Frankl, My Dear : 2

Re: Dick Lipton & Ken Regan(1)(2)

Supplied by the cache of definitions from Post 1, I can return to the passage from (2) that seemed to jog a bit of memory and see if what I imagined I saw there makes any sense.

Let us use {J_{i}(f)} to denote those {x} such that

\displaystyle f(x) \neq f(x^{i}).

Obviously the following is true:

\displaystyle \frac{|J_{i}(f)|}{2^{n}} = I_{i}(f).

There must be a better notation than {J_{i}} — we are open to suggestions. Any?

I responded to their query about a better notation for {J_{i}} in a series of comments along the following lines:

I think {J_{i}(f)} is just the set of places where the partial differential of {f} with respect to {x_i} is {1}.

See Tables A9 and A10 in my article on Differential Logic and Dynamic Systems.

For example, look at {f_8 (x, y),} which is just the logical conjunction {xy}.

In this case, {\partial_x f = y}.

This means that crossing the boundary of {x} will change the value of {f} exactly in those places where {y} is true.

The See a Number, Make a Set principle leads to the following observation, that an arbitrary set of cells in a venn diagram or an arbitrary set of vertices in a {k}-cube is described by a proposition or a boolean function as its fiber of {1}, so the above types of differential operators take us from propositions to propositions, in other words, they stay within the same general datatype.

To be continued …

This entry was posted in Boolean Algebra, Boolean Functions, Computational Complexity, Differential Logic, Frankl Conjecture, Logic, Logical Graphs, Mathematics, Péter Frankl and tagged , , , , , , , , . Bookmark the permalink.

One Response to Frankl, My Dear : 2

  1. Pingback: Survey of Differential Logic • 1 | Inquiry Into Inquiry

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