Differential Analytic Turing Automata • Part 1

Author: Jon Awbrey

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• Overview • Part 1 • Part 2 • Document History •

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Differential Logic

Cactus Language

I will be making use of the cactus language extension of Peirce's Alpha Graphs, so called because it uses a species of graphs that are usually called “cacti” in graph theory.  The last exposition of the cactus syntax that I've written can be found here:

• Propositional Equation Reasoning Systems (PERS)

The representational and computational efficiency of the cactus language for the tasks that are usually associated with boolean algebra and propositional calculus makes it possible to entertain a further extension, to what we may call differential logic, because it develops this basic level of logic in the same way that differential calculus augments analytic geometry to handle change and diversity.  There are several different introductions to differential logic that I have written and distributed across the Internet.  You might start with the following couple of treatments:

• Differential Logic A
• Differential Logic B

I will draw on those previously advertised resources of notation and theory as needed, but right now I sense the need for some concrete examples.

Example 1

Let's say we have a system that is known by the name of its state space $X$ and we have a boolean state variable $x:X\to \mathbb {B} ,$ where $\mathbb {B} =\{0,1\}.$

We observe $X$ for a while, relative to a discrete time frame, and we write down the following sequence of values for $x.$

${\begin{array}{c|c}t&x\\[8pt]0&0\\1&1\\2&0\\3&1\\4&0\\5&1\\6&0\\7&1\\8&0\\9&1\\\ldots &\ldots \end{array}}$

“Aha!” we say, and think we see the way of things, writing down the rule $x'={\texttt {(}}x{\texttt {)}},$ where $x'$ is the next state after $x$ and ${\texttt {(}}x{\texttt {)}}$ is the negation of $x$ in boolean logic.

Another way to detect patterns is to write out a table of finite differences.  For this example, we would get:

${\begin{array}{c|cccc}t&x&\mathrm {d} x&\mathrm {d} ^{2}x&\ldots \\[8pt]0&0&1&0&\ldots \\1&1&1&0&\ldots \\2&0&1&0&\ldots \\3&1&1&0&\ldots \\4&0&1&0&\ldots \\5&1&1&0&\ldots \\6&0&1&0&\ldots \\7&1&1&0&\ldots \\8&0&1&\ldots &\ldots \\9&1&\ldots &\ldots &\ldots \\\ldots &\ldots &\ldots &\ldots &\ldots \end{array}}$

And of course, all the higher order differences are zero.

This leads to thinking of $X$ as having an extended state $(x,\mathrm {d} x,\mathrm {d} ^{2}x,\ldots ,\mathrm {d} ^{k}x),$ and this additional language gives us the facility of describing state transitions in terms of the various orders of differences.  For example, the rule $x'={\texttt {(}}x{\texttt {)}}$ can now be expressed by the rule ${\mathrm {d} x=1}.$

There is a more detailed account of differential logic in the following paper:

• Differential Logic and Dynamic Systems

For future reference, here are a couple of handy rosetta stones for translating back and forth between different notations for the boolean functions $f:\mathbb {B} ^{k}\to \mathbb {B} ,$ where $k=1,2.$

• Tables of Propositional Forms

Example 2

For a slightly more interesting example, let's suppose that we have a dynamic system that is known by its state space $X$ and we have a boolean state variable $x:X\to \mathbb {B} .$  In addition, we are given an initial condition $x=\mathrm {d} x$ and a law ${\mathrm {d} ^{2}x={\texttt {(}}x{\texttt {)}}}.$

The initial condition has two cases:

${\begin{array}{ll}1.&x~=~\mathrm {d} x~=~0\\2.&x~=~\mathrm {d} x~=~1\end{array}}$

Here is a table of the two trajectories or orbits that we get by starting from each of the two permissible initial states and staying within the constraints of the dynamic law ${\mathrm {d} ^{2}x={\texttt {(}}x{\texttt {)}}}.$

${\text{Initial State}}~x\cdot \mathrm {d} x$

${\begin{array}{c|ccc}t&\mathrm {d} ^{0}x&\mathrm {d} ^{1}x&\mathrm {d} ^{2}x\\[8pt]0&1&1&0\\1&0&1&1\\2&1&0&0\\3&1&0&0\\4&1&0&0\\5&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }\end{array}}$

${\text{Initial State}}~{\texttt {(}}x{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} x{\texttt {)}}$

${\begin{array}{c|ccc}t&\mathrm {d} ^{0}x&\mathrm {d} ^{1}x&\mathrm {d} ^{2}x\\[8pt]0&0&0&1\\1&0&1&1\\2&1&0&0\\3&1&0&0\\4&1&0&0\\5&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }\end{array}}$

Note that the state $x{\texttt {(}}\mathrm {d} x{\texttt {)(}}\mathrm {d} ^{2}x{\texttt {)}},$ that is, $(x,\mathrm {d} x,\mathrm {d} ^{2}x)=(1,0,0),$ is a stable attractor for both orbits.

Further discussion of this example, complete with charts and graphs, can be found at this location:

• Example 1. A Square Rigging

Example 3

One more example may serve to suggest how much dynamic complexity can be built on a universe of discourse having but a single logical feature at its base.  But first, there are a few more elements of general notation we'll need to describe finite dimensional universes of discourse and the qualitative dynamics that we envision occurring in them.

Let ${\mathcal {X}}=\{x_{1},\ldots ,x_{n}\}$ be the alphabet of logical features or variables that are used to describe the $n$-dimensional universe of discourse $X^{\bullet }=[{\mathcal {X}}]$ $=[x_{1},\ldots ,x_{n}].$  One may picture a venn diagram whose $n$ overlapping “circles” are labeled with the feature names in ${\mathcal {X}}.$  Staying with this picture, one visualizes the universe of discourse $X^{\bullet }=[{\mathcal {X}}]$ as having two layers:

1. The set $X=\langle {\mathcal {X}}\rangle =\langle x_{1},\dots ,x_{n}\rangle$ of points or cells — the latter used in another sense of the word than when we speak of cellular automata.
2. The set $X^{\uparrow }=(X\to \mathbb {B} )$ of propositions, boolean-valued functions, or maps from $X$ to $\mathbb {B} .$

Thus we picture the universe of discourse ${X^{\bullet }}$ as an ordered pair ${X^{\bullet }=(X,X^{\uparrow })}$ having $2^{n}$ points in the underlying space $X$ and $2^{2^{n}}$ propositions in the function space $X^{\uparrow }.$

A more complete discussion of these notations can be found here:

• A Functional Conception of Propositional Calculus

Now, to the Example.

