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| | | |
| <br> | | <br> |
| + | |
| + | ===Appendix 3. Computational Details=== |
| + | |
| + | ====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(u, v)==== |
| + | |
| + | =====Computation of ε''f''<sub>8</sub>===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F8.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{8}~\!</math> |
| + | | |
| + | <math>\begin{array}{*{10}{l}} |
| + | \boldsymbol\varepsilon f_{8} |
| + | & = && f_{8}(u, v) |
| + | \\[4pt] |
| + | & = && uv |
| + | \\[4pt] |
| + | & = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | & + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v |
| + | & + & uv \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & uv \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | \\[20pt] |
| + | \boldsymbol\varepsilon f_{8} |
| + | & = && uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | \\[4pt] |
| + | && + & uv \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | \\[4pt] |
| + | && + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | \\[4pt] |
| + | && + & uv \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~} |
| + | \end{array}\!</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | =====Computation of E''f''<sub>8</sub>===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F8.2-i} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 1)}\!</math> |
| + | | |
| + | <math>\begin{array}{*{9}{l}} |
| + | \mathrm{E}f_{8} |
| + | & = & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v) |
| + | \\[4pt] |
| + | & = & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)} |
| + | \\[4pt] |
| + | & = & \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)}) |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{8}(\texttt{(} \mathrm{d}u \texttt{)}, \mathrm{d}v) |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{8}(\mathrm{d}u, \texttt{(} \mathrm{d}v \texttt{)}) |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{8}(\mathrm{d}u, \mathrm{d}v) |
| + | \\[4pt] |
| + | & = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | \\[20pt] |
| + | \mathrm{E}f_{8} |
| + | & = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | \\[4pt] |
| + | &&& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v |
| + | \\[4pt] |
| + | &&&&& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} |
| + | \\[4pt] |
| + | &&&&&&& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | \end{array}\!</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math> |
| + | | |
| + | <math>\begin{array}{*{9}{c}} |
| + | \mathrm{E}f_{8} |
| + | & = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v) |
| + | \\[6pt] |
| + | & = & u \cdot v |
| + | & + & u \cdot \mathrm{d}v |
| + | & + & v \cdot \mathrm{d}u |
| + | & + & \mathrm{d}u \cdot \mathrm{d}v |
| + | \\[6pt] |
| + | \mathrm{E}f_{8} |
| + | & = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | \end{array}\!</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | =====Computation of D''f''<sub>8</sub>===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F8.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 1)}\!</math> |
| + | | |
| + | <math>\begin{array}{*{10}{l}} |
| + | \mathrm{D}f_{8} |
| + | & = && \mathrm{E}f_{8} |
| + | & + & \boldsymbol\varepsilon f_{8} |
| + | \\[4pt] |
| + | & = && f_{8}(u + \mathrm{d}u, v + \mathrm{d}v) |
| + | & + & f_{8}(u, v) |
| + | \\[4pt] |
| + | & = && \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & uv |
| + | \\[20pt] |
| + | \mathrm{D}f_{8} |
| + | & = && 0 |
| + | & + & 0 |
| + | & + & 0 |
| + | & + & 0 |
| + | \\[4pt] |
| + | && + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | & + & 0 |
| + | & + & 0 |
| + | \\[4pt] |
| + | && + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~} |
| + | & + & 0 |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | & + & 0 |
| + | \\[4pt] |
| + | && + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~} |
| + | & + & 0 |
| + | & + & 0 |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~} |
| + | \\[20pt] |
| + | \mathrm{D}f_{8} |
| + | & = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{~} |
| + | \end{array}\!</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F8.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 2)}\!</math> |
| + | | |
| + | <math>\begin{array}{*{9}{l}} |
| + | \mathrm{D}f_{8} |
| + | & = & \boldsymbol\varepsilon f_{8} |
| + | & + & \mathrm{E}f_{8} |
| + | \\[6pt] |
| + | & = & f_{8}(u, v) |
| + | & + & f_{8}(u + \mathrm{d}u, v + \mathrm{d}v) |
| + | \\[6pt] |
| + | & = & uv |
| + | & + & \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)(} v \texttt{,} \mathrm{d}v \texttt{)} |
| + | \\[6pt] |
| + | & = & 0 |
| + | & + & u \cdot \mathrm{d}v |
| + | & + & v \cdot \mathrm{d}u |
| + | & + & \mathrm{d}u ~ \mathrm{d}v |
| + | \\[6pt] |
| + | \mathrm{D}f_{8} |
| + | & = & 0 |
| + | & + & u \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v |
