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A '''differential propositional calculus''' is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a [[universe of discourse]] or transformations that map a source universe into a target universe.
A '''differential propositional calculus''' is a [[propositional calculus]] extended by a set of terms for describing aspects of change and difference, for example, processes that take place in a [[universe of discourse]] or transformations that map a source universe into a target universe.
==Casual Introduction==
==1. Casual Introduction==
Consider the situation represented by the venn diagram in Figure 1.
Consider the situation represented by the venn diagram in Figure 1.
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|}
The area of the rectangle represents a universe of discourse, <math>X.\!</math> This might be a population of individuals having various additional properties or it might be a collection of locations that various individuals occupy. The area of the "circle" represents the individuals that have the property <math>q\!</math> or the locations that fall within the corresponding region <math>Q.\!</math> Four individuals, <math>a, b, c, d,\!</math> are singled out by name. It happens that <math>b\!</math> and <math>c\!</math> currently reside in region <math>Q\!</math> while <math>a\!</math> and <math>d\!</math> do not.
The area of the rectangle represents a universe of discourse, <math>X.\!</math> This might be a population of individuals having various additional properties or it might be a collection of locations that various individuals occupy. The area of the “circle” represents the individuals that have the property <math>q\!</math> or the locations that fall within the corresponding region <math>Q.\!</math> Four individuals, <math>a, b, c, d,\!</math> are singled out by name. It happens that <math>b\!</math> and <math>c\!</math> currently reside in region <math>Q\!</math> while <math>a\!</math> and <math>d\!</math> do not.
Now consider the situation represented by the venn diagram in Figure 2.
Now consider the situation represented by the venn diagram in Figure 2.
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|}
Figure 2 differs from Figure 1 solely in the circumstance that the object <math>c\!</math> is outside the region <math>Q\!</math> while the object <math>d\!</math> is inside the region <math>Q.\!</math> So far, there is nothing that says that our encountering these Figures in this order is other than purely accidental, but if we interpret the present sequence of frames as a "moving picture" representation of their natural order in a temporal process, then it would be natural to say that <math>a\!</math> and <math>b\!</math> have remained as they were with regard to quality <math>q\!</math> while <math>c\!</math> and <math>d\!</math> have changed their standings in that respect. In particular, <math>c\!</math> has moved from the region where <math>q\!</math> is <math>true\!</math> to the region where <math>q\!</math> is <math>false\!</math> while <math>d\!</math> has moved from the region where <math>q\!</math> is <math>false\!</math> to the region where <math>q\!</math> is <math>true.\!</math>
Figure 2 differs from Figure 1 solely in the circumstance that the object <math>c\!</math> is outside the region <math>Q\!</math> while the object <math>d\!</math> is inside the region <math>Q.\!</math> So far, there is nothing that says that our encountering these Figures in this order is other than purely accidental, but if we interpret the present sequence of frames as a “moving picture” representation of their natural order in a temporal process, then it would be natural to say that <math>a\!</math> and <math>b\!</math> have remained as they were with regard to quality <math>q\!</math> while <math>c\!</math> and <math>d\!</math> have changed their standings in that respect. In particular, <math>c\!</math> has moved from the region where <math>q\!</math> is <math>\mathrm{true}\!</math> to the region where <math>q\!</math> is <math>\mathrm{false}\!</math> while <math>d\!</math> has moved from the region where <math>q\!</math> is <math>\mathrm{false}\!</math> to the region where <math>q\!</math> is <math>\mathrm{true}.\!</math>
Figure 3 reprises the situation shown in Figure 1, but this time interpolates a new quality that is specifically tailored to account for the relation between Figure 1 and Figure 2.
Figure 3 reprises the situation shown in Figure 1, but this time interpolates a new quality that is specifically tailored to account for the relation between Figure 1 and Figure 2.