Once again, let us begin with a 1-feature alphabet ${\mathcal {X}}=\{x_{1}\}=\{x\}.$  In the discussion that follows I will consider a class of trajectories that are ruled by the constraint that $\mathrm {d} ^{k}x=0$ for all $k$ greater than some fixed $m$ and I will indulge in the use of some picturesque language to describe salient classes of such curves.  Given the finite order condition, there is a highest order non-zero difference $\mathrm {d} ^{m}x$ that is exhibited at each point in the course of any determinate trajectory.  Relative to any point of the corresponding orbit or curve, let us call this highest order differential feature $\mathrm {d} ^{m}x$ the drive at that point.  Curves of constant drive $\mathrm {d} ^{m}x$ are then referred to as $m^{\text{th}}$ gear curves.

One additional piece of notation will be needed here.  Starting from the base alphabet ${\mathcal {X}}=\{x\},$ we define and notate $\mathrm {E} ^{j}{\mathcal {X}}=\{x,\mathrm {d} ^{1}x,\mathrm {d} ^{2}x,\ldots ,\mathrm {d} ^{j}x\}$ as the $j^{\text{th}}$ order extended alphabet over ${\mathcal {X}}.$

Let us now consider the family of 4^th gear curves through the extended space $\mathrm {E} ^{4}X=\langle x,\mathrm {d} x,\mathrm {d} ^{2}x,\mathrm {d} ^{3}x,\mathrm {d} ^{4}x\rangle .$  These are the trajectories that are generated subject to the law $\mathrm {d} ^{4}x=1,$ where it is understood in making such a statement that all higher order differences are equal to $0.$

Since $\mathrm {d} ^{4}x$ and all higher order $\mathrm {d} ^{j}x$ are fixed, the entire dynamics can be plotted in the extended space $\mathrm {E} ^{3}X=\langle x,\mathrm {d} x,\mathrm {d} ^{2}x,\mathrm {d} ^{3}x\rangle .$  Thus, there is just enough room in a planar venn diagram to plot both orbits and to show how they partition the points of $\mathrm {E} ^{3}X.$  As it turns out, there are exactly two possible orbits, of eight points each, as illustrated in Figures 16-a and 16-b.  See here:

• Example 2. Drives and Their Vicissitudes

Here are the 4^th gear curves over the 1-feature universe $X=\langle x\rangle$ arranged in the form of tabular arrays, listing the extended state vectors $(x,\mathrm {d} x,\mathrm {d} ^{2}x,\mathrm {d} ^{3}x,\mathrm {d} ^{4}x)$ as they occur in one cyclic period of each orbit.

${\text{Orbit 1}}$

${\begin{array}{c|ccccc}t&\mathrm {d} ^{0}x&\mathrm {d} ^{1}x&\mathrm {d} ^{2}x&\mathrm {d} ^{3}x&\mathrm {d} ^{4}x\\\\0&0&0&0&0&1\\1&0&0&0&1&1\\2&0&0&1&0&1\\3&0&1&1&1&1\\4&1&0&0&0&1\\5&1&0&0&1&1\\6&1&0&1&0&1\\7&1&1&1&1&1\end{array}}$

${\text{Orbit 2}}$

${\begin{array}{c|ccccc}t&\mathrm {d} ^{0}x&\mathrm {d} ^{1}x&\mathrm {d} ^{2}x&\mathrm {d} ^{3}x&\mathrm {d} ^{4}x\\\\0&1&1&0&0&1\\1&0&1&0&1&1\\2&1&1&1&0&1\\3&0&0&1&1&1\\4&0&1&0&0&1\\5&1&1&0&1&1\\6&0&1&1&0&1\\7&1&0&1&1&1\end{array}}$

In this arrangement, the temporal ordering of states can be reckoned by a kind of parallel round-up rule.  Specifically, if $(a_{k},a_{k+1})$ is any pair of adjacent digits in a state vector ${(a_{0},a_{1},\ldots ,a_{n})},$ then the value of $a_{k}$ in the next state is $a_{k}'=a_{k}+a_{k+1},$ the addition being taken mod 2, of course.

A more complete discussion of this arrangement is given here:

• Example 2. Drives and Their Vicissitudes

Example 4

I am going to tip-toe in silence/consilience past many questions of a philosophical nature/nurture that might be asked at this juncture, no doubt to revisit them at some future opportunity/importunity, however the cases happen to align in the course of their inevitable fall.

Instead, let's follow the adage to “keep it concrete and simple”, taking up the consideration of an incrementally more complex example, but having a slightly more general character than the orders of sequential transformations that we've been discussing up to this point.

The types of logical transformations that I have in mind can be thought of as transformations of discourse because they map a universe of discourse into a universe of discourse by way of logical equations between the qualitative features or logical variables in the source and target universes.

The sequential transformations or state transitions we have been considering so far are actually special cases of these more general logical transformations, specifically, they are the ones that have a single universe of discourse, as it happens to exist at different moments in time, in the role of both the source and the target universes of the transformation in question.

Onward and upward to Flatland, the differential analysis of transformations between 2-dimensional universes of discourse.

Consider the transformation from the universe $U^{\bullet }=[u,v]$ to the universe $X^{\bullet }=[x,y]$ defined by the following system of equations:

${\begin{matrix}x&=&f(u,v)&=&{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\\[8pt]y&=&g(u,v)&=&{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}\end{matrix}}$

The parenthetical expressions on the right are the cactus forms for the boolean functions corresponding to inclusive disjunction and logical equivalence, respectively.  Table 1 summarizes the basic elements of the cactus notation for propositional logic.

