Studies of Euler diagrams/farofe

dummy

This Boolean function has 12 of 16 true spots. It is similar to medusa, but without spot 3. (See here.)

2D Euler diagram with separated cell 0

3D Euler diagram with one intersection point of three planes

symmetry

The Euler diagrams shown above are not good, because they hide the symmetry of this Boolean function.

2D Euler diagram with with rotational symmetry

This code uses the Python library discrete helpers. from discretehelpers.boolf.examples import farofe assert farofe.symmetric_spots.blocks_with_singletons() == [[0, 4], [1, 5], [2, 12], [6, 8], [7, 9], [10, 14]]

filtrates

Each of the 6 pairs is a 2-split (with all 4 quadrants). Only one of the 4 triples is a 3-split (with all 8 octants).

Euler diagram
4 sets A B C D 6 pairs A B A C A D C D B D B C 4 triples B C D 8 cells A C D 7 cells A B D 7 cells A B C 7 cells

graph
complete 4 sets A B C D 6 pairs A B A C A D C D B D B C 4 triples B C D A C D A B D A B C

cells

blue and green   (top)
7      ABC_	6      _BC_	14      _BCD

either blue or green   (middle)
5      A_C_	4      __C_	2      _B__
	12      __CD	10      _B_D

neither blue nor green   (bottom)
0      ____	8      ___D
1      A___	9      A__D

ternary labels