Frankl, My Dear : 10

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


Venn Diagram Frankl Figure 5
(5)

Figure 5 shows the 14 terms of the difference map \mathrm{D}f as arcs, arrows, or directed edges in the venn diagram of the original proposition f(p, q, r) = pqr. The arcs of \mathrm{E}f are directed into the cell where f is true from each of the other cells. The arcs of \boldsymbol\varepsilon f are directed from the cell where f is true into each of the other cells.

The expansion of \mathrm{D}f computed in the previous post is shown again below with the terms arranged by number of positive differential features, from lowest to highest.

\begin{array}{*{4}{l}}  \multicolumn{4}{l}{\mathrm{D}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =}  \\[10pt]  &  \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot  \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}  & + &  \texttt{(} p \texttt{)~} q \texttt{~~} r \texttt{~} \cdot  \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}  \\[10pt]  + &  \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot  \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}  & + &  \texttt{~} p \texttt{~(} q \texttt{)~} r \texttt{~} \cdot  \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}  \\[10pt]  + &  \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot  \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}  & + &  \texttt{~} p \texttt{~~} q \texttt{~(} r \texttt{)} \cdot  \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}  \end{array}
 
\begin{array}{*{4}{l}}  + &  \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot  \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}  & + &  \texttt{(} p \texttt{)(} q \texttt{)~} r \texttt{~} \cdot  \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}  \\[10pt]  + &  \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot  \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}  & + &  \texttt{(} p \texttt{)~} q \texttt{~(} r \texttt{)} \cdot  \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}  \\[10pt]  + &  \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot  \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}  & + &  \texttt{~} p \texttt{~(} q \texttt{)(} r \texttt{)} \cdot  \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}  \\[10pt]  + &  \texttt{~} p \texttt{~~} q \texttt{~~} r \texttt{~} \cdot  \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}  & + &  \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \cdot  \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}  \end{array}

To be continued …

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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 : 10

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

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