${\text{Table 1.}}~~{\text{Syntax and Semantics of a Calculus for Propositional Logic}}$

${\text{Graph}}$	${\text{Expression}}$	${\text{Interpretation}}$	${\text{Other Notations}}$
	$~$	$\mathrm {true}$	$1$
	${\texttt {(}}~{\texttt {)}}$	$\mathrm {false}$	$0$
	$a$	$a$	$a$
	${\texttt {(}}a{\texttt {)}}$	$\mathrm {not} ~a$	$\lnot a\quad {\bar {a}}\quad {\tilde {a}}\quad a^{\prime }$
	$a~b~c$	$a~\mathrm {and} ~b~\mathrm {and} ~c$	$a\land b\land c$
	${\texttt {((}}a{\texttt {)(}}b{\texttt {)(}}c{\texttt {))}}$	$a~\mathrm {or} ~b~\mathrm {or} ~c$	$a\lor b\lor c$
	${\texttt {(}}a{\texttt {(}}b{\texttt {))}}$	${\begin{matrix}a~\mathrm {implies} ~b\\[6pt]\mathrm {if} ~a~\mathrm {then} ~b\end{matrix}}$	$a\Rightarrow b$
	${\texttt {(}}a{\texttt {,}}b{\texttt {)}}$	${\begin{matrix}a~\mathrm {not~equal~to} ~b\\[6pt]a~\mathrm {exclusive~or} ~b\end{matrix}}$	${\begin{matrix}a\neq b\\[6pt]a+b\end{matrix}}$
	${\texttt {((}}a{\texttt {,}}b{\texttt {))}}$	${\begin{matrix}a~\mathrm {is~equal~to} ~b\\[6pt]a~\mathrm {if~and~only~if} ~b\end{matrix}}$	${\begin{matrix}a=b\\[6pt]a\Leftrightarrow b\end{matrix}}$
	${\texttt {(}}a{\texttt {,}}b{\texttt {,}}c{\texttt {)}}$	${\begin{matrix}\mathrm {just~one~of} \\a,b,c\\\mathrm {is~false} .\end{matrix}}$	${\begin{matrix}&{\bar {a}}~b~c\\\lor &a~{\bar {b}}~c\\\lor &a~b~{\bar {c}}\end{matrix}}$
	${\texttt {((}}a{\texttt {),(}}b{\texttt {),(}}c{\texttt {))}}$	${\begin{matrix}\mathrm {just~one~of} \\a,b,c\\\mathrm {is~true} .\\[6pt]\mathrm {partition~all} \\\mathrm {into} ~a,b,c.\end{matrix}}$	${\begin{matrix}&a~{\bar {b}}~{\bar {c}}\\\lor &{\bar {a}}~b~{\bar {c}}\\\lor &{\bar {a}}~{\bar {b}}~c\end{matrix}}$
	${\texttt {(}}a{\texttt {,(}}b{\texttt {,}}c{\texttt {))}}$	${\begin{matrix}\mathrm {oddly~many~of} \\a,b,c\\\mathrm {are~true} .\end{matrix}}$	$a+b+c$ ${\begin{matrix}&a~b~c\\\lor &a~{\bar {b}}~{\bar {c}}\\\lor &{\bar {a}}~b~{\bar {c}}\\\lor &{\bar {a}}~{\bar {b}}~c\end{matrix}}$
	${\texttt {(}}x{\texttt {,(}}a{\texttt {),(}}b{\texttt {),(}}c{\texttt {))}}$	${\begin{matrix}\mathrm {partition} ~x\\\mathrm {into} ~a,b,c.\\[6pt]\mathrm {genus} ~x~\mathrm {comprises} \\\mathrm {species} ~a,b,c.\end{matrix}}$	${\begin{matrix}&{\bar {x}}~{\bar {a}}~{\bar {b}}~{\bar {c}}\\\lor &x~a~{\bar {b}}~{\bar {c}}\\\lor &x~{\bar {a}}~b~{\bar {c}}\\\lor &x~{\bar {a}}~{\bar {b}}~c\end{matrix}}~$

The component notation $F=(F_{1},F_{2})=(f,g):U^{\bullet }\to X^{\bullet }$ allows us to give a name and a type to this transformation, and permits us to define it by means of the following compact description:

${\begin{matrix}(x,y)&=&F(u,v)&=&(~~{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}~~,~~{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}~~).\end{matrix}}$

The information defining the logical transformation $F$ can be represented in the form of a truth table, as shown below.

$u$	$v$	$f$	$g$
${\begin{matrix}0\\[4pt]0\\[4pt]1\\[4pt]1\end{matrix}}$	${\begin{matrix}0\\[4pt]1\\[4pt]0\\[4pt]1\end{matrix}}$	${\begin{matrix}0\\[4pt]1\\[4pt]1\\[4pt]1\end{matrix}}$	${\begin{matrix}1\\[4pt]0\\[4pt]0\\[4pt]1\end{matrix}}$
$u$	$v$	${\texttt {((}}u{\texttt {)(}}v{\texttt {))}}$	${\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}$

A more complete framework of discussion and a fuller development of this example can be found in the neighborhood of the following site:

• Transformations of Type B² → B²

Consider the transformation of textual elements (TOTE) in progress:

${\begin{matrix}x&=&f(u,v)&=&{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\\[8pt]y&=&g(u,v)&=&{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}\\[8pt](x,y)&=&F(u,v)&=&(~~{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}~~,~~{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}~~)\end{matrix}}$

Taken as a transformation from the universe $U^{\bullet }=[u,v]$ to the universe $X^{\bullet }=[x,y],$ this is a particular type of formal object, and it can be studied at that level of abstraction until the chickens come home to roost, as they say, but when the time comes to count those chickens, if you will, the terms of artifice that we use to talk about abstract objects, almost as if we actually knew what we were talking about, need to be fully fledged or fleshed out with extra bits of interpretive data (BOIDs).

And so, to decompress the story, the TOTE we use to convey the FOMA has to be interpreted before it can be applied to anything actually putting supper on the table, so to speak.

What are some of the ways that an abstract logical transformation like $F$ gets interpreted in the setting of a concrete application?

Mathematical parlance comes part way to the rescue here and tosses us the line that a transformation of syntactic signs can be interpreted in either one of two ways, as an alias or as an alibi.

