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Directory talk:Jon Awbrey/Papers/Differential Logic : Introduction (view source)

Revision as of 18:36, 29 November 2015

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We may understand the enlarged proposition <math>\mathrm{E}f\!</math> as telling us all the different ways to reach a model of the proposition <math>f\!</math> from each point of the universe <math>X.\!</math>
 
We may understand the enlarged proposition <math>\mathrm{E}f\!</math> as telling us all the different ways to reach a model of the proposition <math>f\!</math> from each point of the universe <math>X.\!</math>
 +
 +
==Propositional Forms on Two Variables==
 +
 +
To broaden our experience with simple examples, let us examine the sixteen functions of concrete type <math>P \times Q \to \mathbb{B}\!</math> and abstract type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.\!</math>  A&nbsp;few Tables are set here that detail the actions of <math>\mathrm{E}\!</math> and <math>\mathrm{D}\!</math> on each of these functions, allowing us to view the results in several different ways.
 +
 +
Tables&nbsp;A1 and A2 show two ways of arranging the 16 boolean functions on two variables, giving equivalent expressions for each function in several different systems of notation.
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Table A1.}~~\text{Propositional Forms on Two Variables}\!</math>
 +
|- style="background:#f0f0ff"
 +
| width="15%" |
 +
<p><math>\mathcal{L}_1\!</math></p>
 +
<p><math>\text{Decimal}\!</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_2\!</math></p>
 +
<p><math>\text{Binary}\!</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_3\!</math></p>
 +
<p><math>\text{Vector}\!</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_4\!</math></p>
 +
<p><math>\text{Cactus}\!</math></p>
 +
| width="25%" |
 +
<p><math>\mathcal{L}_5\!</math></p>
 +
<p><math>\text{English}\!</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_6~\!</math></p>
 +
<p><math>\text{Ordinary}\!</math></p>
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|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>p\colon\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>q\colon\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_0
 +
\\[4pt]
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_3
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_5
 +
\\[4pt]
 +
f_6
 +
\\[4pt]
 +
f_7
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0000}
 +
\\[4pt]
 +
f_{0001}
 +
\\[4pt]
 +
f_{0010}
 +
\\[4pt]
 +
f_{0011}
 +
\\[4pt]
 +
f_{0100}
 +
\\[4pt]
 +
f_{0101}
 +
\\[4pt]
 +
f_{0110}
 +
\\[4pt]
 +
f_{0111}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~0~0
 +
\\[4pt]
 +
0~0~0~1
 +
\\[4pt]
 +
0~0~1~0
 +
\\[4pt]
 +
0~0~1~1
 +
\\[4pt]
 +
0~1~0~0
 +
\\[4pt]
 +
0~1~0~1
 +
\\[4pt]
 +
0~1~1~0
 +
\\[4pt]
 +
0~1~1~1
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
(p)[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])(q)
 +
\\[4pt]
 +
(p,~q)
 +
\\[4pt]
 +
(p~~q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{false}
 +
\\[4pt]
 +
\text{neither}~ p ~\text{nor}~ q
 +
\\[4pt]
 +
q ~\text{without}~ p
 +
\\[4pt]
 +
\text{not}~ p
 +
\\[4pt]
 +
p ~\text{without}~ q
 +
\\[4pt]
 +
\text{not}~ q
 +
\\[4pt]
 +
p ~\text{not equal to}~ q
 +
\\[4pt]
 +
\text{not both}~ p ~\text{and}~ q