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This new quality, <math>\mathrm{d}q,\!</math> is an example of a ''differential quality'', since its absence or presence qualifies the absence or presence of change occurring in another quality. As with any other quality, it is represented in the venn diagram by means of a "circle" that distinguishes two halves of the universe of discourse, in this case, the portions of <math>X\!</math> outside and inside the region <math>\mathrm{d}Q.\!</math>
This new quality, <math>\mathrm{d}q,\!</math> is an example of a ''differential quality'', since its absence or presence qualifies the absence or presence of change occurring in another quality. As with any other quality, it is represented in the venn diagram by means of a “circle” that distinguishes two halves of the universe of discourse, in this case, the portions of <math>X\!</math> outside and inside the region <math>\mathrm{d}Q.\!</math>
Figure 1 represents a universe of discourse, <math>X,\!</math> together with a basis of discussion, <math>\{ q \},\!</math> for expressing propositions about the contents of that universe. Once the quality <math>q\!</math> is given a name, say, the symbol "<math>q\!</math>", we have the basis for a formal language that is specifically cut out for discussing <math>X\!</math> in terms of <math>q,\!</math> and this formal language is more formally known as the ''propositional calculus'' with alphabet <math>\{\!</math>"<math>q\!</math>"<math>\}.\!</math>
Figure 1 represents a universe of discourse, <math>X,\!</math> together with a basis of discussion, <math>\{ q \},\!</math> for expressing propositions about the contents of that universe. Once the quality <math>q\!</math> is given a name, say, the symbol <math>{}^{\backprime\backprime} q {}^{\prime\prime},\!</math> we have the basis for a formal language that is specifically cut out for discussing <math>X\!</math> in terms of <math>q,\!</math> and this formal language is more formally known as the ''propositional calculus'' with alphabet <math>\{ {}^{\backprime\backprime} q {}^{\prime\prime} \}.\!</math>
In the context marked by <math>X\!</math> and <math>\{ q \}\!</math> there are but four different pieces of information that can be expressed in the corresponding propositional calculus, namely, the propositions: <math>false,\!</math> <math>\lnot q,\!</math> <math>q,\!</math> <math>true.\!</math> Referring to the sample of points in Figure 1, <math>false\!</math> holds of no points, <math>\lnot q\!</math> holds of <math>a\!</math> and <math>d,\!</math> <math>q\!</math> holds of <math>b\!</math> and <math>c,\!</math> and <math>true\!</math> holds of all points in the sample.
In the context marked by <math>X\!</math> and <math>\{ q \}\!</math> there are but four different pieces of information that can be expressed in the corresponding propositional calculus, namely, the propositions: <math>\mathrm{false}, ~ \lnot q, ~ q, ~ \mathrm{true}.\!</math> Referring to the sample of points in Figure 1, the constant proposition <math>\mathrm{false}\!</math> holds of no points, the proposition <math>\lnot q\!</math> holds of <math>a\!</math> and <math>d,\!</math> the proposition <math>q\!</math> holds of <math>b\!</math> and <math>c,\!</math> and the constant proposition <math>\mathrm{true}\!</math> holds of all points in the sample.
Figure 3 preserves the same universe of discourse and extends the basis of discussion to a set of two qualities, <math>\{ q, \mathrm{d}q \}.\!</math> In parallel fashion, the initial propositional calculus is extended by means of the enlarged alphabet, <math>\{\!</math>"<math>q\!</math>"<math>,\!</math> "<math>\mathrm{d}q\!</math>"<math>\}.\!</math> Any propositional calculus over two basic propositions allows for the expression of 16 propositions all together. Just by way of salient examples in the present setting, we can pick out the most informative propositions that apply to each of our sample points. Using overlines to express logical negation, these are given as follows:
Figure 3 preserves the same universe of discourse and extends the basis of discussion to a set of two qualities, <math>\{ q, \mathrm{d}q \}.\!</math> In parallel fashion, the initial propositional calculus is extended by means of the enlarged alphabet, <math>\{ {}^{\backprime\backprime} q {}^{\prime\prime}, {}^{\backprime\backprime} \mathrm{d}q {}^{\prime\prime} \}.\!</math> Any propositional calculus over two basic propositions allows for the expression of 16 propositions all together. Just by way of salient examples in the present setting, we can pick out the most informative propositions that apply to each of our sample points. Using overlines to express logical negation, these are given as follows:
:* <p><math>\overline{q} ~ \overline{\mathrm{d}q}\!</math> describes <math>a\!</math></p>
:* <p><math>\overline{q} ~ \overline{\mathrm{d}q}\!</math> describes <math>a\!</math></p>
<br>
<br>
==2. Cactus Calculus==
==Cactus Calculus==
Table 5 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.