When we consider a transformation in the alias interpretation, we are merely changing the terms we use to describe what may very well be, to some approximation, the very same things.

For example, in some applications the discursive universes $U^{\bullet }=[u,v]$ and $X^{\bullet }=[x,y]$ are best understood as diverse frames, instruments, reticules, scopes, or templates we adopt for the sake of viewing from variant perspectives what we conceive to be roughly the same underlying objects.

When we consider a transformation in the alibi interpretation, we are thinking of the objective things as objectively moving around in space or changing their qualitative characteristics.  There are times when we think of this alibi transformation as taking place in a dimension of time, and then there are times when time is not an object.

For example, in some applications the discursive universes $U^{\bullet }=[u,v]$ and $X^{\bullet }=[x,y]$ are actually the same universe, and what we have is a frame where $x$ is the next state of $u$ and $y$ is the next state of $v,$ notated as $x=u'$ and $y=v'.$  This permits us to rewrite the transformation $F$ as follows:

${\begin{matrix}u'&=&f(u,v)&=&{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\\[8pt]v'&=&g(u,v)&=&{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}\\[8pt](u',v')&=&F(u,v)&=&(~~{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}~~,~~{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}~~)\end{matrix}}$

All in all, then, we have three different ways in general of applying or interpreting a transformation of discourse, that we might sum up as one brand of alias and two brands of alibi, all together, the Elseword, the Elsewhere, and the Elsewhen.

No more angels on pinheads, the brass tacks next time.

Differential Analysis

It is time to formulate the differential analysis of a logical transformation, or a mapping of discourse. It is wise to begin with the first order differentials.

We are considering an abstract logical transformation $F=(f,g):[u,v]\to [x,y]$ which can be interpreted in a number of different ways.  Let's fix on a couple of major variants which might be indicated as follows:

${\begin{array}{lccccc}{\text{Alias Map}}&(x,y)&=&F(u,v)&=&(~~{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}~~,~~{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}~~)\\[8pt]{\text{Alibi Map}}&(u',v')&=&F(u,v)&=&(~~{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}~~,~~{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}~~)\end{array}}$

$F$ is just one example among — well, now that I think of it — how many other logical transformations from the same source to the same target universe?  In the light of that question, maybe it would be advisable to contemplate the character of $F$ within the fold of its most closely akin transformations.

Given the alphabets ${\mathcal {U}}=\{u,v\}$ and ${\mathcal {X}}=\{x,y\}$ along with the corresponding universes of discourse $U^{\bullet }$ and $X^{\bullet }=[\mathbb {B} ^{2}],$ how many logical transformations of the general form $G=(G_{1},G_{2}):U^{\bullet }\to X^{\bullet }$ are there?

Since $G_{1}$ and $G_{2}$ can be any propositions of the type $\mathbb {B} ^{2}\to \mathbb {B} ,$ there are $2^{4}=16$ choices for each of the maps $G_{1}$ and $G_{2},$ and thus there are $2^{4}\cdot 2^{4}=2^{8}=256$ different mappings altogether of the form $G:U^{\bullet }\to X^{\bullet }.$

The set of all functions of a given type is customarily denoted by placing its type indicator in parentheses, in the present instance writing $(U^{\bullet }\to X^{\bullet })=\{G:U^{\bullet }\to X^{\bullet }\},$ and so the cardinality of this function space can most conveniently be summed up by writing:

$|(U^{\bullet }\to X^{\bullet })|~=~|(\mathbb {B} ^{2}\to \mathbb {B} ^{2})|~=~4^{4}~=~256.$

Given any transformation of this type, $G:U^{\bullet }\to X^{\bullet },$ the (first order) differential analysis of $G$ is based on the definition of a couple of further transformations, derived by way of operators on $G,$ that ply between the (first order) extended universes, $\mathrm {E} U^{\bullet }=[u,v,du,dv]$ and $\mathrm {E} X^{\bullet }=[x,y,dx,dy],$ of $G{\text{'s}}$ own source and target universes.

First, the enlargement map (or the secant transformation) $\mathrm {E} G=(\mathrm {E} G_{1},\mathrm {E} G_{2}):\mathrm {E} U^{\bullet }\to \mathrm {E} X^{\bullet }$ is defined by the following pair of component equations:

${\begin{array}{lll}\mathrm {E} G_{1}&=&G_{1}(u+\mathrm {d} u,v+\mathrm {d} v)\\[8pt]\mathrm {E} G_{2}&=&G_{2}(u+\mathrm {d} u,v+\mathrm {d} v)\end{array}}$

Second, the difference map (or the chordal transformation) ${\mathrm {D} G=(\mathrm {D} G_{1},\mathrm {D} G_{2}):\mathrm {E} U^{\bullet }\to \mathrm {E} X^{\bullet }}$ is defined in a component-wise fashion as the boolean sum of the initial proposition $G_{j}$ and the enlarged or shifted proposition $\mathrm {E} G_{j},$ for $j=1,2,$ in accord with following pair of equations:

${\begin{array}{lllll}\mathrm {D} G_{1}&=&G_{1}(u,v)&+&\mathrm {E} G_{1}(u,v,\mathrm {d} u,\mathrm {d} v)\\[8pt]&=&G_{1}(u,v)&+&G_{1}(u+\mathrm {d} u,v+\mathrm {d} v)\\[8pt]\mathrm {D} G_{2}&=&G_{2}(u,v)&+&\mathrm {E} G_{2}(u,v,\mathrm {d} u,\mathrm {d} v)\\[8pt]&=&G_{2}(u,v)&+&G_{2}(u+\mathrm {d} u,v+\mathrm {d} v)\end{array}}$

Maintaining a strict analogy with ordinary difference calculus would perhaps have us write $\mathrm {D} G_{j}=\mathrm {E} G_{j}-G_{j},$ but the sum and difference operations are the same thing in boolean arithmetic.  It is more often natural in the logical context to consider an initial proposition $q,$ then to compute the enlargement $\mathrm {E} q,$ and finally to determine the difference $\mathrm {D} q=q+\mathrm {E} q,$ so we let the variant order of terms reflect this sequence of considerations.

