Changes

This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage.  I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems.  The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors.  The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.

This much preparation is enough to begin introducing my subject, if I excuse myself from giving full arguments for my definitional choices until some later stage.  I am trying to develop a ''differential theory of qualitative equations'' that parallels the application of differential geometry to dynamic systems.  The idea of a tangent vector is key to this work and a major goal is to find the right logical analogues of tangent spaces, bundles, and functors.  The strategy is taken of looking for the simplest versions of these constructions that can be discovered within the realm of propositional calculus, so long as they serve to fill out the general theme.

===Differential Propositions : The Qualitative Analogues of Differential Equations===

===Differential Propositions : Qualitative Analogues of Differential Equations===

In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math>  Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.

In order to define the differential extension of a universe of discourse <math>[\mathcal{A}],</math> the initial alphabet <math>\mathcal{A}</math> must be extended to include a collection of symbols for ''differential features'', or basic ''changes'' that are capable of occurring in <math>[\mathcal{A}].</math>  Intuitively, these symbols may be construed as denoting primitive features of change, qualitative attributes of motion, or propositions about how things or points in the universe may be changing or moving with respect to the features that are noted in the initial alphabet.

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The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least,

The finite character of the extended universe <math>[\mathrm{E}\mathcal{A}]</math> makes the problem of solving differential propositions relatively straightforward, at least, in principle.  The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)\!</math> in <math>\mathrm{E}A.</math>  Finding all the models of <math>q,\!</math> the extended interpretations in <math>\mathrm{E}A</math> that satisfy <math>q,\!</math> can be carried out by a finite search.  Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely.  While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.

in principle.  The solution set of the differential proposition <math>q : \mathrm{E}A \to \mathbb{B}</math> is the set of models <math>q^{-1}(1)\!</math> in <math>\mathrm{E}A.</math>  Finding all the models of <math>q,\!</math> the extended interpretations in <math>\mathrm{E}A</math> that satisfy <math>q,\!</math> can be carried out by a finite search.  Being in possession of complete algorithms for propositional calculus modeling, theorem checking, or theorem proving makes the analytic task fairly simple in principle, though the question of efficiency in the face of arbitrary complexity may always remain another matter entirely.  While the fact that propositional satisfiability is NP-complete may be discouraging for the prospects of a single efficient algorithm that covers the whole space <math>[\mathrm{E}\mathcal{A}]</math> with equal facility, there appears to be much room for improvement in classifying special forms and in developing algorithms that are tailored to their practical processing.

In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications.  In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus.  But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.

In view of these constraints and contingencies, our focus shifts to the tasks of approximation and interpretation that support intuition, especially in dealing with the natural kinds of differential propositions that arise in applications, and in the effort to understand, in succinct and adaptive forms, their dynamic implications.  In the absence of direct insights, these tasks are partially carried out by forging analogies with the familiar situations and customary routines of ordinary calculus.  But the indirect approach, going by way of specious analogy and intuitive habit, forces us to remain on guard against the circumstance that occurs when the word ''forging'' takes on its shadier nuance, indicting the constant risk of a counterfeit in the proportion.

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Earlier we defined the tacit extension operators <math>\boldsymbol\varepsilon : X^\bullet \to Y^\bullet\!</math> as maps embedding each proposition of a given universe <math>X^\bullet~\!</math> in a more generously given universe <math>Y^\bullet \supset X^\bullet.\!</math>  Of immediate interest are the tacit extensions <math>\boldsymbol\epsilon : U^\bullet \to \mathrm{E}U^\bullet,\!</math> that locate each proposition of <math>U^\bullet\!</math> in the enlarged context of <math>\mathrm{E}U^\bullet.\!</math>  In its application to the propositional conjunction <math>J = u\!\cdot\!v</math> in <math>[u, v],\!</math> the tacit extension operator <math>\boldsymbol\varepsilon\!</math> yields the proposition <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v].\!</math>  The extended proposition <math>\boldsymbol\varepsilon J\!</math> may be computed according to the scheme in Table&nbsp;36, in effect doing nothing more that conjoining a tautology of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to <math>J\!</math> in <math>U^\bullet.\!</math>

Earlier we defined the tacit extension operators <math>\boldsymbol\varepsilon : X^\bullet \to Y^\bullet\!</math> as maps embedding each proposition of a given universe <math>X^\bullet~\!</math> in a more generously given universe <math>Y^\bullet \supset X^\bullet.\!</math>  Of immediate interest are the tacit extensions <math>\boldsymbol\varepsilon : U^\bullet \to \mathrm{E}U^\bullet,\!</math> that locate each proposition of <math>U^\bullet\!</math> in the enlarged context of <math>\mathrm{E}U^\bullet.\!</math>  In its application to the propositional conjunction <math>J = u\!\cdot\!v</math> in <math>[u, v],\!</math> the tacit extension operator <math>\boldsymbol\varepsilon\!</math> yields the proposition <math>\boldsymbol\varepsilon J\!</math> in <math>\mathrm{E}U^\bullet = [u, v, \mathrm{d}u, \mathrm{d}v].\!</math>  The extended proposition <math>\boldsymbol\varepsilon J\!</math> may be computed according to the scheme in Table&nbsp;36, in effect doing nothing more that conjoining a tautology of <math>[\mathrm{d}u, \mathrm{d}v]\!</math> to <math>J\!</math> in <math>U^\bullet.\!</math>

