can see that the binary representation of the circuit accords with NOT at rows (1) and (4): when both switches are off (false ), the bulb is on ( true ). And when both switches are on (true ), the bulb is off (false ).
• Rows (2) and (3) of the binary truth table accord with rows (2) and (3) of the AND truth table: if one of the switches is true but the other is false , the output is false (the bulb remains on).

More complex outputs from combining NANDS

The example we have looked at so far is fairly simple because there is just one NAND gate corresponding to two inputs (the two switches) and one output (the bulb).

When we add more NAND gates and combine them with each other in different ways we can create more complex output sequences and these two will have corresponding truth tables.

NOT gate

This gate corresponds to the NOT Boolean or negation logical connective. It is really simple and derived from the trivial logical fact that true is true and false is false also known as logical identity.

Natural language

The negation operator (¬ or ~ ) switches the value of a proposition from true to false. When we put ~ before true it becomes false and when we put ~ before false it becomes true .

Truth table

This corresponds to a simple on-off switch.

In terms of logic gates we would create this by using a single NAND gate. Although it can take a total of two inputs, it would be controlled by a single switch, so both inputs would be set to 1 1 or 0 0 when the switch is activated and deactivated. This would remove the AND aspect of NAND and reduce it to NOT .

A NAND gate simulating NOT logic

Symbol for NOT gate

NOT has its own electrical signal to distinguish it from a NAND:

AND gate

Just as we can create NOT logic from a NAND gate, without the AND conditions, we can create a circuit that exemplifies the truth conditions of AND without including those of NOT.

When we attach two NAND gates in sequence connected to two switches as input this creates the following binary conditions:

A    B   Output
_    _   _____

0    0     0       (1)
1    0     0       (2)
0    1     0       (3)
1    1     1       (4)

Which is identical to the truth table for AND :

p    q   p & q
_    _   _____

t    t     t      (1)
t    f     f      (2)
f    t     f      (3)
f    f     f      (4)  

Natural language

AND (&) is true when both constituent propositions are true and false in all other circumstances viz. false false (¬P & ¬Q / 0 0 ), true false (P & ¬Q / 1 0 ), false true (¬P & Q / 0 1 )

AND at 0 0

AND at 1 0 or 0 1

Symbol for AND gate

It's very similar to NAND so be careful not to confuse it

OR

OR (in logic known as disjunction) in its non-exclusive form is true if either of its propositions are true or both are true . It is false otherwise.

p    q   p V q
_    _   _____

t    t     t      (1)
t    f     t      (2)
f    t     t      (3)
f    f     f      (4)  

XOR

XOR stands for exclusive or, also known as exclusive conjunction. This means it can only be true if one of its propositions are true . If both are true this doesn't exclude one of the propositions so the overall statement has to be false . This is the only change in the truth conditions from OR .

Electrical symbol for XOR

p    q   p X V q
_    _   ________

t    t     f      (1)
t    f     t      (2)
f    t     t      (3)
f    f     f      (4)  

**NOR**

This is equivalent to saying 'neither' in natural language. It is only true both propositions are false . If either one of the propositions is true the outcome is false . If both are true it is false

XNOR

This one is confusing. I can see the truth conditions but don't understand them. It is true if both propositions are false like NOR or if both propositions are true and false otherwise.

p    q   p ¬V q
_    _   ________

t    t     f      (1)
t    f     f      (2)
f    t     f      (3)
f    f     t      (4)  

p    q   p X¬V q
_    _   ________

t    t     t      (1)
t    f     f      (2)
f    t     f      (3)
f    f     t      (4)