Frankl, My Dear : 9

“It doesn’t matter what one does,” the Man Without Qualities said to himself, shrugging his shoulders. “In a tangle of forces like this it doesn’t make a scrap of difference.” He turned away like a man who has learned renunciation, almost indeed like a sick man who shrinks from any intensity of contact. And then, striding through his adjacent dressing-room, he passed a punching-ball that hung there; he gave it a blow far swifter and harder than is usual in moods of resignation or states of weakness.

Robert Musil • The Man Without Qualities

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

We continue with the differential analysis of the proposition in Example 1.

Example 1


Venn Diagram PQR
(1)

The difference operator \mathrm{D} is defined as the difference \mathrm{E} - \boldsymbol\varepsilon between the enlargement operator \mathrm{E} and the tacit extension operator \boldsymbol\varepsilon.

The difference map \mathrm{D}f is the result of applying the difference operator \mathrm{D} to the function f. When the sense is clear, we may refer to \mathrm{D}f simply as the difference of f.

In boolean spaces there is no difference between the sum (+) and the difference (-) so the difference operator \mathrm{D} is equally well expressed as the exclusive disjunction or symmetric difference \mathrm{E} + \boldsymbol\varepsilon. In this case the difference map \mathrm{D}f can be computed according to the formula \mathrm{D}f = (\mathrm{E} + \boldsymbol\varepsilon)f = \mathrm{E}f + \boldsymbol\varepsilon f.

The action of \mathrm{D} on our present example, f(p, q, r) = pqr, can be computed from the data on hand according to the following prescription.

The enlargement map \mathrm{E}f, computed in Post 5 and graphed in Post 6, is shown again here:

\mathrm{E}f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =

\begin{smallmatrix}  & p q r \,\cdot\,  \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}  & + & p q \texttt{(} r \texttt{)} \,\cdot\,  \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)} \mathrm{d}r  & + & p \texttt{(} q \texttt{)} r \,\cdot\,  \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}  & + & p \texttt{(} q \texttt{)(} r \texttt{)} \,\cdot\,  \texttt{(} \mathrm{d}p \texttt{)} \mathrm{d}q \, \mathrm{d}r  \\[4pt]  + & \texttt{(} p \texttt{)} q r \,\cdot\,  \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}  & + & \texttt{(} p \texttt{)} q \texttt{(} r \texttt{)} \,\cdot\,  \mathrm{d}p \texttt{(} \mathrm{d}q \texttt{)} \mathrm{d}r  & + & \texttt{(} p \texttt{)(} q \texttt{)} r \,\cdot\,  \mathrm{d}p \, \mathrm{d}q \texttt{(} \mathrm{d}r \texttt{)}  & + & \texttt{(} p \texttt{)(} q \texttt{)(} r \texttt{)} \,\cdot\,  \mathrm{d}p \, \mathrm{d}q \, \mathrm{d}r  \end{smallmatrix}

The tacit extension \boldsymbol\varepsilon f, computed in Post 7 and graphed in Post 8, is shown again here:

\boldsymbol\varepsilon f(p, q, r, \mathrm{d}p, \mathrm{d}q, \mathrm{d}r) =

\begin{array}{*{8}{l}}  &  p q r ~  \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}  & + &  p q r ~  \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}  & + &  p q r ~  \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}  & + &  p q r ~  \texttt{(} \mathrm{d}p \texttt{)~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}  \\[4pt]  + &  p q r ~  \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)}  & + &  p q r ~  \texttt{~} \mathrm{d}p \texttt{~(} \mathrm{d}q \texttt{)~} \mathrm{d}r \texttt{~}  & + &  p q r ~  \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~(} \mathrm{d}r \texttt{)}  & + &  p q r ~  \texttt{~} \mathrm{d}p \texttt{~~} \mathrm{d}q \texttt{~~} \mathrm{d}r \texttt{~}  \end{array}

The difference map \mathrm{D}f is the sum of the enlargement map \mathrm{E}f and the tacit extension \boldsymbol\varepsilon f.

Here we adopt a paradigm of computation for \mathrm{D}f that aids not only in organizing the stages of the work but also in highlighting the diverse facets of logical meaning that may be read off the result.

The terms of the enlargement map \mathrm{E}f are obtained from the table below by multiplying the base factor at the head of each column by the differential factor that appears beneath it in the body of the table.

Enlargement Map • Conjunction PQR

The terms of the tacit extension \boldsymbol\varepsilon f are obtained from the next table below by multiplying the base factor at the head of the first column by each of the differential factors that appear beneath it in the body of the table.

Tacit Extension • Conjunction PQR

Finally, the terms of the difference map \mathrm{D}f are obtained by overlaying the displays for \mathrm{E}f and \boldsymbol\varepsilon f and taking their boolean sum entry by entry.

Difference Map • Conjunction PQR

Notice that the “loop” or “no change” term p q r \cdot \texttt{(} \mathrm{d}p \texttt{)(} \mathrm{d}q \texttt{)(} \mathrm{d}r \texttt{)} cancels out, leaving 14 terms in the end.

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.

3 Responses to Frankl, My Dear : 9

  1. Pingback: Frankl, My Dear : 10 | Inquiry Into Inquiry

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

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