Given these general considerations about the operators $\mathrm {E}$ and $\mathrm {D} ,$ let's return to particular cases, and carry out the first order analysis of the transformation $F(u,v)=(~~{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}~~,~~{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}~~).$

By way of getting our feet back on solid ground, let's crank up our current case of a transformation of discourse, $F:U^{\bullet }\to X^{\bullet },$ with concrete type $[u,v]\to [x,y]$ or abstract type $\mathbb {B} ^{2}\to \mathbb {B} ^{2},$ and let it spin through a sufficient number of turns to see how it goes, as viewed under the scope of what is probably its most straightforward view, as an elsewhen map $F:[u,v]\to [u',v'].$

${\begin{array}{ccc}u'&=&{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\\v'&=&{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}\end{array}}$

${\begin{matrix}{\text{Orbit 1}}\\{\text{Initial Point :}}~(u,v)=(1,1)\end{matrix}}$

${\begin{array}{c|cc}t&u&v\\[8pt]0&1&1\\1&1&1\\2&{}^{\shortparallel }&{}^{\shortparallel }\end{array}}$

${\begin{matrix}{\text{Orbit 2}}\\{\text{Initial Point :}}~(u,v)=(0,0)\end{matrix}}$

${\begin{array}{c|cc}t&u&v\\[8pt]0&0&0\\1&0&1\\2&1&0\\3&1&0\\4&{}^{\shortparallel }&{}^{\shortparallel }\end{array}}$

In the upshot there are two basins of attraction, the state $(1,1)$ and the state $(1,0),$ with the orbit $(1,1)$ making up an isolated basin and the orbit $(0,0),(0,1),(1,0)$ leading to the basin $(1,0).$

On first examination of our present example we made a likely guess at a form of rule that would account for the finite protocol of states that we observed the system $X$ passing through, as spied in the light of its boolean state variable $x:X\to \mathbb {B} ,$ and that rule is well-formulated in any of these styles of notation:

${\begin{array}{ll}1.1.&f:\mathbb {B} \to \mathbb {B} ~{\text{such that}}~f:x\mapsto {\texttt {(}}x{\texttt {)}}\\1.2.&x'~=~{\texttt {(}}x{\texttt {)}}\\1.3.&x~:=~{\texttt {(}}x{\texttt {)}}\\1.4.&\mathrm {d} x~=~1\end{array}}$

In the current example, we already know in advance the program that generates the state transitions, and it is a rule of the following equivalent and easily derivable forms:

${\begin{array}{ll}2.1.&F:\mathbb {B} ^{2}\to \mathbb {B} ^{2}~{\text{such that}}~F:(u,v)\mapsto (~~{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}~~,~~{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}~~)\\2.2.&u'~=~{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\quad ~,~\quad v'~=~{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}\\2.3.&u~:=~{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}\quad ~,~\quad v~:=~{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}\\2.4.&?\end{array}}$

Well, the last one is not such a fall off a log, but that is exactly the purpose for which we have been developing all of the foregoing machinations.

Here is what I got when I just went ahead and calculated the finite differences willy-nilly:

${\text{Orbit 1. Intitial Point :}}~(u,v)=(1,1)$

${\begin{array}{c|cc|cc|cc|cc|cc|c}t&u&v&\mathrm {d} u&\mathrm {d} v&\mathrm {d} ^{2}u&\mathrm {d} ^{2}v&\mathrm {d} ^{3}u&\mathrm {d} ^{3}v&\mathrm {d} ^{4}u&\mathrm {d} ^{4}v&\ldots \\[8pt]0&1&1&0&0&0&0&0&0&0&0&\ldots \\1&1&1&0&0&0&0&0&0&0&0&\ldots \\2&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&\ldots \end{array}}$

${\text{Orbit 2. Intitial Point :}}~(u,v)=(0,0)$

${\begin{array}{c|cc|cc|cc|cc|cc|c}t&u&v&\mathrm {d} u&\mathrm {d} v&\mathrm {d} ^{2}u&\mathrm {d} ^{2}v&\mathrm {d} ^{3}u&\mathrm {d} ^{3}v&\mathrm {d} ^{4}u&\mathrm {d} ^{4}v&\ldots \\[8pt]0&0&0&0&1&1&0&0&1&1&0&\ldots \\1&0&1&1&1&1&1&1&1&1&1&\ldots \\2&1&0&0&0&0&0&0&0&0&0&\ldots \\3&1&0&0&0&0&0&0&0&0&0&\ldots \\4&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&\ldots \end{array}}$

To be honest, I have never thought of trying to hack the problem in such a brute-force way until just now, and so I know enough to expect a not inappreciable probability of error about all that I've taken the risk to write out here, but let me forge ahead and see what I can see.

What we are looking for is — one rule to rule them all, a rule that applies to every state and works every time.

What we see at first sight in the tables above are patterns of differential features that attach to the states in each orbit of the dynamics. Looked at locally to these orbits, the isolated fixed point at $(1,1)$ is no problem, as the rule $\mathrm {d} u=\mathrm {d} v=0$ describes it pithily enough.  When it comes to the other orbit, the first thing that comes to mind is to write out the law $\mathrm {d} u=v~,~\mathrm {d} v={\texttt {(}}u{\texttt {)}}.$

Symbolic Method

It ought to be clear at this point that we need a more systematic symbolic method for computing the differentials of logical transformations, using the term differential in a loose way at present for all sorts of finite differences and derivatives, leaving it to another discussion to sharpen up its more exact technical senses.

For ease of reference, let's recast our current example in the following form:

${\begin{matrix}F&=&(f,g)&=&(~~{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}~~,~~{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}~~).\end{matrix}}$

In their application to this logical transformation the operators $\mathrm {E}$ and $\mathrm {D}$ respectively produce the enlarged map $\mathrm {E} F=(\mathrm {E} f,\mathrm {E} g)$ and the difference map $\mathrm {D} F=(\mathrm {D} f,\mathrm {D} g),$ whose components can be given as follows.