| + | & + & v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F8.3-iii} ~~ \text{Computation of}~ \mathrm{D}f_{8} ~\text{(Method 3)}\!</math> |
| + | | |
| + | <math>\begin{array}{c*{9}{l}} |
| + | \mathrm{D}f_{8} |
| + | & = & \boldsymbol\varepsilon f_{8} ~+~ \mathrm{E}f_{8} |
| + | \\[20pt] |
| + | \boldsymbol\varepsilon f_{8} |
| + | & = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v |
| + | & + & ~ u \,\cdot\, v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & ~ u \;\cdot\; v \;\cdot\; \mathrm{d}u ~ \mathrm{d}v |
| + | \\[6pt] |
| + | \mathrm{E}f_{8} |
| + | & = & u \,\cdot\, v \,\cdot\, \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & u ~ \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)} ~ v \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)} \texttt{(} v \texttt{)} \cdot\, \mathrm{d}u ~ \mathrm{d}v |
| + | \\[20pt] |
| + | \mathrm{D}f_{8} |
| + | & = & ~~~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & ~~~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v |
| + | & + & ~~~~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | \end{array}\!</math> |
| + | |} |
| + | |
| + | =====Computation of d''f''<sub>8</sub>===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F8.4} ~~ \text{Computation of}~ \mathrm{d}f_{8}\!</math> |
| + | | |
| + | <math>\begin{array}{c*{8}{l}} |
| + | \mathrm{D}f_{8} |
| + | & = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} |
| + | & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | \\[6pt] |
| + | \Downarrow |
| + | \\[6pt] |
| + | \mathrm{d}f_{8} |
| + | & = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0 |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | =====Computation of r''f''<sub>8</sub>===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F8.5} ~~ \text{Computation of}~ \mathrm{r}f_{8}\!</math> |
| + | | |
| + | <math>\begin{array}{c*{8}{l}} |
| + | \mathrm{r}f_{8} & = & \mathrm{D}f_{8} ~+~ \mathrm{d}f_{8} |
| + | \\[20pt] |
| + | \mathrm{D}f_{8} |
| + | & = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} |
| + | & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | \\[6pt] |
| + | \mathrm{d}f_{8} |
| + | & = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0 |
| + | \\[20pt] |
| + | \mathrm{r}f_{8} |
| + | & = & uv \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | =====Computation Summary for Conjunction===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F8.6} ~~ \text{Computation Summary for}~ f_{8}(u, v) = uv\!</math> |
| + | | |
| + | <math>\begin{array}{c*{8}{l}} |
| + | \boldsymbol\varepsilon f_{8} |
| + | & = & uv \cdot 1 |
| + | & + & u \texttt{(} v \texttt{)} \cdot 0 |
| + | & + & \texttt{(} u \texttt{)} v \cdot 0 |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0 |
| + | \\[6pt] |
| + | \mathrm{E}f_{8} |
| + | & = & uv \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | \\[6pt] |
| + | \mathrm{D}f_{8} |
| + | & = & uv \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} |
| + | & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | \\[6pt] |
| + | \mathrm{d}f_{8} |
| + | & = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0 |
| + | \\[6pt] |
| + | \mathrm{r}f_{8} |
| + | & = & uv \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | ====Operator Maps for the Logical Equality ''f''<sub>9</sub>(u, v)==== |
| + | |
| + | =====Computation of ε''f''<sub>9</sub>===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F9.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{9}\!</math> |
| + | | |
| + | <math>\begin{array}{*{10}{l}} |
| + | \boldsymbol\varepsilon f_{9} |
| + | & = && f_{9}(u, v) |
| + | \\[4pt] |
| + | & = && \texttt{((} u \texttt{,~} v \texttt{))} |
| + | \\[4pt] |
| + | & = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{9}(1, 1) |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{9}(1, 0) |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{9}(0, 1) |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{9}(0, 0) |
| + | \\[4pt] |
| + | & = && u v & + & 0 & + & 0 & + & \texttt{(} u \texttt{)(} v \texttt{)} |
| + | \\[20pt] |
| + | \boldsymbol\varepsilon f_{9} |
| + | & = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | & + & 0 |
| + | & + & 0 |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | \\[4pt] |
| + | && + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | & + & 0 |
| + | & + & 0 |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | \\[4pt] |
| + | && + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | & + & 0 |
| + | & + & 0 |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | \\[4pt] |
| + | && + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~} |
| + | & + & 0 |
| + | & + & 0 |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~} |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | =====Computation of E''f''<sub>9</sub>===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F9.2} ~~ \text{Computation of}~ \mathrm{E}f_{9}\!</math> |
