making Venn diagrams for mathematics

[gusl]

I'm looking for a way to quickly (automatically) draw diagrams like this:





The user would input a set of formulas like the above. (Does anyone know what logic this corresponds to? I don't think plain propositional logic would cut it.)

The system would:
* draw a Venn diagram corresponding to these formulas, choosing appropriate colors/patterns and intersections.
* index its regions

The above diagram illustrates a well-known theorem: "every complete sufficient statistic is minimally sufficient".


(1): complete: F, sufficient: F, minimally sufficient: F
(2): complete: T, sufficient: F, minimally sufficient: F
(3): complete: T, sufficient: T, minimally sufficient: T
(4): complete: F, sufficient: T, minimally sufficient: T (the most interesting counter-example)
(5): complete: F, sufficient: T, minimally sufficient: F

Comments

[robbat2.livejournal.com]

If you do find something nice for Linux to do it, I'd love to have it too.

[gustavolacerda.livejournal.com]

sure thing.

seems like a nice little project for a CS undergrad, so maybe I should ask someone in the Columbia Computer Graphics group.

[gustavolacerda.livejournal.com]

http://www.informatik.uni-ulm.de/ni/staff/HKestler/vennm/doc.html

[henriknordmark.livejournal.com]

> Does anyone know what logic this corresponds to? I don't think plain propositional logic would cut it.

The logic you are referring to is simply called Set Algebra.

[gustavolacerda.livejournal.com]

does Set Algebra correspond to a modal logic?

I want to be able to do model-checking, etc.

[henriknordmark.livejournal.com]

Hmmmmm... propositional logic would almost cut it:

union becomes disjunction
intersection becomes intersection
complement becomes negation
subsethood becomes implication
empty set becomes falsedom

The only problem is that you have no way of expressing strict subsethood.

"A is a subset of B", becomes "a --> b"

but, "A is a strict subset of B", becomes ?????


You want to say something like "a --> b" and b does not necessarily imply a.

If you do this in the most naive way possible...

b does not imply a, becomes "not ( b --> a )"
which is equivalent to "b & not a".

But this is clearly a bad translation as we would go from "A is a strict subset of B" to:

"( a --> b ) & ( b & not a )"

which is equivalent to "b & not a"

which if you translate back into set algebra would be "complement of A intersected with B" which isn't even a statement!

So you are right, it would seem as if you need something slightly stronger than propositional logic.

Would modal logic in some form do it?
Who knows...

How would you choose to translate "A is a strict subset of B"?

"( a --> b ) & ( b --> ( <>a & <> not a ) )"

perhaps???

I have no idea if that would work... and I am too lazy to think this through and see why it may not work or verify that it does work in fact work in a systematic and meaningful manner.

Just rambling on at this point...

[gustavolacerda.livejournal.com]

A -> B           ## $A \subset B$
<> ~(B -> A)     ## possibly a proper subset

that gives us 3 possible worlds, as desired. Am I making sense?

[henriknordmark.livejournal.com]

That does sorta seem to make sense...

"A is a subset of B" becomes

( a --> b ) & <> ( b & not a )


I think that works!

And yes you end up with three possible worlds

not a & not b

a & b

b & not a

the only world that is excluded is "a & not b" which is perfect!

That would seem to do the trick!