${\begin{array}{lll}\mathrm {E} f&=&{\texttt {((}}u+\mathrm {d} u{\texttt {)(}}v+\mathrm {d} v{\texttt {))}}\\[8pt]\mathrm {E} g&=&{\texttt {((}}u+\mathrm {d} u{\texttt {,}}~~v+\mathrm {d} v{\texttt {))}}\\[8pt]\mathrm {D} f&=&{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}~+~{\texttt {((}}u+\mathrm {d} u{\texttt {)(}}v+\mathrm {d} v{\texttt {))}}\\[8pt]\mathrm {D} g&=&{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}~+~{\texttt {((}}u+\mathrm {d} u{\texttt {,}}~~v+\mathrm {d} v{\texttt {))}}\end{array}}$

But these initial formulas are purely definitional, and help us little to understand either the purpose of the operators or the significance of the results.  Working symbolically, let's apply a more systematic method to the separate components of the mapping $F.$

A sketch of this work is presented in the following series of Figures, where each logical proposition is expanded over the basic cells $uv,u{\texttt {(}}v{\texttt {)}},{\texttt {(}}u{\texttt {)}}v,{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}$ of the 2-dimensional universe of discourse $U^{\bullet }=[u,v].$

Computation Summary for Logical Disjunction

The venn diagram in Figure 1.1 shows how the proposition $f={\texttt {((}}u{\texttt {)(}}v{\texttt {))}}$ can be expanded over the universe of discourse $[u,v]$ to produce a logically equivalent exclusive disjunction, namely, $uv+u{\texttt {(}}v{\texttt {)}}+{\texttt {(}}u{\texttt {)}}v.$

o---------------------------------------o | | | o | | /%\ | | /%%%\ | | /%%%%%\ | | o%%%%%%%o | | /%\%%%%%/%\ | | /%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o | | /%\%%%%%/%\%%%%%/%\ | | /%%%\%%%/%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o%%%%%%%o | | /%\%%%%%/%\%%%%%/%\%%%%%/%\ | | /%%%\%%%/%%%\%%%/%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\%/%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o%%%%%%%o%%%%%%%o | | |\%%%%%/%\%%%%%/ \%%%%%/%\%%%%%/| | | | \%%%/%%%\%%%/ \%%%/%%%\%%%/ | | | | \%/%%%%%\%/ \%/%%%%%\%/ | | | | o%%%%%%%o o%%%%%%%o | | | | |\%%%%%/ \ / \%%%%%/| | | | | | \%%%/ \ / \%%%/ | | | | | u | \%/ \ / \%/ | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 1.1. f = ((u)(v))

Figure 1.2 expands $\mathrm {E} f={\texttt {((}}u+\mathrm {d} u{\texttt {)(}}v+\mathrm {d} v{\texttt {))}}$ over $[u,v]$ to give:

${\begin{matrix}\mathrm {E} {\texttt {((}}u{\texttt {)(}}v{\texttt {))}}&=&uv\cdot {\texttt {(}}\mathrm {d} u~\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {))}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {((}}\mathrm {d} u{\texttt {)}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}\end{matrix}}$

o---------------------------------------o | | | o | | /%\ | | /%%%\ | | /%%%%%\ | | o%%%%%%%o | | /%\%%%%%/%\ | | /%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o | | /%\%%%%%/ \%%%%%/%\ | | /%%%\%%%/ \%%%/%%%\ | | /%%%%%\%/ \%/%%%%%\ | | o%%%%%%%o o%%%%%%%o | | /%\%%%%%/%\ /%\%%%%%/%\ | | /%%%\%%%/%%%\ /%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\ /%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o%%%%%%%o%%%%%%%o | | |\%%%%%/ \%%%%%/%\%%%%%/ \%%%%%/| | | | \%%%/ \%%%/%%%\%%%/ \%%%/ | | | | \%/ \%/%%%%%\%/ \%/ | | | | o o%%%%%%%o o | | | | |\ /%\%%%%%/%\ /| | | | | | \ /%%%\%%%/%%%\ / | | | | | u | \ /%%%%%\%/%%%%%\ / | v | | | o---+---o%%%%%%%o%%%%%%%o---+---o | | | \%%%%%/ \%%%%%/ | | | | \%%%/ \%%%/ | | | | du \%/ \%/ dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 1.2. Ef = ((u + du)(v + dv))

Figure 1.3 expands $\mathrm {D} f=f+\mathrm {E} f$ over $[u,v]$ to produce:

${\begin{matrix}\mathrm {D} {\texttt {((}}u{\texttt {)(}}v{\texttt {))}}&=&uv\cdot \mathrm {d} u~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}\end{matrix}}$

o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | o o o | | / \ /%\ / \ | | / \ /%%%\ / \ | | / \ /%%%%%\ / \ | | o o%%%%%%%o o | | / \ / \%%%%%/ \ / \ | | / \ / \%%%/ \ / \ | | / \ / \%/ \ / \ | | o o o o o | | |\ /%\ /%\ /%\ /| | | | \ /%%%\ /%%%\ /%%%\ / | | | | \ /%%%%%\ /%%%%%\ /%%%%%\ / | | | | o%%%%%%%o%%%%%%%o%%%%%%%o | | | | |\%%%%%/%\%%%%%/%\%%%%%/| | | | | | \%%%/%%%\%%%/%%%\%%%/ | | | | | u | \%/%%%%%\%/%%%%%\%/ | v | | | o---+---o%%%%%%%o%%%%%%%o---+---o | | | \%%%%%/ \%%%%%/ | | | | \%%%/ \%%%/ | | | | du \%/ \%/ dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 1.3. Df = f + Ef

I'll break this here in case anyone wants to try and do the work for $g$ on their own.