Table 5 outlines a syntax for propositional calculus based on two types of logical connectives, both of variable <math>k\!</math>-ary scope.
For more information about this syntax for propositional calculus, see the entries on [[minimal negation operator]]s, [[zeroth order logic]], and [[Differential Propositional Calculus#Table A1. Propositional Forms on Two Variables|Table A1 in Appendix 1]].
For more information about this syntax for propositional calculus, see the entries on [[minimal negation operator]]s, [[zeroth order logic]], and [[Differential Propositional Calculus#Table A1. Propositional Forms on Two Variables|Table A1 in Appendix 1]].
==3. Formal Development==
==Formal Development==
The preceding discussion outlined the ideas leading to the differential extension of propositional logic. The next task is to lay out the concepts and terminology that are needed to describe various orders of differential propositional calculi.
The preceding discussion outlined the ideas leading to the differential extension of propositional logic. The next task is to lay out the concepts and terminology that are needed to describe various orders of differential propositional calculi.
===3.1. Elementary Notions===
===Elementary Notions===
Logical description of a universe of discourse begins with a set of logical signs. For the sake of simplicity in a first approach, assume that these logical signs are collected in the form of a finite alphabet, <math>\mathfrak{A} = \{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}.\!</math> Each of these signs is interpreted as denoting a logical feature, for instance, a property that objects in the universe of discourse may have or a proposition about objects in the universe of discourse. Corresponding to the alphabet <math>\mathfrak{A}\!</math> there is then a set of logical features, <math>\mathcal{A} = \{ a_1, \ldots, a_n \}.\!</math>
Logical description of a universe of discourse begins with a set of logical signs. For the sake of simplicity in a first approach, assume that these logical signs are collected in the form of a finite alphabet, <math>\mathfrak{A} = \{ {}^{\backprime\backprime} a_1 {}^{\prime\prime}, \ldots, {}^{\backprime\backprime} a_n {}^{\prime\prime} \}.\!</math> Each of these signs is interpreted as denoting a logical feature, for instance, a property that objects in the universe of discourse may have or a proposition about objects in the universe of discourse. Corresponding to the alphabet <math>\mathfrak{A}\!</math> there is then a set of logical features, <math>\mathcal{A} = \{ a_1, \ldots, a_n \}.\!</math>
<br>
<br>
===3.2. Special Classes of Propositions===
===Special Classes of Propositions===
A ''basic proposition'', ''coordinate proposition'', or ''simple proposition'' in the universe of discourse <math>[a_1, \ldots, a_n]</math> is one of the propositions in the set <math>\{ a_1, \ldots, a_n \}.</math>
A ''basic proposition'', ''coordinate proposition'', or ''simple proposition'' in the universe of discourse <math>[a_1, \ldots, a_n]</math> is one of the propositions in the set <math>\{ a_1, \ldots, a_n \}.</math>
Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical basis <math>\mathcal{A} = \{ a_1, \ldots, a_n \}.\!</math> For example, a singular proposition with respect to the basis <math>\mathcal{A}\!</math> will not remain singular if <math>\mathcal{A}\!</math> is extended by a number of new and independent features. Even if one keeps to the original set of pairwise options <math>\{ a_i \} \cup \{ \texttt{(} a_i \texttt{)} \}\!</math> to pick out a new basis, the sets of linear propositions and positive propositions are both determined by the choice of basic propositions, and this whole determination is tantamount to the purely conventional choice of a cell as origin.