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
0
 +
\\[4pt]
 +
\lnot p \land \lnot q
 +
\\[4pt]
 +
\lnot p \land q
 +
\\[4pt]
 +
\lnot p
 +
\\[4pt]
 +
p \land \lnot q
 +
\\[4pt]
 +
\lnot q
 +
\\[4pt]
 +
p \ne q
 +
\\[4pt]
 +
\lnot p \lor \lnot q
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_8
 +
\\[4pt]
 +
f_9
 +
\\[4pt]
 +
f_{10}
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{12}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\\[4pt]
 +
f_{15}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{1000}
 +
\\[4pt]
 +
f_{1001}
 +
\\[4pt]
 +
f_{1010}
 +
\\[4pt]
 +
f_{1011}
 +
\\[4pt]
 +
f_{1100}
 +
\\[4pt]
 +
f_{1101}
 +
\\[4pt]
 +
f_{1110}
 +
\\[4pt]
 +
f_{1111}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
1~0~0~0
 +
\\[4pt]
 +
1~0~0~1
 +
\\[4pt]
 +
1~0~1~0
 +
\\[4pt]
 +
1~0~1~1
 +
\\[4pt]
 +
1~1~0~0
 +
\\[4pt]
 +
1~1~0~1
 +
\\[4pt]
 +
1~1~1~0
 +
\\[4pt]
 +
1~1~1~1
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~~p~~q~~
 +
\\[4pt]
 +
((p,~q))
 +
\\[4pt]
 +
18:36, 29 November 2015 (UTC)q~~
 +
\\[4pt]
 +
~(p~(q))
 +
\\[4pt]
 +
~~p18:36, 29 November 2015 (UTC)
 +
\\[4pt]
 +
((p)~q)~
 +
\\[4pt]
 +
((p)(q))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
p ~\text{and}~ q
 +
\\[4pt]
 +
p ~\text{equal to}~ q
 +
\\[4pt]
 +
q
 +
\\[4pt]
 +
\text{not}~ p ~\text{without}~ q
 +
\\[4pt]
 +
p
 +
\\[4pt]
 +
\text{not}~ q ~\text{without}~ p
 +
\\[4pt]
 +
p ~\text{or}~ q
 +
\\[4pt]
 +
\text{true}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
p \land q
 +
\\[4pt]
 +
p = q
 +
\\[4pt]
 +
q
 +
\\[4pt]
 +
p \Rightarrow q
 +
\\[4pt]
 +
p
 +
\\[4pt]
 +
p \Leftarrow q
 +
\\[4pt]
 +
p \lor q
 +
\\[4pt]
 +
1
 +
\end{matrix}\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Table A2.}~~\text{Propositional Forms on Two Variables}\!</math>
 +
|- style="background:#f0f0ff"
 +
| width="15%" |
 +
<p><math>\mathcal{L}_1\!</math></p>
 +
<p><math>\text{Decimal}\!</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_2\!</math></p>
 +
<p><math>\text{Binary}\!</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_3\!</math></p>
 +
<p><math>\text{Vector}\!</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_4\!</math></p>
 +
<p><math>\text{Cactus}\!</math></p>
 +
| width="25%" |
 +
<p><math>\mathcal{L}_5\!</math></p>
 +
<p><math>\text{English}\!</math></p>
 +
| width="15%" |
 +
<p><math>\mathcal{L}_6~\!</math></p>
 +
<p><math>\text{Ordinary}\!</math></p>
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>p\colon\!</math>
 +
| <math>1~1~0~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|- style="background:#f0f0ff"
 +
| &nbsp;
 +
| align="right" | <math>q\colon\!</math>
 +
| <math>1~0~1~0\!</math>
 +
| &nbsp;
 +
| &nbsp;
 +
| &nbsp;
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>f_{0000}\!</math>
 +
| <math>0~0~0~0\!</math>
 +
| <math>(~)\!</math>
 +
| <math>\text{false}\!</math>
 +
| <math>0\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0001}
 +
\\[4pt]
 +
f_{0010}
 +
\\[4pt]
 +
f_{0100}
 +
\\[4pt]
 +
f_{1000}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~0~1
 +
\\[4pt]
 +
0~0~1~0
 +
\\[4pt]
 +
0~1~0~0
 +
\\[4pt]
 +
1~0~0~0
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{neither}~ p ~\text{nor}~ q
 +
\\[4pt]
 +
q ~\text{without}~ p
 +
\\[4pt]
 +
p ~\text{without}~ q
 +
\\[4pt]
 +
p ~\text{and}~ q
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot p \land \lnot q
