Re: Boolean math: Fill in the blanks.

[repellent]

Very interesting topic!

I see an overall pattern, which I would like to share :-)

First, a few basic observations:

• There seems to be a certain symmetry evident from how R & R results in 8 bits set on average (halfway between 1 and 16) versus R | R resulting in 24 bits set on average (halfway between 16 and 32). I'll be sure to make my hypothesis symmetrical, in that regard.
• If we apply AND, the average bits set goes down due to higher likelihood of 0 bits swallowing the 1 bits. Conversely, if we apply OR, the average bits set goes up due to higher likelihood of 1 bits overriding the 0 bits.
• Non-associative rules apply for AND and OR:
  
  
  ( (A & B) | C )  !=  ( A & (B | C) )
  
  So evaluating the expression from left-to-right is important! Update: Perl gives precedence to the & operator over |

Hypothesis:

Each AND/OR operation is evaluated in the following way:

( <left_gained_avg_bits_set> <op> <right_modifier> ) = <resulting_avg_bits_set>

Whenever an AND/OR <op> is applied, it induces a change to the already gained average bits on the left-side. That change is a multiplicative factor determined by the average bits set on the right-side:

right_mod_factor = right_avg_set_bits / total_bits

Cases:

For <op> = AND, we would take right_mod_factor and multiply it with left_avg_bits_set to get <resulting_avg_bits_set>. Examples:

R & (R & R) => 16 * (8 / 32) = 4 bits set on average

(R | R) & (R & R & R) => 24 * (4 / 32) = 3 bits set on average

That is, if there are more bits set on average on the AND right-side, there will be less chance of swallowing, so more bits are set on average in the result.

Conversely, for <op> = OR, we would take right_mod_factor and multiply it with left_avg_unset_bits. Then, take that result and add it to left_avg_set_bits to get <resulting_avg_bits_set>. Examples:

R | (R | R) => 16 + 16 * (24 / 32) = 28 bits set on average [second 16 is # of avg unset bits]

(R | R) | (R & R & R) => 24 + 8 * (4 / 32) = 25 bits set on average

R & R & R | R | R = (R & R & R) | (R | R) => 4 + 28 * (24 / 32) = 25 bits set on average
or
R & R & R | R | R = (R & R & R) | R | R => 4 + 28 * (16 / 32) + 14 * (16 / 32) = 25 bits set on average [got 14 from 32 - ( 4 + 28 * (16 / 32) )]

That is, if there are more bits set on average on the OR right-side, there will be more chance of overriding, so more bits are set on average in the result.

Conclusion:

So, following my pattern hypothesis, I see some inconsistent results in your "full set". For instance, 7 of ( R & R | R & R ) & R evaluates to 5 bits set on average. Attention has to be paid to precedence of AND/OR evaluations. Also, 22 & 26 are the same, when 22 of R & R | R | R evaluates to 26 bits set on average.

There may be some fallacy in my hypothesis, so feedback is welcomed :)

So, given the following legend:

n  = total # bits per sample
L1 = # bits   set on average on  left-side of AND/OR
L0 = # bits unset on average on  left-side of AND/OR
   = n - L1
R1 = # bits   set on average on right-side of AND/OR

we can derive a set of equations to calculate the resulting average bits set for each AND/OR operation:

AND : L1 * R1 / n

OR  : L1 + L0 * R1 / n = L1 + (n - L1) * R1 / n
                       = L1 + R1 - L1  * R1 / n
                       = L1 * ( 1 - R1 / n ) + R1