Computation Summary for Logical Equality

The venn diagram in Figure 2.1 shows how the proposition $g={\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}$ can be expanded over the universe of discourse $[u,v]$ to produce a logically equivalent exclusive disjunction, namely, $uv+{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}.$

o---------------------------------------o | | | o | | /%\ | | /%%%\ | | /%%%%%\ | | o%%%%%%%o | | /%\%%%%%/%\ | | /%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o | | / \%%%%%/%\%%%%%/ \ | | / \%%%/%%%\%%%/ \ | | / \%/%%%%%\%/ \ | | o o%%%%%%%o o | | / \ / \%%%%%/ \ / \ | | / \ / \%%%/ \ / \ | | / \ / \%/ \ / \ | | o o o o o | | |\ / \ /%\ / \ /| | | | \ / \ /%%%\ / \ / | | | | \ / \ /%%%%%\ / \ / | | | | o o%%%%%%%o o | | | | |\ /%\%%%%%/%\ /| | | | | | \ /%%%\%%%/%%%\ / | | | | | u | \ /%%%%%\%/%%%%%\ / | v | | | o---+---o%%%%%%%o%%%%%%%o---+---o | | | \%%%%%/%\%%%%%/ | | | | \%%%/%%%\%%%/ | | | | du \%/%%%%%\%/ dv | | | o-------o%%%%%%%o-------o | | \%%%%%/ | | \%%%/ | | \%/ | | o | | | o---------------------------------------o Figure 2.1. g = ((u, v))

Figure 2.2 expands $\mathrm {E} g={\texttt {((}}u+\mathrm {d} u{\texttt {,}}~~v+\mathrm {d} v{\texttt {))}}$ over $[u,v]$ to give:

${\begin{matrix}\mathrm {E} {\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}&=&uv\cdot {\texttt {((}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {))}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {))}}\end{matrix}}$

o---------------------------------------o | | | o | | /%\ | | /%%%\ | | /%%%%%\ | | o%%%%%%%o | | / \%%%%%/ \ | | / \%%%/ \ | | / \%/ \ | | o o o | | /%\ /%\ /%\ | | /%%%\ /%%%\ /%%%\ | | /%%%%%\ /%%%%%\ /%%%%%\ | | o%%%%%%%o%%%%%%%o%%%%%%%o | | / \%%%%%/ \%%%%%/ \%%%%%/ \ | | / \%%%/ \%%%/ \%%%/ \ | | / \%/ \%/ \%/ \ | | o o o o o | | |\ /%\ /%\ /%\ /| | | | \ /%%%\ /%%%\ /%%%\ / | | | | \ /%%%%%\ /%%%%%\ /%%%%%\ / | | | | o%%%%%%%o%%%%%%%o%%%%%%%o | | | | |\%%%%%/ \%%%%%/ \%%%%%/| | | | | | \%%%/ \%%%/ \%%%/ | | | | | u | \%/ \%/ \%/ | v | | | o---+---o o o---+---o | | | \ /%\ / | | | | \ /%%%\ / | | | | du \ /%%%%%\ / dv | | | o-------o%%%%%%%o-------o | | \%%%%%/ | | \%%%/ | | \%/ | | o | | | o---------------------------------------o Figure 2.2. Eg = ((u + du, v + dv))

Figure 2.3 expands $\mathrm {D} g=g+\mathrm {E} g$ over $[u,v]$ to yield the form:

${\begin{matrix}\mathrm {D} {\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}\end{matrix}}$

o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | /%\ /%\ | | /%%%\ /%%%\ | | /%%%%%\ /%%%%%\ | | o%%%%%%%o%%%%%%%o | | /%\%%%%%/ \%%%%%/%\ | | /%%%\%%%/ \%%%/%%%\ | | /%%%%%\%/ \%/%%%%%\ | | o%%%%%%%o o%%%%%%%o | | / \%%%%%/ \ / \%%%%%/ \ | | / \%%%/ \ / \%%%/ \ | | / \%/ \ / \%/ \ | | o o o o o | | |\ /%\ / \ /%\ /| | | | \ /%%%\ / \ /%%%\ / | | | | \ /%%%%%\ / \ /%%%%%\ / | | | | o%%%%%%%o o%%%%%%%o | | | | |\%%%%%/%\ /%\%%%%%/| | | | | | \%%%/%%%\ /%%%\%%%/ | | | | | u | \%/%%%%%\ /%%%%%\%/ | v | | | o---+---o%%%%%%%o%%%%%%%o---+---o | | | \%%%%%/ \%%%%%/ | | | | \%%%/ \%%%/ | | | | du \%/ \%/ dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 2.3. Dg = g + Eg

Differential as Locally Linear Approximation

	'Tis a derivative from me to mine, And only that I stand for.
	— Winter's Tale, 3.2.43–44

We've talked about differentials long enough that I think it's way past time we met with some.

When the term is being used with its more exact sense, a differential is a locally linear approximation to a function, in the context of this logical discussion, then, a locally linear approximation to a proposition.

Recall the form of the current example:

${\begin{array}{lllll}F&=&(f,g)&=&(~~{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}~~,~~{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}~~)\end{array}}$

To speed things along, I will skip a mass of motivating discussion and just exhibit the simplest form of a differential $\mathrm {d} F$ for the current example of a logical transformation $F,$ after which the majority of the easiest questions will have been answered in visually intuitive terms.

For $F=(f,g)$ we have $\mathrm {d} F=(\mathrm {d} f,\mathrm {d} g),$ and so we can proceed componentwise, patching the pieces back together at the end.

We have prepared the ground already by computing these terms:

${\begin{array}{lll}\mathrm {E} f&=&{\texttt {((}}u+\mathrm {d} u{\texttt {)(}}v+\mathrm {d} v{\texttt {))}}\\[8pt]\mathrm {E} g&=&{\texttt {((}}u+\mathrm {d} u{\texttt {,}}~~v+\mathrm {d} v{\texttt {))}}\\[8pt]\mathrm {D} f&=&{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}~+~{\texttt {((}}u+\mathrm {d} u{\texttt {)(}}v+\mathrm {d} v{\texttt {))}}\\[8pt]\mathrm {D} g&=&{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}~+~{\texttt {((}}u+\mathrm {d} u{\texttt {,}}~~v+\mathrm {d} v{\texttt {))}}\end{array}}$

As a matter of fact, computing the symmetric differences $\mathrm {D} f=f+\mathrm {E} f$ and $\mathrm {D} g=g+\mathrm {E} g$ has already taken care of the localizing part of the task by subtracting out the forms of $f$ and $g$ from the forms of $\mathrm {E} f$ and $\mathrm {E} g,$ respectively.  Thus all we have left to do is to decide what linear propositions best approximate the difference maps ${\mathrm {D} f}$ and ${\mathrm {D} g},$ respectively.