 +
\\[4pt]
 +
\lnot p \land q
 +
\\[4pt]
 +
p \land \lnot q
 +
\\[4pt]
 +
p \land q
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0011}
 +
\\[4pt]
 +
f_{1100}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
0~0~1~1
 +
\\[4pt]
 +
1~1~0~0
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ p
 +
\\[4pt]
 +
p
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot p
 +
\\[4pt]
 +
p
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0110}
 +
\\[4pt]
 +
f_{1001}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~1~0
 +
\\[4pt]
 +
1~0~0~1
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
p ~\text{not equal to}~ q
 +
\\[4pt]
 +
p ~\text{equal to}~ q
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
p \ne q
 +
\\[4pt]
 +
p = q
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0101}
 +
\\[4pt]
 +
f_{1010}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~0~1
 +
\\[4pt]
 +
1~0~1~0
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not}~ q
 +
\\[4pt]
 +
q
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot q
 +
\\[4pt]
 +
q
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
f_{0111}
 +
\\[4pt]
 +
f_{1011}
 +
\\[4pt]
 +
f_{1101}
 +
\\[4pt]
 +
f_{1110}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
0~1~1~1
 +
\\[4pt]
 +
1~0~1~1
 +
\\[4pt]
 +
1~1~0~1
 +
\\[4pt]
 +
1~1~1~0
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p~~q)~
 +
\\[4pt]
 +
~(p~(q))
 +
\\[4pt]
 +
((p)~q)~
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\text{not both}~ p ~\text{and}~ q
 +
\\[4pt]
 +
\text{not}~ p ~\text{without}~ q
 +
\\[4pt]
 +
\text{not}~ q ~\text{without}~ p
 +
\\[4pt]
 +
p ~\text{or}~ q
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\lnot p \lor \lnot q
 +
\\[4pt]
 +
p \Rightarrow q
 +
\\[4pt]
 +
p \Leftarrow q
 +
\\[4pt]
 +
p \lor q
 +
\end{matrix}\!</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>f_{1111}\!</math>
 +
| <math>1~1~1~1\!</math>
 +
| <math>((~))\!</math>
 +
| <math>\text{true}\!</math>
 +
| <math>1\!</math>
 +
|}
 +
 +
<br>
 +
 +
===Transforms Expanded over Differential Features===
 +
 +
The next four Tables expand the expressions of <math>\mathrm{E}f\!</math> and <math>\mathrm{D}f~\!</math> in two different ways, for each of the sixteen functions.  Notice that the functions are given in a different order, partitioned into seven natural classes by a group action.
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Table A3.}~~\mathrm{E}f ~\text{Expanded over Differential Features}~ \{ \mathrm{d}p, \mathrm{d}q \}\!</math>
 +
|- style="background:#f0f0ff"
 +
| width="10%" | &nbsp;
 +
| width="18%" | <math>f\!</math>
 +
| width="18%" |
 +
<p><math>\mathrm{T}_{11} f\!</math></p>
 +
<p><math>\mathrm{E}f|_{\mathrm{d}p~\mathrm{d}q}\!</math></p>
 +
| width="18%" |
 +
<p><math>\mathrm{T}_{10} f\!</math></p>
 +
<p><math>\mathrm{E}f|_{\mathrm{d}p(\mathrm{d}q)}\!</math></p>
 +
| width="18%" |
 +
<p><math>\mathrm{T}_{01} f\!</math></p>
 +
<p><math>\mathrm{E}f|_{(\mathrm{d}p)\mathrm{d}q}\!</math></p>
 +
| width="18%" |
 +
<p><math>\mathrm{T}_{00} f\!</math></p>
 +
<p><math>\mathrm{E}f|_{(\mathrm{d}p)(\mathrm{d}q)}\!</math></p>
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
(p)(q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\\[4pt]
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)~q~
 +
\\[4pt]
 +
(p)(q)
 +
\\[4pt]
 +
~p~~q~
 +
\\[4pt]
