oeis.org/a000616/list]]>

It turns out the number of irreducible boolean functions of 4 variables is 402.

So the sequence I found on

the web is 3, 6, 22, 402, 1228158, 400507806843728,

527471432057653004017274030725792, (edit: fixed typo on seven variable number.--J.E.F.)

and for eight variables it is very long number according to the internet and is:

11218076601767519586965281984173341005925142853855481024470471657123840.

11218076601767519586965281984173341005925142853855481024470471657123840.

I typed it twice to hope I did it right. (Looks right).

Notice that for 2 variables, the number is 6 "shapes" or 6 irreducible boolean functions.

00

00

01

00

11

00

10

01

11

10

11

11

And that's the six shapes because you can

rotate them to get all sixteen.

(end of post)

]]>The 8 corners of a cube designate the 8 areas in a Venn diagram. You must count the outside area

as the eighth area in a Venn diagram with 3 circles. Now if you have this cube with 8 corners, the

corners can have colored beads on the corners. Since this is boolean algebra, and not ternary, there

are 2 colors (boolean = 2 values, black and white, orange and pear, etc.) Now the input variables

are usually A, B, and C. The colors of the corners is the output of true or false (boolean, true or false),

such as the outputs on a "truth table". Now if you choose the left or right side of the cube, that could

be the variable A. (Any way you choose is okay, left/right, top/bottom, front/back) Now the B

variable could be the top and the negation of B would then be the bottom of the cube, 4 corners on

the bottom. Finally the C variable could the front 4 corners of the cube and the negation of C (called

"C not") could be the back 4 corners of the cube. Now the fun begins if you combine the different

halves of the cube together with boolean operators such as NAND, AND, OR, NOR, XOR, EQUALITYGATE,

IMPLICATIONGATE, etc.... When you combine two halves of the cube together, then you get a

different shape or pattern of black and white beads in the corners. What I did is write a short

computer program to iterate all possible combinations of operators and keep track of the complexity

of the equation, by counting the operators in the equation. Operators is called "gates" in digital

electronics. I ran the 3 variable case on a PC, and it runs in under a second. Then I changed

the program (written in C) to 4 variable case (hypercube) and that runs in 20 minutes. The

result is an equation for all 65536 four variable cases... Anyway, it was a lot of the fun.

That's all for now, maybe I'll write more later on this in a couple months... Bye.]]>

```
0 2 6 4
1 3 7 5
2**0 = 1 2**2=4 2**6=64=40Hex 2**4=16=10Hex
2**1 = 2 2**3=8 2**7=128=80Hex 2**5=32=20Hex
01 04 40 10
02 08 80 20
Add these numbers up.
FF = tautology, always true
00 = falsehood, contradiciton, always false.
F0 = A
0F = not A = /A
55 = B
AA = not B = /B
CC = C
33 = not C = /C
...Note A, B, and C are interchangeable to your liking when starting the definitions...(it depends on the definition of your axes.)
The variables A, B, and C are inputs that create the output conditions when combined together with AND, OR, and XOR,
and also IMP and others...
```

]]>01 = 33 & 05, where 05 = 55 & 0F

03 = 33 & 0F

09 = 99 & 0F, where 99 = CC XOR 55

81 = C3 & A5, where C3 = F0 XOR 33, and where A5 = F0 XOR 55

07 = 0F & 77, where 77 = 55 OR 33

83 = F0 XOR 73, where 73 = 50 OR 33, where 50 = F0 & 55

49 = AA XOR E3, where E3 = C3 OR A0, and where C3 = F0 XOR 33, and A0 = F0 & AA

0F = trivial case, a boolean variable.

96 = CC XOR 5A, where 5A = F0 XOR AA

35 = C0 XOR F5, where C0 = F0 & CC and where F5 = F0 OR 55

39 = CC XOR F5, where F5 = F0 OR 55

17 = C0 XOR D7, where C0 = F0 & CC and where D7 = C3 OR 55, and where C3 = F0 XOR 33

99 = CC XOR 55

These are 3-variable cases above.

Use a cube by folding the 2 by 4 Karnaugh map in half.

There are 22 shapes for 3 variables.

And Logic Circuit Minimization.]]>

```
<?php
function canOpener ($can) {
$canOpened = false;
$args = func_get_args();
unset ($args[0]); //that's $can; we have that one
foreach ($args as $compareTo) {
if ($can == $compareTo) { $canOpened = true; }
}
return $canOpened;
}
?>
```

]]>```
canopener ($problem) {
$result = solved;
}
```

]]>OK. Too right. I still wish their were a language construct for compacting conditions wherein a variable could equal one of several things that trigger the condition.

Isn't that the "select" statement (aso called "case" or "switch") ?

Example (Visual Basic code to replace any non-alphanumeric with "_"):

```
Select Case c
Case "a" To "z", "A" To "Z", "0" To "9"
Mid$(x, i, 1) = c
Case Else
Mid$(x, i, 1) = "_"
End Select
```

I chose VB because it has such a rich case statement, PHP can only have one value per "case":

```
switch ($MatchType) {
case "=":
$s .= $MatchVar . "=" . $MatchVal;
break;
case "Like":
$s .= $MatchVar . ' like "%' . $MatchVal . '%"';
break;
default:
$s .= $MatchVar . $MatchVal; break;
}
```

But PHP, like C, will continue to run statements below it without the "break;", so this can be used to put several "case" statments in sequence:

```
switch ($MatchType) {
case "=":
case "equal":
case "same":
$s .= $MatchVar . "=" . $MatchVal;
break;
case "Like":
$s .= $MatchVar . ' like "%' . $MatchVal . '%"';
break;
default:
$s .= $MatchVar . $MatchVal; break;
}
```

But there is a trick in PHP: use switch (TRUE) and then the "case"s can be conditions like "($x >5)" or "in_array()" etc.

Sorry, didn't mean to ramble on.

]]>Really an if statement returns true if the statement is true, and false if the statement is false, but you can use simple values instead of relational opererators such as ==, >, <, etc. A variable with a value of zero evaluates false. All others evaluate true.

If for instance you were to say:

if(5) { statement; }

it will evaluate true and always will since 5 does not equal zero, and the statement will be excecuted.

if(0) { statement; }

this will not be executed. Zero returns false.

When you use the logical operators, you can use the "or" and "and" operaters to execute a statement if either, or only if both are true.

you could say

if(0 || 1) { statement; }

that would return true. ( "||" means or)

if (0 && 1) { statement; }

that would not return true since both must be true.

But when you say if ( x == (5 || 7)) { statement;}

I believe this is incorrect use of the or operator. The or operator is used between two expressions that are evaluated for their truth or falsity. Not their value.

]]>I'm starting to learn Ruby, which is supposedly nice enough to make PHP blush with shame.

]]>Why the heck is it 8 bits on some compilers? I thought the purpose was to save memory for switch variables. Peculiar...

This is a bit of a complex question (pun intended). Computers run much faster if all data is stored in whole bytes. To understand why, you need to understand the underlying architecture, which would take me a while to describe.

]]>