Understanding binary coding with FORTH

published: 29 April 2024 / updated 29 April 2024

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article: 15 juin 2019 / mis à jour 01 mai 2020

When exploiting logical functions, such as AND OR XOR or the test result = > < <>, it becomes essential to understand the binary coding which is far from being a simple arithmetic representation a bit exotic.

The binary code

The modern binary number system, based on the binary code, was invented by Gottfried LEIBNITZ in 1689 and appears in his article Explanation of Binary Arithmetic in 1703.

In 1847, Georges BOOLE published an article entitled Mathematical Analysis of Logic which explains an algebraic system of binary logic. His works were forgotten until 1937, when Claude SHANNON discovered that BOOLE algebra could apply to electrical circuits.

Electrical circuits

Here is the simplest electrical circuit you can imagine:

We have a power source, the battery. Then a switch, which we will abbreviate in C. And finally, a lamp, which we will abbreviate in L:

• if C is activated, the lamp L is on
• if C is deactivated, the lamp L is off

The state of L depends directly on the state of C. We can table this. We code the state inactive with the number 0 and the active state with the number 1:

Using 0 and 1 is a binary representation. Coding binary only knows two states: 0 or 1.

Combinatorial states

Combinatorial states designate the management of a device from several separate states. The best example, the circuit comes and goes for domestic lighting:

we will describe the state of the lamp L managed by these two switches:

We could number the states of c1 and c2 like this:

The first state has the number zero, the next the number 1 .. until the last state which has the number 3. For our two switches, there are only four possible states, therefore numbered from 0 to 3.

But why start numbering at zero and not at 1?

As Mr SPOCK would say in STAR TREK: "there is a logical reason..."

And let's explore this logic.

Binary decoding

Take state 3 of the circuit back and forth whose combination of states C1 and C2 is 11.

In decimal notation, we mention the order of the numbers in units, tens, hundreds, etc. Take a year: 2019. Mathematically, each digit of this number 2019 can be broken down like this. By convention, we note the power 'exp':

2019 = (9 * 10 exp 0) + (1 * 10 exp 1) + (0 * 10 exp 2) + (2 * 10 exp 3)

In mathematics:

• 10 power 0, which we noted 10 exp 0 gives 1
• 10 power 1, which we noted 10 exp 1 gives 10
• 10 power 2, which we noted 10 exp 2 gives 100
• 10 power 3, which we noted 10 exp 3 gives 1000

So we see that the exponent of 10 starts at 0 and goes up to 3. Again, we start from zero.

For binary coding, we will simply replace 10 by 2:

11 = (1 * 2 exp 0) + (1 * 2 exp 1)

• 2 power 0, noted 2 exp 0 gives 1
• 2 power 1, noted 2 exp 1 gives 2

The sum 1 + 2 gives in decimal 3, which corresponds to the decoding of the binary value 11.

First names randomly have a binary value 10010111 and let's see how to decode this in decimal. We will start with the "units", that is to say the rightmost bit:

• 1 * 2 exp 0 = 1 * 1
• 1 * 2 exp 1 = 1 * 2
• 1 * 2 exp 2 = 1 * 4
• 0 * 2 exp 3 = 0 * 8
• 1 * 2 exp 4 = 1 * 16
• 0 * 2 exp 5 = 0 * 32
• 0 * 2 exp 6 = 0 * 64
• 1 * 2 exp 7 = 1 * 128

10010111 is recomposed into decimal: 1 + 2 + 4 + 16 + 128, which gives 151 in decimal.

$24 constant DDRB       \ Direction Data Register PORT B 
$25 constant PORTB      \ data in PORT B 
255 DDRB c! 
128 PORTB c!            \ turn LED on 
  0 PORTB c!            \ turn LED off 

%10000000 PORTB c!      \ turn LED on 

Decimal to binary conversion and vice versa

The FORTH language knows how to manage numbers in almost any numerical base, from 2 to N, N can be any value. We can therefore use numbers in binary, in base 3, base 5, base 8, base 128, etc ...

A little exercise with gForth:

: numLoop ( ---) 
    1000 0 do i . loop ; 
128 base ! 
numLoop 

Display:

numLoop 0 1 2 3 4 5 6 7 8 9 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z
[ \ ] ^ _ ` a b c d e f g h i j k l m n o p q r s t u v w x y z { | } ~  ?  ? '
ƒ " . ┼ ╬ ^ % S < O ? Z ? ? ' ' " "  - - ~ T s > o ? z Y   ¡ ¢ £ $ ¥ ¦ § ¨ © ª
« ¬ ­ ® ¯ ° ± ² ³ ´ µ ¶ 10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F 1G 1H 1I
1J 1K 1L 1M 1N 1O 1P 1Q 1R 1S 1T 1U 1V 1W 1X 1Y 1Z 1[ 1\ 1] 1^ 1_ 1` 1a 1b 1c 1d
1e 1f 1g 1h 1i 1j 1k 1l 1m 1n 1o 1p 1q 1r 1s 1t 1u 1v 1w 1x 1y 1z 1{ 1| 1} 1~ 1
1? 1? 1' 1ƒ 1" 1. 1┼ 1╬ 1^ 1% 1S 1< 1O 1? 1Z 1? 1? 1' 1' 1" 1" 1 1- 1- 1~ 1T 1s
1> 1o 1? 1z 1Y 1  1¡ 1¢ 1£ 1$ 1¥ 1¦ 1§ 1¨ 1© 1ª 1« 1¬ 1­ 1® 1¯ 1° 1± 1² 1³ 1´ 1µ
1¶ 20 21 22 23 24 25 26 27 28 29 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 2K 2L 2M 2N 2O 2P
2Q 2R 2S 2T 2U 2V 2W 2X 2Y 2Z 2[ 2\ 2] 2^ 2_ 2` 2a 2b 2c 2d 2e 2f 2g 2h 2i 2j 2k
2l 2m 2n 2o 2p 2q 2r 2s 2t 2u 2v 2w 2x 2y 2z 2{ 2| 2} 2~ 2 2? 2? 2' 2ƒ 2" 2. 2┼
2╬ 2^ 2% 2S 2< 2O 2? 2Z 2? 2? 2' 2' 2" 2" 2 2- 2- 2~ 2T 2s 2> 2o 2? 2z 2Y 2  2¡
2¢ 2£ 2$ 2¥ 2¦ 2§ 2¨ 2© 2ª 2« 2¬ 2­ 2® 2¯ 2° 2± 2² 2³ 2´ 2µ 2¶ 30 31 32 33 34 35
36 37 38 39 3A 3B 3C 3D 3E 3F 3G 3H 3I 3J 3K 3L 3M 3N 3O 3P 3Q 3R 3S 3T 3U 3V 3W
3X 3Y 3Z 3[ 3\ 3] 3^ 3_ 3` 3a 3b 3c 3d .......

In FORTH, it is the BASE variable which thus determines in which base Numeric are entered and processed numbers. The binary coding being in base 2, to go to binary, you just have to put 2 in this BASE variable:

2 base ! 

And you're in binary! We can no longer enter other numbers than those made up of "0" and "1":

2 base ! 
101 110 + .   \ display 1011 

To return to decimal, just type decimal:

2 base ! 
1011 decimal . \ display 1011 

So, with the FORTH language, we have simple tools to display any value in decimal or binary. Let's create two words:

: bin. ( n ---) 
    base @ >r 
    2 base ! 
    0 <# # # # # # # # # #> type 
    r> base ! ; 
: dec. ( n ---) 
    base @ >r 
    decimal . 
    r> base ! ; 

For the word bin., we chose to display the binary value on 8 digits:

decimal 
235 bin.   \ affiche 11101011 

We are equipped to understand what happens when we carry out operations arithmetic or logic on binary values.

Logical operations in binary

In binary logic, there are only two logical operators: OR and AND. The operator OR is combined in two versions: OR which is inclusive, and XOR which is exclusive.

The negation of a logical value is carried out by NOT. Let's see in detail the states two-bit logic when we execute these different logical operators.

The AND operator

It performs a logical AND operation:

Dans un circuit électrique, l'opérateur AND est semblable à deux interrupteurs en série:

In FORTH language, the word for performing a logical AND operation is and.

The OR operator

It performs a logical OR operation:

In an electrical circuit, the operator OR is similar to two parallel switches:

In FORTH language, the word for performing a logical OR operation is or.

The NOT operator

The NOT operator reverses the logical state:

In FORTH language, the word used to perform a logical NOT operation is invert.

With the three binary operators AND, OR and NOT, we can process all combinations binaries. This is what we will see with some praptic examples.

Example of the NAND logic gate

The NAND logic function corresponds to the inverse of the AND operator:

This NAND logic function does not exist in FORTH language. But as it was written earlier, with the logical operators AND, OR and NOT, we have the words to create this function:

: nand ( fl1 fl2 --- fl) 
    and invert 
  ; 

Pfiouuu....! It's complicated huh?

Little exercise: managing three states

We are going to simulate a very special, totally imaginary circuit, which works like this:

• an output S which depends on three inputs A B C
• if no entry is active, S is inactive
• if only one entry A B or C is active, S is active
• if two entries among A B or C, S is active
• if the three entries A B and C are active, S is inactive

Here is Karnaugh's chart which describes the state of S:

If we look at this table, we can make four groupings of states to 1.

First grouping

In the rest of the article:
 the logical operator OR will be noted +
 the AND operator will be noted x
 the NOT operator will be noted !

For this first grouping, the common states of A B C are formulated: !A x C

Second grouping

For this second grouping, the common states of A B C are formulated: B x !C

Third grouping

For this third grouping, the common states of A B C are formulated: A x !B

State formula of S

The logical equation for the state of output S will be:

  ( !A x C ) + ( B x !C ) + ( A x !B )

If we had to write a FORTH definition processing the output S, we could write:

variable A 
variable B 
variable C 
: S-resolve ( --- fl) 
  A @ invert   C @         and     \ ( !A x C )  
  B @          C @ invert  and     \ ( B x !C ) 
  A @          B @ invert  and     \ ( A x B! ) 
  or or 
  ; 

Is there another way to set the state of the S output?

Yes. For that, let's go back to our first table. In this table we have only two O values to define the state of our S output:

By taking exclusively into account these two state values 0 of the output S, the logical equation of the state of the output S could be rewritten in this form:

  !( ( !A x !B x !C ) + ( A x B x C ) )

Which would give this definition in FORTH language:

variable A 
variable B 
variable C 
: S-resolve ( --- fl) 
  A @ invert  B @ invert  C @ invert  and and  \ ( !A x !B x !C ) 
  A @         B @         C @         and and  \ ( A x B C ) 
  or invert 
  ;