 +
~p~(q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~
 +
\\[4pt]
 +
(p)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~
 +
\\[4pt]
 +
(p)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((p)(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
(~p~~q~)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\\[4pt]
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~(q))
 +
\\[4pt]
 +
(~p~~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}\!</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
|- style="background:#f0f0ff"
 +
| colspan="2" | <math>\text{Fixed Point Total}\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>4\!</math>
 +
| <math>16\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Table A4.}~~\mathrm{D}f ~\text{Expanded over Differential Features}~ \{ \mathrm{d}p, \mathrm{d}q \}\!</math>
 +
|- style="background:#f0f0ff"
 +
| width="10%" | &nbsp;
 +
| width="18%" | <math>f\!</math>
 +
| width="18%" |
 +
<math>\mathrm{D}f|_{\mathrm{d}p~\mathrm{d}q}\!</math>
 +
| width="18%" |
 +
<math>\mathrm{D}f|_{\mathrm{d}p(\mathrm{d}q)}\!</math>
 +
| width="18%" |
 +
<math>\mathrm{D}f|_{(\mathrm{d}p)\mathrm{d}q}\!</math>
 +
| width="18%" |
 +
<math>\mathrm{D}f|_{(\mathrm{d}p)(\mathrm{d}q)}\!</math>
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\\[4pt]
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}~\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}~\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}~\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}~\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}~\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((~))
 +
\\[4pt]
 +
((~))
 +
\end{matrix}~\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p~~q)~
 +
\\[4pt]
 +
~(p~(q))
 +
\\[4pt]
 +
((p)~q)~
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((p,~q))
 +
\\[4pt]
 +
~(p,~q)~
 +
\\[4pt]
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\\[4pt]
 +
(q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~p~
 +
\\[4pt]
 +
~p~
 +
\\[4pt]
 +
(p)
 +
\\[4pt]
 +
(p)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\\[4pt]
 +
(~)
 +
\end{matrix}\!</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>((~))\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
|}
 +
 +
<br>
 +
 +
===Transforms Expanded over Ordinary Features===
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Table A5.}~~\mathrm{E}f ~\text{Expanded over Ordinary Features}~ \{ p, q \}\!</math>
 +
|- style="background:#f0f0ff"
 +
| width="10%" | &nbsp;
 +
| width="18%" | <math>f\!</math>
 +
| width="18%" | <math>\mathrm{E}f|_{pq}\!</math>
 +
| width="18%" | <math>\mathrm{E}f|_{p(q)}\!</math>
 +
| width="18%" | <math>\mathrm{E}f|_{(p)q}\!</math>
 +
| width="18%" | <math>\mathrm{E}f|_{(p)(q)}\!</math>
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~\mathrm{d}p~~\mathrm{d}q~
 +
\\[4pt]
 +
~\mathrm{d}p~(\mathrm{d}q)
 +
\\[4pt]
 +
(\mathrm{d}p)~\mathrm{d}q~
 +
\\[4pt]
 +
(\mathrm{d}p)(\mathrm{d}q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~\mathrm{d}p~(\mathrm{d}q)
 +
\\[4pt]
 +
~\mathrm{d}p~~\mathrm{d}q~
 +
\\[4pt]
 +
(\mathrm{d}p)(\mathrm{d}q)
 +
\\[4pt]
 +
(\mathrm{d}p)~\mathrm{d}q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p)~\mathrm{d}q~
 +
\\[4pt]
 +
(\mathrm{d}p)(\mathrm{d}q)
 +
\\[4pt]
 +
~\mathrm{d}p~~\mathrm{d}q~
 +
\\[4pt]
 +