Finally, it is important to note that all of the above distinctions are relative to the choice of a particular logical basis <math>\mathcal{A} = \{ a_1, \ldots, a_n \}.\!</math> For example, a singular proposition with respect to the basis <math>\mathcal{A}\!</math> will not remain singular if <math>\mathcal{A}\!</math> is extended by a number of new and independent features. Even if one keeps to the original set of pairwise options <math>\{ a_i \} \cup \{ \texttt{(} a_i \texttt{)} \}\!</math> to pick out a new basis, the sets of linear propositions and positive propositions are both determined by the choice of basic propositions, and this whole determination is tantamount to the purely conventional choice of a cell as origin.
===3.3. Differential Extensions===
===Differential Extensions===
An initial universe of discourse, <math>A^\bullet,</math> supplies the groundwork for any number of further extensions, beginning with the ''first order differential extension'', <math>\mathrm{E}A^\bullet.</math> The construction of <math>\mathrm{E}A^\bullet</math> can be described in the following stages:
An initial universe of discourse, <math>A^\bullet,</math> supplies the groundwork for any number of further extensions, beginning with the ''first order differential extension'', <math>\mathrm{E}A^\bullet.</math> The construction of <math>\mathrm{E}A^\bullet</math> can be described in the following stages:
\texttt{(} x \texttt{)~} y \texttt{~}
\texttt{(} x \texttt{)~} y \texttt{~}
\\
\\
\texttt{(} x \texttt{)~~~}
\texttt{(} x \texttt{)~ ~}
\\
\\
\texttt{~} x \texttt{~(} y \texttt{)}
\texttt{~} x \texttt{~(} y \texttt{)}
\\
\\
\texttt{~~~(} y \texttt{)}
\texttt{~ ~(} y \texttt{)}
\\
\\
\texttt{(} x \texttt{,~} y \texttt{)}
\texttt{(} x \texttt{,~} y \texttt{)}
\texttt{((} x \texttt{,~} y \texttt{))}
\texttt{((} x \texttt{,~} y \texttt{))}
\\
\\
\texttt{~~~~~} y \texttt{~~}
\texttt{~ ~ ~} y \texttt{~~}
\\
\\
\texttt{~(} x \texttt{~(} y \texttt{))}
\texttt{~(} x \texttt{~(} y \texttt{))}
\\
\\
\texttt{~~} x \texttt{~~~~~}
\texttt{~~} x \texttt{~ ~ ~}
\\
\\
\texttt{((} x \texttt{)~} y \texttt{)~}
\texttt{((} x \texttt{)~} y \texttt{)~}
<br>
<br>
==References==
===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>
[[Category:Adaptive Systems]]
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
[[Category:Artificial Intelligence]]
|+ style="height:30px" | <math>\text{Table F8.2-ii} ~~ \text{Computation of}~ \mathrm{E}f_{8} ~\text{(Method 2)}\!</math>
[[Category:Combinatorics]]
|
[[Category:Computer Science]]
<math>\begin{array}{*{9}{c}}
[[Category:Cybernetics]]
\mathrm{E}f_{8}
[[Category:Differential Logic]]
& = & (u + \mathrm{d}u) \cdot (v + \mathrm{d}v)
[[Category:Discrete Systems]]
\\[6pt]
[[Category:Dynamical Systems]]
& = & u \cdot v
[[Category:Formal Languages]]
& + & u \cdot \mathrm{d}v
[[Category:Formal Sciences]]
& + & v \cdot \mathrm{d}u
[[Category:Formal Systems]]
& + & \mathrm{d}u \cdot \mathrm{d}v
[[Category:Graph Theory]]
\\[6pt]
[[Category:Group Theory]]
\mathrm{E}f_{8}
[[Category:Inquiry]]
& = & \texttt{ } u \texttt{ } v \texttt{ } \cdot \texttt{(} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{)}
[[Category:Linguistics]]
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{)} ~ \mathrm{d}v
[[Category:Logic]]
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot \mathrm{d}u ~ \texttt{(} \mathrm{d}v \texttt{)}
[[Category:Mathematics]]
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u ~ \mathrm{d}v
[[Category:Mathematical Systems Theory]]
\end{array}\!</math>
[[Category:Science]]
|}
[[Category:Semiotics]]
<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 Implication ''f''<sub>11</sub>(u, v)====
