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Directory:Jon Awbrey/Papers/Differential Propositional Calculus (view source)

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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.
   −
<div class="nonumtoc">__TOC__</div>
+
==Casual Introduction==
−
 
  −
==1. Casual Introduction==
      
Consider the situation represented by the venn diagram in Figure&nbsp;1.
 
Consider the situation represented by the venn diagram in Figure&nbsp;1.
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<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.
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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>
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<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>
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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:
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\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{)}
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\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{)~}
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<br>
 
<br>
   −
==References==
+
===Appendix 3. Computational Details===
 +
 
 +
====Operator Maps for the Logical Conjunction ''f''<sub>8</sub>(u, v)====
 +
 
 +
=====Computation of &epsilon;''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>
 +
|}
   −
* Ashby, William Ross (1956/1964), ''An Introduction to Cybernetics'', Chapman and Hall, London, UK, 1956.  Reprinted, Methuen and Company, London, UK, 1964.
+
<br>
   −
* 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].
+
=====Computation of E''f''<sub>8</sub>=====
   −
* Edelman, Gerald M. (1988), ''Topobiology : An Introduction to Molecular Embryology'', Basic Books, New York, NY.
+
<br>
   −
* 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&ndash;1890, Routledge and Kegan Paul, London, UK, 1951. Reprinted, Open Court, La Salle, IL, 1985.
+
{| 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>
 +
|}
   −
* McClelland, James L., and Rumelhart, David E. (1988), ''Explorations in Parallel Distributed Processing : A Handbook of Models, Programs, and Exercises'', MIT Press, Cambridge, MA.
+
<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 &epsilon;''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 &epsilon;''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 &epsilon;''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&ndash;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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