~\mathrm{d}p~(\mathrm{d}q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p)(\mathrm{d}q)
 +
\\[4pt]
 +
(\mathrm{d}p)~\mathrm{d}q~
 +
\\[4pt]
 +
~\mathrm{d}p~(\mathrm{d}q)
 +
\\[4pt]
 +
~\mathrm{d}p~~\mathrm{d}q~
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~\mathrm{d}p~
 +
\\[4pt]
 +
(\mathrm{d}p)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~\mathrm{d}p~
 +
\\[4pt]
 +
(\mathrm{d}p)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p)
 +
\\[4pt]
 +
~\mathrm{d}p~
 +
\end{matrix}~\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p)
 +
\\[4pt]
 +
~\mathrm{d}p~
 +
\end{matrix}~\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\mathrm{d}p,~\mathrm{d}q)~
 +
\\[4pt]
 +
((\mathrm{d}p,~\mathrm{d}q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((\mathrm{d}p,~\mathrm{d}q))
 +
\\[4pt]
 +
~(\mathrm{d}p,~\mathrm{d}q)~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((\mathrm{d}p,~\mathrm{d}q))
 +
\\[4pt]
 +
~(\mathrm{d}p,~\mathrm{d}q)~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\mathrm{d}p,~\mathrm{d}q)~
 +
\\[4pt]
 +
((\mathrm{d}p,~\mathrm{d}q))
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~\mathrm{d}q~
 +
\\[4pt]
 +
(\mathrm{d}q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}q)
 +
\\[4pt]
 +
~\mathrm{d}q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~\mathrm{d}q~
 +
\\[4pt]
 +
(\mathrm{d}q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}q)
 +
\\[4pt]
 +
~\mathrm{d}q~
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\\[4pt]
 +
((\mathrm{d}p)~\mathrm{d}q~)
 +
\\[4pt]
 +
(~\mathrm{d}p~(\mathrm{d}q))
 +
\\[4pt]
 +
(~\mathrm{d}p~~\mathrm{d}q~)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((\mathrm{d}p)~\mathrm{d}q~)
 +
\\[4pt]
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\\[4pt]
 +
(~\mathrm{d}p~~\mathrm{d}q~)
 +
\\[4pt]
 +
(~\mathrm{d}p~(\mathrm{d}q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~\mathrm{d}p~(\mathrm{d}q))
 +
\\[4pt]
 +
(~\mathrm{d}p~~\mathrm{d}q~)
 +
\\[4pt]
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\\[4pt]
 +
((\mathrm{d}p)~\mathrm{d}q~)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~\mathrm{d}p~~\mathrm{d}q~)
 +
\\[4pt]
 +
(~\mathrm{d}p~(\mathrm{d}q))
 +
\\[4pt]
 +
((\mathrm{d}p)~\mathrm{d}q~)
 +
\\[4pt]
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\end{matrix}\!</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
|}
 +
 +
<br>
 +
 +
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:90%"
 +
|+ <math>\text{Table A6.}~~\mathrm{D}f ~\text{Expanded over Ordinary Features}~ \{ p, q \}\!</math>
 +
|- style="background:#f0f0ff"
 +
| width="10%" | &nbsp;
 +
| width="18%" | <math>f\!</math>
 +
| width="18%" | <math>\mathrm{D}f|_{pq}\!</math>
 +
| width="18%" | <math>\mathrm{D}f|_{p(q)}\!</math>
 +
| width="18%" | <math>\mathrm{D}f|_{(p)q}\!</math>
 +
| width="18%" | <math>\mathrm{D}f|_{(p)(q)}\!</math>
 +
|-
 +
| <math>f_0\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
| <math>(~)\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_1
 +
\\[4pt]
 +
f_2
 +
\\[4pt]
 +
f_4
 +
\\[4pt]
 +
f_8
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)(q)
 +
\\[4pt]
 +
(p)~q~
 +
\\[4pt]
 +
~p~(q)
 +
\\[4pt]
 +
~p~~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\mathrm{d}p~~\mathrm{d}q~~
 +
\\[4pt]
 +
~~\mathrm{d}p~(\mathrm{d}q)~
 +
\\[4pt]
 +
~(\mathrm{d}p)~\mathrm{d}q~~
 +
\\[4pt]