=====Computation of ε''f''<sub>11</sub>=====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.1} ~~ \text{Computation of}~ \boldsymbol\varepsilon f_{11}\!</math>
|
<math>\begin{array}{*{10}{l}}
\boldsymbol\varepsilon f_{11}
& = && f_{11}(u, v)
\\[4pt]
& = && \texttt{(} u \texttt{(} v \texttt{))}
\\[4pt]
& = && \texttt{ } u \texttt{ } v \texttt{ } \cdot f_{11}(1, 1)
& + & \texttt{ } u \texttt{ (} v \texttt{)} \cdot f_{11}(1, 0)
& + & \texttt{(} u \texttt{) } v \texttt{ } \cdot f_{11}(0, 1)
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot f_{11}(0, 0)
\\[4pt]
& = && \texttt{ } u \texttt{ } v \texttt{ }
& + & 0
& + & \texttt{(} u \texttt{) } v \texttt{ }
& + & \texttt{(} u \texttt{)(} v \texttt{)}
\\[20pt]
\boldsymbol\varepsilon f_{11}
& = && \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]
&& + & \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]
&& + & \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]
&& + & \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{~}
\end{array}\!</math>
|}
<br>
=====Computation of E''f''<sub>11</sub>=====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.2} ~~ \text{Computation of}~ \mathrm{E}f_{11}\!</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{E}f_{11}
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)
\\[4pt]
& = &&
\texttt{(}
\\
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
\\
&&& \texttt{(}
\\
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\\
&&& \texttt{))}
\\[4pt]
& = &&
u 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{))}
& + &
\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{))}
\\[4pt]
& = &&
u 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{))}
& + &
\texttt{(} u \texttt{)} v
\!\cdot\!
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}
& + &
\texttt{(} u \texttt{)(} v \texttt{)}
\!\cdot\!
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
\\[20pt]
\mathrm{E}f_{11}
& = && \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{~}
\\[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{~}
& + & 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>11</sub>=====
<br>
{| align="center" border="1" cellpadding="10" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.3-i} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 1)}\!</math>
|
<math>\begin{array}{*{10}{l}}
\mathrm{D}f_{11}
& = && \mathrm{E}f_{11}
& + & \boldsymbol\varepsilon f_{11}
\\[4pt]
& = && f_{11}(u + \mathrm{d}u, v + \mathrm{d}v)
& + & f_{11}(u, v)
\\[4pt]
& = &&
\texttt{(} \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
\texttt{(} \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\texttt{))}
& + &
\texttt{(} u \texttt{(} v \texttt{))}
\\[20pt]
\mathrm{D}f_{11}
& = && 0
& + & 0
& + & 0
& + & 0
\\[4pt]
&& + & u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~(} \mathrm{d}u \texttt{)~} \mathrm{d}v \texttt{~~}
& + & 0
& + & 0
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~(} \mathrm{d}v \texttt{)~}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[4pt]
&& + & 0
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{~~} \mathrm{d}u \texttt{~~} \mathrm{d}v \texttt{~~}
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v
& + & 0
\\[20pt]
\mathrm{D}f_{11}
& = && u v \!\cdot\! \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \!\cdot\! \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \!\cdot\! \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \!\cdot\! \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 F11.3-ii} ~~ \text{Computation of}~ \mathrm{D}f_{11} ~\text{(Method 2)}\!</math>
|
<math>\begin{array}{c*{9}{l}}
\mathrm{D}f_{11}
& = & \boldsymbol\varepsilon f_{11} ~+~ \mathrm{E}f_{11}
\\[20pt]
\boldsymbol\varepsilon f_{11}