| + | | |
| + | <math>\begin{array}{*{10}{l}} |
| + | \mathrm{E}f_{9} |
| + | & = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v) |
| + | \\[4pt] |
| + | & = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))} |
| + | \\[4pt] |
| + | & = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)}) |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{9}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ }) |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)}) |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{9}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ }) |
| + | \\[4pt] |
| + | & = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) } |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{ (} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{) } |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))} |
| + | \\[20pt] |
| + | \mathrm{E}f_{9} |
| + | & = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | & + & 0 |
| + | & + & 0 |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | \\[4pt] |
| + | && + & 0 |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | & + & 0 |
| + | \\[4pt] |
| + | && + & 0 |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | & + & 0 |
| + | \\[4pt] |
| + | && + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~} |
| + | & + & 0 |
| + | & + & 0 |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~} |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | =====Computation of D''f''<sub>9</sub>===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F9.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 1)}\!</math> |
| + | | |
| + | <math>\begin{array}{*{10}{l}} |
| + | \mathrm{D}f_{9} |
| + | & = && \mathrm{E}f_{9} |
| + | & + & \boldsymbol\varepsilon f_{9} |
| + | \\[4pt] |
| + | & = && f_{9}(u + \mathrm{d}u, v + \mathrm{d}v) |
| + | & + & f_{9}(u, v) |
| + | \\[4pt] |
| + | & = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{),(} v \texttt{,} \mathrm{d}v \texttt{)))} |
| + | & + & \texttt{((} u \texttt{,} v \texttt{))} |
| + | \\[20pt] |
| + | \mathrm{D}f_{9} |
| + | & = && 0 |
| + | & + & 0 |
| + | & + & 0 |
| + | & + & 0 |
| + | \\[4pt] |
| + | && + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | \\[4pt] |
| + | && + & \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | \\[4pt] |
| + | && + & 0 |
| + | & + & 0 |
| + | & + & 0 |
| + | & + & 0 |
| + | \\[20pt] |
| + | \mathrm{D}f_{9} |
| + | & = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | \end{array}\!</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F9.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{9} ~\text{(Method 2)}\!</math> |
| + | | |
| + | <math>\begin{array}{*{9}{l}} |
| + | \mathrm{D}f_{9} |
| + | & = & 0 \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & 1 \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & 1 \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v |
| + | & + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | =====Computation of d''f''<sub>9</sub>===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F9.4} ~~ \text{Computation of}~ \mathrm{d}f_{9}\!</math> |
| + | | |
| + | <math>\begin{array}{c*{8}{l}} |
| + | \mathrm{D}f_{9} |
| + | & = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | \\[6pt] |
| + | \Downarrow |
| + | \\[6pt] |
| + | \mathrm{d}f_{9} |
| + | & = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | =====Computation of r''f''<sub>9</sub>===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F9.5} ~~ \text{Computation of}~ \mathrm{r}f_{9}\!</math> |
| + | | |
| + | <math>\begin{array}{c*{8}{l}} |
| + | \mathrm{r}f_{9} & = & \mathrm{D}f_{9} ~+~ \mathrm{d}f_{9} |
| + | \\[20pt] |
| + | \mathrm{D}f_{9} |
| + | & = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | \\[6pt] |
| + | \mathrm{d}f_{9} |
| + | & = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | \\[20pt] |
| + | \mathrm{r}f_{9} |
| + | & = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0 |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot 0 |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot 0 |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0 |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | =====Computation Summary for Equality===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F9.6} ~~ \text{Computation Summary for}~ f_{9}(u, v) = \texttt{((} u \texttt{,} v \texttt{))}\!</math> |
| + | | |
| + | <math>\begin{array}{c*{8}{l}} |
| + | \boldsymbol\varepsilon f_{9} |
| + | & = & uv \cdot 1 |
| + | & + & u \texttt{(} v \texttt{)} \cdot 0 |
| + | & + & \texttt{(} u \texttt{)} v \cdot 0 |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1 |
| + | \\[6pt] |
| + | \mathrm{E}f_{9} |
| + | & = & uv \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))} |
| + | & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{))} |
| + | \\[6pt] |
| + | \mathrm{D}f_{9} |
| + | & = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | \\[6pt] |