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<br>

& = & ~~ 0 ~~ \,\cdot\, ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}

& = & ~~ 0 ~~ \,\cdot\, ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}

& + & ~~ u ~  \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v

& + & ~~ u ~  \,\cdot\, ~~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v

& + & ~~~ v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}

& + & [[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) v ~~ \,\cdot\, \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}

& + & \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>

\end{array}\!</math>

<math>\begin{array}{*{9}{l}}

<math>\begin{array}{*{9}{l}}

\mathrm{r}J ~

\mathrm{r}J ~

& = & u \!\cdot\! v \cdot ~ \mathrm{d}u \cdot \mathrm{d}v ~~~~~~

& = & u \!\cdot\! v \cdot ~ \mathrm{d}u \cdot \mathrm{d}v 21:00, 28 August 2015 (UTC)~

& + & u \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,

& + & u \texttt{(} v \texttt{)} \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,

& + & \texttt{(} u \texttt{)} v \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,

& + & \texttt{(} u \texttt{)} v \cdot \, \mathrm{d}u \cdot \mathrm{d}v \,

\texttt{(} x \texttt{)~} y \texttt{~}

\texttt{(} x \texttt{)~} y \texttt{~}

\\[2pt]

\\[2pt]

\texttt{(} x \texttt{)~~~}

\texttt{(} x \texttt{)[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])}

\\[2pt]

\\[2pt]

\texttt{~} x \texttt{~(} y \texttt{)}

\texttt{~} x \texttt{~(} y \texttt{)}

\\[2pt]

\\[2pt]

\texttt{~~~(} y \texttt{)}

\texttt{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])(} y \texttt{)}

\\[2pt]

\\[2pt]

\texttt{(} x \texttt{,~} y \texttt{)}

\texttt{(} x \texttt{,~} y \texttt{)}

\texttt{((} x \texttt{,~} y \texttt{))}

\texttt{((} x \texttt{,~} y \texttt{))}

\\[2pt]

\\[2pt]

\texttt{~~~~~} y \texttt{~~}

\texttt{21:00, 28 August 2015 (UTC)} y \texttt{~~}

\\[2pt]

\\[2pt]

\texttt{~(} x \texttt{~(} y \texttt{))}

\texttt{~(} x \texttt{~(} y \texttt{))}

\\[2pt]

\\[2pt]

\texttt{~~} x \texttt{~~~~~}

\texttt{~~} x \texttt{21:00, 28 August 2015 (UTC)}

\\[2pt]

\\[2pt]

\texttt{((} x \texttt{)~} y \texttt{)~}

\texttt{((} x \texttt{)~} y \texttt{)~}

\texttt{(} x \texttt{)~} y \texttt{~}

\texttt{(} x \texttt{)~} y \texttt{~}

\\

\\

\texttt{(} x \texttt{)~~~}

\texttt{(} x \texttt{)[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])}

\\

\\

\texttt{~} x \texttt{~(} y \texttt{)}

\texttt{~} x \texttt{~(} y \texttt{)}

\\

\\

\texttt{~~~(} y \texttt{)}

\texttt{[[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]])(} 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{21:00, 28 August 2015 (UTC)} y \texttt{~~}

\\

\\

\texttt{~(} x \texttt{~(} y \texttt{))}

\texttt{~(} x \texttt{~(} y \texttt{))}

\\

\\

\texttt{~~} x \texttt{~~~~~}

\texttt{~~} x \texttt{21:00, 28 August 2015 (UTC)}

\\

\\

\texttt{((} x \texttt{)~} y \texttt{)~}

\texttt{((} x \texttt{)~} y \texttt{)~}

\\[20pt]

\\[20pt]

\mathrm{D}f_{8}

\mathrm{D}f_{8}

& = & ~~~ 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}

& = & [[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 0 ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \texttt{(} \mathrm{d}v \texttt{)}

& + & ~~~ u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v

& + & [[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) u ~~ \cdot ~ \texttt{(} \mathrm{d}u \texttt{)} \mathrm{d}v

& + & ~~~~ v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}

& + & [[User:Jon Awbrey|Jon Awbrey]] ([[User talk:Jon Awbrey|talk]]) 21:00, 28 August 2015 (UTC) v ~~ \cdot ~ \mathrm{d}u \texttt{(} \mathrm{d}v \texttt{)}

& + & \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>

\end{array}\!</math>

[[Category:Adaptive Systems]]

[[Category:Adaptive Systems]]

[[Category:Artificial Intelligence]]

[[Category:Artificial Intelligence]]

[[Category:Boolean Functions]]

[[Category:Boolean Functions]]

[[Category:Combinatorics]]

[[Category:Combinatorics]]

[[Category:Computer Science]]

[[Category:Computer Science]]