z = xor a b

Human hand coded minimum information form.

1. ~(ab)(a+b)
             __
         +--|  \
         |  |   |--+
       +-|--|__/   |  |\       __
       | |   __    +--| >o----|  \
  b ---|-+--\  \      |/      |   |---- z
       |     |  |-------------|__/
  a ---+----/__/


2. ~(ab)a+~(ab)b  : destributivity

                                 __
  b ----+-----------------------|  \
        |     __                |   |--+   __
        +----|  \     |\     +--|__/   +--\  \
             |   |----| >o---+   __        |  |---- z
        +----|__/     |/     +--|  \   +--/__/
        |                       |   |--+
  a ----+-----------------------|__/


3. (~a + ~b)a + (~a + ~b)b  : De Morgan

                                 __
  b ----+-----------------------|  \
        |    |\                 |   |--+
        +----| >o--+   __    +--|__/   |   __
             |/    +--\  \   |         +--\  \
                       |  |--+             |  |---- z
             |\    +--/__/   |   __    +--/__/
        +----| >o--+         +--|  \   |
        |    |/                 |   |--+
  a ----+-----------------------|__/


4. uncommonizes ~a + ~b
                                   __
  b ----+-------------------------|  \
        |    |\          __       |   |--+
        +----| >o----+--\  \   +--|__/   |
             |/      |   |  |--+         |   __
                   +-|--/__/             +---\  \
                   | |   __                   |  |---- z
                   | +--\  \             +---/__/
             |\    |     |  |--+   __    |
        +----| >o--+----/__/   +--|  \   |
        |    |/                   |   |--+
  a ----+-------------------------|__/


5. ~a a + ~b a + ~a b + ~b b  : destributivity
                                   __
  b ----+---------------------+---|  \
        |   |\                |   |   |--+  __
        +---| >o------+-------|---|__/   +--\  \
            |/        |       |    __        |  |--+
                      |       +---|  \   +--/__/   |
                      |           |   |--+         |  __
                   +--|-----------|__/             +--\  \
                   |  |            __                  |  |---- z
                   |  +-----------|  \             +--/__/
                   |              |   |--+  __     |
                   |          +---|__/   +--\  \   |
            |\     |          |    __        |  |--+
        +---| >o---+----------|---|  \   +--/__/
        |   |/                |   |   |--+
  a ----+---------------------+---|__/


6. ~b a + ~a b  : complements

  b ----+---------------------+
        |   |\                |             __
        +---| >o------+       |         0 --\  \
            |/        |       |    __        |  |--+
                      |       +---|  \   +--/__/   |
                      |           |   |--+         |  __
                   +--|-----------|__/             +--\  \
                   |  |            __                  |  |---- z
                   |  +-----------|  \             +--/__/
                   |              |   |--+  __     |
                   |          +---|__/   +--\  \   |
            |\     |          |              |  |--+
        +---| >o---+          |         0 --/__/
        |   |/                |
  a ----+---------------------+


7. absorption

                              __
  b ---+---------------------|  \
       |    |\               |   |--+
       +----| >o------+  +---|__/   |  __
            |/        |  |          +--\  \
                   +--|--+              |  |---- z
            |\     |  |       __    +--/__/
       +----| >o---+  +------|  \   |
       |    |/               |   |--+
  a ---+---------------------|__/



7. --> 6.  a limited inverted absorption

optional: permuability
((?i-0 ?i-1) (?n-0 or2) (?o-0))
-->
((?i-1 ?i-0) (?n-0 or2) (?o-0))

((?i-0 ?i-1) (?n-0 and2)   (?m-0))
((?i-2 ?m-0) (?n-1 or2)    (?o-0))
-->
((?i-0 ?i-1) (?n-a and2)   (?m-a))
(nil         (?n-b const0) (?m-b))
((?i-2 ?m-b) (?n-c or2)    (?m-c))
((?m-a ?m-c) (?n-d or2)    (?o-0))


6. --> 5.  a limited inverted complements

((?i-0)      (?n-0 not1)   (?m-0))
(?is-0       (?n-1 const0) (?o-0))
-->
((?i-0)      (?n-a not1)   (?m-0))
((?i-0 ?m-0) (?n-b and2)   (?o-0))


5. --> 4.  distributivity

((?i-0 ?i-1) (?n-0 and2) (?m-0))
((?i-0 ?i-2) (?n-1 and2) (?m-1))
((?m-0 ?m-1) (?n-2 or2)  (?o-0))
-->
((?i-1 ?i-2) (?n-a and2) (?m-a))
((?i-0 ?m-a) (?n-b or2)  (?o-0))


4. --> 3.  commonize or2

((?i-0 ?i-1) (?n-0 or2) (?m-0))
((?i-0 ?i-1) (?n-1 or2) (?m-1))
((?i-2 ?m-0) ?n-2       ?no-2)
((?i-3 ?m-1) ?n-3       ?no-3)
-->
((?i-0 ?i-1) (?n-a or2) (?m-a))
((?i-2 ?m-a) ?n-2       ?no-2)
((?i-3 ?m-a) ?n-3       ?no-3)

3. --> 2.  De Morgan's law

((?i-0)      (?n-0 not1) (?m-0))
((?i-1)      (?n-1 not1) (?m-1))
((?m-0 ?m-1) (?n-2 or2)  (?o-0))
-->
((?i-0 ?i-1) (?n-a and2) (?m-a))
((?m-a)      (?n-b not1) (?o-0))


2. --> 1.  same as 5. --> 4.
