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'''''NOTE.'''  This section is currently under construction.  In the meantime, see [[Logical Graph]].''

'''''NOTE.'''  This section is currently under construction.  In the meantime, see [[Logical Graph]].''

The development of differential logic is greatly facilitated by having a conceptually efficient calculus in place at the level of [[boolean-valued functions]] and elementary logical propositions.  A calculus that is very efficient from both conceptual and computational standpoints is based on just two types of logical connectives, both of variable <math>k\!</math>-ary scope.  The formulas of this calculus map into a species of graph-theoretical structures called ''painted and rooted cacti'' (PARCs) that lend visual representation to their functional structures and smooth the path to efficient computation.

The development of differential logic is greatly facilitated by having a conceptually efficient calculus in place at the level of [[boolean-valued functions]] and elementary logical propositions.  A calculus that is very efficient from both conceptual and computational standpoints is based on just two types of logical connectives, both of variable <math>k\!</math>-ary scope.  The formulas of this calculus map into a species of graph-theoretical structures called ''painted and rooted cacti'' (PARCs) that lend visual representation to their functional structure and smooth the path to efficient computation.

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| The first kind of propositional expression takes the form of a parenthesized sequence of propositional expressions, <math>\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)},</math> and is taken to indicate that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false, in other words, that their [[minimal negation]] is true.  A clause of this form may also be exhibited as a graph-theoretical structure calle

| The first kind of propositional expression is a parenthesized sequence of propositional expressions, written as <math>\texttt{(} e_1 \texttt{,} e_2 \texttt{,} \ldots \texttt{,} e_{k-1} \texttt{,} e_k \texttt{)}</math> and read to say that exactly one of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> is false, in other words, that their [[minimal negation]] is true.  A clause of this form maps into a PARC structure called a ''lobe'', in this case, one that is ''painted'' with the colors <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> as shown below.

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{| align="center" cellpadding="6" width="90%"

| The second kind of propositional expression takes the form of a concatenated sequence of propositional expressions, <math>e_1\ e_2\ \ldots\ e_{k-1}\ e_k,</math> and is taken to indicate that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.

| The second kind of propositional expression is a concatenated sequence of propositional expressions, written as <math>e_1\ e_2\ \ldots\ e_{k-1}\ e_k</math> and read to say that all of the propositions <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> are true, in other words, that their [[logical conjunction]] is true.  A clause of this form maps into a PARC structure called a ''node'', in this case, one that is ''painted'' with the colors <math>e_1, e_2, \ldots, e_{k-1}, e_k</math> as shown below.

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