& = & u v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 1
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
\\[6pt]
\mathrm{E}f_{11}
& = &
u 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{))}
& + &
\texttt{(} u \texttt{)} v
\cdot
\texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}
& + &
\texttt{(} u \texttt{)(} v \texttt{)}
\cdot
\texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
\\[20pt]
\mathrm{D}f_{11}
& = &
u 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{))}
& + &
\texttt{(} u \texttt{)} v
\cdot
\texttt{~} \mathrm{d}u ~ \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>11</sub>=====
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.4} ~~ \text{Computation of}~ \mathrm{d}f_{11}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{D}f_{11}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[6pt]
\Downarrow
\\[6pt]
\mathrm{d}f_{11}
& = & u v \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
\end{array}</math>
|}
<br>
=====Computation of r''f''<sub>11</sub>=====
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.5} ~~ \text{Computation of}~ \mathrm{r}f_{11}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\mathrm{r}f_{11} & = & \mathrm{D}f_{11} ~+~ \mathrm{d}f_{11}
\\[20pt]
\mathrm{D}f_{11}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{d}f_{11}
& = & u v \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
\\[20pt]
\mathrm{r}f_{11}
& = & u v \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 Implication=====
<br>
{| align="center" border="1" cellpadding="20" cellspacing="0" style="text-align:left; width:90%"
|+ style="height:30px" | <math>\text{Table F11.6} ~~ \text{Computation Summary for}~ f_{11}(u, v) = \texttt{(} u \texttt{(} v \texttt{))}\!</math>
|
<math>\begin{array}{c*{8}{l}}
\boldsymbol\varepsilon f_{11}
& = & u v \cdot 1
& + & u \texttt{(} v \texttt{)} \cdot 0
& + & \texttt{(} u \texttt{)} v \cdot 1
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot 1
\\[6pt]
\mathrm{E}f_{11}
& = & u 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{))}
& + & \texttt{(} u \texttt{)} v \cdot \texttt{(} \mathrm{d}u ~ \mathrm{d}v \texttt{)}
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{))}
\\[6pt]
\mathrm{D}f_{11}
& = & u v \cdot \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{((} \mathrm{d}u \texttt{)(} \mathrm{d}v \texttt{))}
& + & \texttt{(} u \texttt{)} v \cdot \mathrm{d}u ~ \mathrm{d}v
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}
\\[6pt]
\mathrm{d}f_{11}
& = & u v \cdot \mathrm{d}v
& + & u \texttt{(} v \texttt{)} \cdot \texttt{(} \mathrm{d}u \texttt{,} \mathrm{d}v \texttt{)}
& + & 0
& + & \texttt{(} u \texttt{)(} v \texttt{)} \cdot \mathrm{d}u
\\[6pt]
\mathrm{r}f_{11}
& = & 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 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{((}
\\
&&& \qquad \texttt{(} u \texttt{,} \mathrm{d}u \texttt{)}
\\
&&& \texttt{)(}
\\
&&& \qquad \texttt{(} v \texttt{,} \mathrm{d}v \texttt{)}
\\
&&& \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==
* 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://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.
* Leibniz, Gottfried Wilhelm, Freiherr von, ''Theodicy : Essays on the Goodness of God, The Freedom of Man, and The Origin of Evil'', Austin Farrer (ed.), E.M. Huggard (trans.), based on C.J. Gerhardt (ed.), ''Collected Philosophical Works'', 1875–1890, Routledge and Kegan Paul, London, UK, 1951. Reprinted, Open Court, La Salle, IL, 1985.
* McClelland, James L., and Rumelhart, David E. (1988), ''Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises'', MIT Press, Cambridge, MA.
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