[seqfan] Animated Proofs & Cactus Logic

Jon Awbrey jawbrey at att.net
Tue Mar 23 16:22:56 CET 2010

```Hi SeqFans --

I finally got around to doing something I've
been thinking about doing since the late 60's,
animating proofs in Peirce's logical graphs.

Here's a rough cut where all I did was flip through
in storyboard form:

http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems#Majority_function_example

Background:

There's an extension of Peirce's logical graphs for propositional calculus
that uses "minimal negation operators" -- the family of connectives that
assert "just one false" of the propositions in their argument lists.
This generalizes the corresponding syntactical trees to cacti and
represents a minimal negation operator of k arguments by means
of a cactus lobe of k+1 nodes.

For instance, XOR(x, y) is graphed as follows, where * is the root node:

x   y
o---o
\ /
*

This provides us with a nice way of representing boolean expansions
and using them as disjunctive normal forms to establish results.

For example, here's a graphical DNF way of proving Leibniz's
Praeclarum Theorema -- with a proof animation at the end:

http://mywikibiz.com/Directory:Jon_Awbrey/Papers/Propositional_Equation_Reasoning_Systems#Praeclarum_theorema_:_Proof_by_CAST

The species of cacti that arises is known as "painted and rooted cacti" --
this means that you can assign any set of labels ("paints") from
a given finite set of boolean variable names to each node.

I remember that enumerating cacti was one of my favorite problems in Harary & Palmer,
and I've always thought that there ought to be some nice counting problems here,
but I fear that exceeds my current powers, so maybe someone else can see one.

Cheers,

Jon Awbrey

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