| + | \mathrm{d}f_{9} |
| + | & = & uv \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | \\[6pt] |
| + | \mathrm{r}f_{9} |
| + | & = & uv \cdot 0 |
| + | & + & u \texttt{(} v \texttt{)} \cdot 0 |
| + | & + & \texttt{(} u \texttt{)} v \cdot 0 |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0 |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | ====Operator Maps for the Logical Disjunction ''f''<sub>14</sub>(u, v)==== |
| + | |
| + | =====Computation of ε''f''<sub>14</sub>===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F14.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{14}\!</math> |
| + | | |
| + | <math>\begin{array}{*{10}{l}} |
| + | \boldsymbol\varepsilon f_{14} |
| + | & = && f_{14}(u, v) |
| + | \\[4pt] |
| + | & = && \texttt{((} u \texttt{)(} v \texttt{))} |
| + | \\[4pt] |
| + | & = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{14}(1, 1) |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{14}(1, 0) |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{14}(0, 1) |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{14}(0, 0) |
| + | \\[4pt] |
| + | & = && \texttt{ } u \texttt{ } v \texttt{ } |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } |
| + | & + & 0 |
| + | \\[20pt] |
| + | \boldsymbol\varepsilon f_{14} |
| + | & = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | & + & 0 |
| + | \\[4pt] |
| + | && + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | & + & 0 |
| + | \\[4pt] |
| + | && + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | & + & 0 |
| + | \\[4pt] |
| + | && + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~} |
| + | & + & 0 |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | =====Computation of E''f''<sub>14</sub>===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F14.2} ~~ \text{Computation of}~ \mathrm{E}f_{14}\!</math> |
| + | | |
| + | <math>\begin{array}{*{10}{l}} |
| + | \mathrm{E}f_{14} |
| + | & = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v) |
| + | \\[4pt] |
| + | & = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))} |
| + | \\[4pt] |
| + | & = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{(} \mathrm{d}v \texttt{)}) |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! f_{14}(\texttt{(} \mathrm{d}u \texttt{)}, \texttt{ } \mathrm{d}v \texttt{ }) |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{(} \mathrm{d}v \texttt{)}) |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! f_{14}(\texttt{ } \mathrm{d}u \texttt{ }, \texttt{ } \mathrm{d}v \texttt{ }) |
| + | \\[4pt] |
| + | & = && \texttt{ } u \texttt{ } v \texttt{ } \!\cdot\! \texttt{(} \mathrm{d}u \texttt{~} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \!\cdot\! \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} |
| + | \\[20pt] |
| + | \mathrm{E}f_{14} |
| + | & = && \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | & + & 0 |
| + | \\[4pt] |
| + | && + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | & + & 0 |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~} |
| + | \\[4pt] |
| + | && + & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | & + & 0 |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)} |
| + | \\[4pt] |
| + | && + & 0 |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~} |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | =====Computation of D''f''<sub>14</sub>===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F14.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 1)}\!</math> |
| + | | |
| + | <math>\begin{array}{*{10}{l}} |
| + | \mathrm{D}f_{14} |
| + | & = && \mathrm{E}f_{14} |
| + | & + & \boldsymbol\varepsilon f_{14} |
| + | \\[4pt] |
| + | & = && f_{14}(u + \mathrm{d}u, v + \mathrm{d}v) |
| + | & + & f_{14}(u, v) |
| + | \\[4pt] |
| + | & = && \texttt{(((} u \texttt{,} \mathrm{d}u \texttt{))((} v \texttt{,} \mathrm{d}v \texttt{)))} |
| + | & + & \texttt{((} u \texttt{)(} v \texttt{))} |
| + | \\[20pt] |
| + | \mathrm{D}f_{14} |
| + | & = && 0 |
| + | & + & 0 |
| + | & + & 0 |
| + | & + & 0 |
| + | \\[4pt] |
| + | && + & 0 |
| + | & + & 0 |
| + | & + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~} |
| + | \\[4pt] |
| + | && + & 0 |
| + | & + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & 0 |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~} |
| + | \\[4pt] |
| + | && + & uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v |
| + | & + & 0 |
| + | & + & 0 |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~} |
| + | \\[20pt] |
| + | \mathrm{D}f_{14} |
| + | & = && uv \!\cdot\! \mathrm{d}u ~ \mathrm{d}v |
| + | & + & u \texttt{(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)} v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F14.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{14} ~\text{(Method 2)}\!</math> |
| + | | |
| + | <math>\begin{array}{*{9}{l}} |
| + | \mathrm{D}f_{14} |