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\mathrm{d}p~(\mathrm{d}q)~
 +
\\[4pt]
 +
~~\mathrm{d}p~~\mathrm{d}q~~
 +
\\[4pt]
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\\[4pt]
 +
~(\mathrm{d}p)~\mathrm{d}q~~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\mathrm{d}p)~\mathrm{d}q~~
 +
\\[4pt]
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\\[4pt]
 +
~~\mathrm{d}p~~\mathrm{d}q~~
 +
\\[4pt]
 +
~~\mathrm{d}p~(\mathrm{d}q)~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\\[4pt]
 +
~(\mathrm{d}p)~\mathrm{d}q~~
 +
\\[4pt]
 +
~~\mathrm{d}p~(\mathrm{d}q)~
 +
\\[4pt]
 +
~~\mathrm{d}p~~\mathrm{d}q~~
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_3
 +
\\[4pt]
 +
f_{12}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(p)
 +
\\[4pt]
 +
~p~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}p
 +
\\[4pt]
 +
\mathrm{d}p
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}p
 +
\\[4pt]
 +
\mathrm{d}p
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}p
 +
\\[4pt]
 +
\mathrm{d}p
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}p
 +
\\[4pt]
 +
\mathrm{d}p
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_6
 +
\\[4pt]
 +
f_9
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(p,~q)~
 +
\\[4pt]
 +
((p,~q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\\[4pt]
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\\[4pt]
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\\[4pt]
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\\[4pt]
 +
(\mathrm{d}p,~\mathrm{d}q)
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_5
 +
\\[4pt]
 +
f_{10}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(q)
 +
\\[4pt]
 +
~q~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}q
 +
\\[4pt]
 +
\mathrm{d}q
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}q
 +
\\[4pt]
 +
\mathrm{d}q
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}q
 +
\\[4pt]
 +
\mathrm{d}q
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
\mathrm{d}q
 +
\\[4pt]
 +
\mathrm{d}q
 +
\end{matrix}\!</math>
 +
|-
 +
|
 +
<math>\begin{matrix}
 +
f_7
 +
\\[4pt]
 +
f_{11}
 +
\\[4pt]
 +
f_{13}
 +
\\[4pt]
 +
f_{14}
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
(~p~~q~)
 +
\\[4pt]
 +
(~p~(q))
 +
\\[4pt]
 +
((p)~q~)
 +
\\[4pt]
 +
((p)(q))
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\\[4pt]
 +
~(\mathrm{d}p)~\mathrm{d}q~~
 +
\\[4pt]
 +
~~\mathrm{d}p~(\mathrm{d}q)~
 +
\\[4pt]
 +
~~\mathrm{d}p~~\mathrm{d}q~~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~(\mathrm{d}p)~\mathrm{d}q~~
 +
\\[4pt]
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\\[4pt]
 +
~~\mathrm{d}p~~\mathrm{d}q~~
 +
\\[4pt]
 +
~~\mathrm{d}p~(\mathrm{d}q)~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\mathrm{d}p~(\mathrm{d}q)~
 +
\\[4pt]
 +
~~\mathrm{d}p~~\mathrm{d}q~~
 +
\\[4pt]
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\\[4pt]
 +
~(\mathrm{d}p)~\mathrm{d}q~~
 +
\end{matrix}\!</math>
 +
|
 +
<math>\begin{matrix}
 +
~~\mathrm{d}p~~\mathrm{d}q~~
 +
\\[4pt]
 +
~~\mathrm{d}p~(\mathrm{d}q)~
 +
\\[4pt]
 +
~(\mathrm{d}p)~\mathrm{d}q~~
 +
\\[4pt]
 +
((\mathrm{d}p)(\mathrm{d}q))
 +
\end{matrix}\!</math>
 +
|-
 +
| <math>f_{15}\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
| <math>((~))\!</math>
 +
|}
 +
 +
<br>
    
==Logical Cacti==
 
==Logical Cacti==
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/466973"