How can we create Randolph diagrams in Mathematica?

Question

I have become fascinated with Randolph Diagrams and was wondering how we would create the diagrams from logical sequences using Mathematica?

It looks like it would be a combination of logic and graphing. I have no idea where to start. I need some suggestions to get me going.

Edit Randolph's paper on JStor

Answer 1

This could provide a good starting point, since the structure of the diagrams is simply a cross with four regions that themselves can contain similar crosses, you can simply define a structure to represent this nesting and a recursive function to draw such structures. In my implementation I just use the head c to indicate a cross:

dirs = {{1, 0}, {0, 1}, {-1, 0}, {0, -1}};

cross[mid_, scale_] := {
 Thickness[0.02 scale],
 Line[{scale {-1, -1} + mid, scale {1, 1} + mid}],
 Line[{scale {1, -1} + mid, scale {-1, 1} + mid}]
}

rDraw[True, mid_, scale_] := Disk[mid, scale]
rDraw[False, ___] := {}
rDraw[c[a__], mid_: {0, 0}, scale_: 1] := {cross[mid, scale], 
 Sequence @@ MapIndexed[
   rDraw[#1, mid + scale  dirs[[#2[[1]]]], 0.45 scale] &, {a}]}

So then you can create a diagram via:

With[{t = True, f = False},
 rDraw@c[t, f, c[f, c[f, t, c[f, t, f, t], f], t, f], f] // Graphics
]

Now it is just a question of converting expressions like (A&B) or !C into this c[...] structure.

Here's a small graph of some simple operations:

With[{t = True, f = False},
 Graphics[rDraw[#2], PlotLabel -> #1] & @@@ {
    {"and", c[f, t]},
    {"Nand", c[t, f, t, t]},
    {"or", c[t, t, t]},
    {"Nor", c[f, f, f, t]}
    } // GraphicsRow
]