| + | & = & \texttt{((} u \texttt{,} v \texttt{))} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v |
| + | & + & 0 \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)} |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | =====Computation of d''f''<sub>14</sub>===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F14.4} ~~ \text{Computation of}~ \mathrm{d}f_{14}\!</math> |
| + | | |
| + | <math>\begin{array}{c*{8}{l}} |
| + | \mathrm{D}f_{14} |
| + | & = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} |
| + | \\[6pt] |
| + | \Downarrow |
| + | \\[6pt] |
| + | \mathrm{d}f_{14} |
| + | & = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0 |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | =====Computation of r''f''<sub>14</sub>===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F14.5} ~~ \text{Computation of}~ \mathrm{r}f_{14}\!</math> |
| + | | |
| + | <math>\begin{array}{c*{8}{l}} |
| + | \mathrm{r}f_{14} & = & \mathrm{D}f_{14} ~+~ \mathrm{d}f_{14} |
| + | \\[20pt] |
| + | \mathrm{D}f_{14} |
| + | & = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} |
| + | \\[6pt] |
| + | \mathrm{d}f_{14} |
| + | & = & \texttt{ } u \texttt{ } v \texttt{ } \cdot 0 |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | \\[20pt] |
| + | \mathrm{r}f_{14} |
| + | & = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | =====Computation Summary for Disjunction===== |
| + | |
| + | <br> |
| + | |
| + | {| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%" |
| + | |+ style="height:30px" | <math>\text{Table F14.6} ~~ \text{Computation Summary for}~ f_{14}(u, v) = \texttt{((} u \texttt{)(} v \texttt{))}\!</math> |
| + | | |
| + | <math>\begin{array}{c*{8}{l}} |
| + | \boldsymbol\varepsilon f_{14} |
| + | & = & uv \cdot 1 |
| + | & + & u \texttt{(} v \texttt{)} \cdot 1 |
| + | & + & \texttt{(} u \texttt{)} v \cdot 1 |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 0 |
| + | \\[6pt] |
| + | \mathrm{E}f_{14} |
| + | & = & uv \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)} |
| + | & + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))} |
| + | & + & \texttt{(} u \texttt{)} v \cdot \texttt{((} \mathrm{d}u \texttt{)} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} |
| + | \\[6pt] |
| + | \mathrm{D}f_{14} |
| + | & = & uv \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)} |
| + | & + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))} |
| + | \\[6pt] |
| + | \mathrm{d}f_{14} |
| + | & = & uv \cdot 0 |
| + | & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u |
| + | & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)} |
| + | \\[6pt] |
| + | \mathrm{r}f_{14} |
| + | & = & uv \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & u \texttt{(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | & + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v |
| + | \end{array}</math> |
| + | |} |
| + | |
| + | <br> |
| + | |
| + | ===Appendix 4. Source Materials=== |
| + | |
| + | ===Appendix 5. Various Definitions of the Tangent Vector=== |
| | | |
| ==References== | | ==References== |
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| * Ashby, William Ross (1956/1964), ''An Introduction to Cybernetics'', Chapman and Hall, London, UK, 1956. Reprinted, Methuen and Company, London, UK, 1964. | | * Ashby, William Ross (1956/1964), ''An Introduction to Cybernetics'', Chapman and Hall, London, UK, 1956. Reprinted, Methuen and Company, London, UK, 1964. |
| | | |
− | * Awbrey, J., and Awbrey, S. (1989), "Theme One : A Program of Inquiry", Unpublished Manuscript, 09 Aug 1989. [http://www.iki.fi/~kartturi/Awbrey/Theme1Prog/Theme1Guide.doc Microsoft Word Document]. | + | * Awbrey, J., and Awbrey, S. (1989), "Theme One : A Program of Inquiry", Unpublished Manuscript, 09 Aug 1989. [http://web.archive.org/web/20071021145200/http://ndirty.cute.fi/~karttu/Awbrey/Theme1Prog/Theme1Guide.doc Microsoft Word Document]. |
| | | |
| * Edelman, Gerald M. (1988), ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY. | | * Edelman, Gerald M. (1988), ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY. |
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| [[Category:Adaptive Systems]] | | [[Category:Adaptive Systems]] |
| [[Category:Artificial Intelligence]] | | [[Category:Artificial Intelligence]] |
| + | [[Category:Boolean Functions]] |
| [[Category:Combinatorics]] | | [[Category:Combinatorics]] |
| [[Category:Computer Science]] | | [[Category:Computer Science]] |
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| [[Category:Mathematics]] | | [[Category:Mathematics]] |
| [[Category:Mathematical Systems Theory]] | | [[Category:Mathematical Systems Theory]] |
| + | [[Category:Philosophy]] |
| [[Category:Science]] | | [[Category:Science]] |
| [[Category:Semiotics]] | | [[Category:Semiotics]] |
− | [[Category:Philosophy]]
| |
| [[Category:Systems Science]] | | [[Category:Systems Science]] |
| [[Category:Visualization]] | | [[Category:Visualization]] |