## Frankl, My Dear : 3

Here’s a few pages on differential logic, whose ideas I’ll be trying out in the present setting:

I next need to look at the following “key lemma” from (2) and see if I can wrap my head, or at least my own formalism, around what it says.

Lemma.  Let ${f(x_{1}, \dots, x_{n})}$ be a Boolean function and let ${x_{i}}$ be fixed.  Then the Boolean inputs can be partitioned into six sets:

$\displaystyle A_{0}, A_{1}, B_{0}, B_{1}, C_{0}, C_{1}.$

These sets have the following properties:

1. The variable ${x_{i}}$ is equal to ${0}$ on ${A_{0} \cup B_{0} \cup C_{0}}.$
2. The variable ${x_{i}}$ is equal to ${1}$ on ${A_{1} \cup B_{1} \cup C_{1}}.$
3. The union ${A_{0} \cup A_{1}}$ is equal to ${J_{i}}.$
4. The function is always ${0}$ on ${B_{0} \cup B_{1}}.$
5. The function is always ${1}$ on ${C_{0} \cup C_{1}}.$
6. Finally ${|A_{0}| = |A_{1}|}$  and  ${|B_{0}| = |B_{1}|}$  and  ${|C_{0}| = |C_{1}|}.$

This may take a while …