This raises the question:  What is a linear proposition?

The answer that makes the most sense in this context is this:  A proposition is just a boolean-valued function, so a linear proposition is a linear function into the boolean space $\mathbb {B} .$

In particular, the linear functions we want will be linear functions in the differential variables $\mathrm {d} u$ and $\mathrm {d} v.$

As it turns out, there are just four linear propositions in the associated differential universe $\mathrm {d} U^{\bullet }=[\mathrm {d} u,\mathrm {d} v].$  These are the propositions commonly denoted:  ${0,~\mathrm {d} u,~\mathrm {d} v,~\mathrm {d} u+\mathrm {d} v},$ in other words:  ${\texttt {(}}~{\texttt {)}},~\mathrm {d} u,~\mathrm {d} v,~{\texttt {(}}\mathrm {d} u{\texttt {,}}~\mathrm {d} v{\texttt {)}}.$

Notions of Approximation

	for equalities are so weighed that curiosity in neither can make choice of either's moiety.
	— King Lear, Sc.1.5–7 (Quarto)
	for qualities are so weighed that curiosity in neither can make choice of either's moiety.
	— King Lear, 1.1.5–6 (Folio)

Justifying a notion of approximation is a little more involved in general, and especially in these discrete logical spaces, than it would be expedient for people in a hurry to tangle with right now.  I will just say that there are naive or obvious notions and there are sophisticated or subtle notions that we might choose among.  The later would engage us in trying to construct proper logical analogues of Lie derivatives, and so let's save that for when we have become subtle or sophisticated or both.  Against or toward that day, as you wish, let's begin with an option in plain view.

Figure 1.4 illustrates one way of ranging over the cells of the underlying universe $U^{\bullet }=[u,v]$ and selecting at each cell the linear proposition in $\mathrm {d} U^{\bullet }=[\mathrm {d} u,\mathrm {d} v]$ that best approximates the patch of the difference map ${\mathrm {D} f}$ located there, yielding the following formula for the differential $\mathrm {d} f.$

${\begin{array}{*{11}{c}}\mathrm {d} f&=&\mathrm {d} {\texttt {((}}u{\texttt {)(}}v{\texttt {))}}&=&uv\cdot 0&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}\end{array}}$

o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | o o o | | / \ / \ / \ | | / \ / \ / \ | | / \ / \ / \ | | o o o o | | / \ /%\ /%\ / \ | | / \ /%%%\ /%%%\ / \ | | / \ /%%%%%\ /%%%%%\ / \ | | o o%%%%%%%o%%%%%%%o o | | |\ /%\%%%%%/ \%%%%%/%\ /| | | | \ /%%%\%%%/ \%%%/%%%\ / | | | | \ /%%%%%\%/ \%/%%%%%\ / | | | | o%%%%%%%o o%%%%%%%o | | | | |\%%%%%/%\ /%\%%%%%/| | | | | | \%%%/%%%\ /%%%\%%%/ | | | | | u | \%/%%%%%\ /%%%%%\%/ | v | | | o---+---o%%%%%%%o%%%%%%%o---+---o | | | \%%%%%/ \%%%%%/ | | | | \%%%/ \%%%/ | | | | du \%/ \%/ dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 1.4. df = linear approx to Df

Figure 2.4 illustrates one way of ranging over the cells of the underlying universe $U^{\bullet }=[u,v]$ and selecting at each cell the linear proposition in $\mathrm {d} U^{\bullet }=[du,dv]$ that best approximates the patch of the difference map $\mathrm {D} g$ located there, yielding the following formula for the differential $\mathrm {d} g.$

${\begin{array}{*{11}{c}}\mathrm {d} g&=&\mathrm {d} {\texttt {((}}u{\texttt {,}}v{\texttt {))}}&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}\mathrm {d} v{\texttt {)}}\end{array}}$

o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | /%\ /%\ | | /%%%\ /%%%\ | | /%%%%%\ /%%%%%\ | | o%%%%%%%o%%%%%%%o | | /%\%%%%%/ \%%%%%/%\ | | /%%%\%%%/ \%%%/%%%\ | | /%%%%%\%/ \%/%%%%%\ | | o%%%%%%%o o%%%%%%%o | | / \%%%%%/ \ / \%%%%%/ \ | | / \%%%/ \ / \%%%/ \ | | / \%/ \ / \%/ \ | | o o o o o | | |\ /%\ / \ /%\ /| | | | \ /%%%\ / \ /%%%\ / | | | | \ /%%%%%\ / \ /%%%%%\ / | | | | o%%%%%%%o o%%%%%%%o | | | | |\%%%%%/%\ /%\%%%%%/| | | | | | \%%%/%%%\ /%%%\%%%/ | | | | | u | \%/%%%%%\ /%%%%%\%/ | v | | | o---+---o%%%%%%%o%%%%%%%o---+---o | | | \%%%%%/ \%%%%%/ | | | | \%%%/ \%%%/ | | | | du \%/ \%/ dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 2.4. dg = linear approx to Dg

Well, $g,$ that was easy, seeing as how $\mathrm {D} g$ is already linear at each locus, $\mathrm {d} g=\mathrm {D} g.$

Analytic Series

We have been conducting the differential analysis of the logical transformation $F:[u,v]\mapsto [u,v]$ defined as $F:(u,v)\mapsto (~{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}~,{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}~),$ and this means starting with the extended transformation $\mathrm {E} F:[u,v,\mathrm {d} u,\mathrm {d} v]\to [u,v,\mathrm {d} u,\mathrm {d} v]$ and breaking it into an analytic series, $\mathrm {E} F=F+\mathrm {d} F+\mathrm {d} ^{2}F+\ldots ,$ and so on until there is nothing left to analyze any further.

As a general rule, one proceeds by way of the following stages:

${\begin{array}{*{6}{l}}1.&\mathrm {E} F&=&\mathrm {d} ^{0}F&+&\mathrm {r} ^{0}F\\2.&\mathrm {r} ^{0}F&=&\mathrm {d} ^{1}F&+&\mathrm {r} ^{1}F\\3.&\mathrm {r} ^{1}F&=&\mathrm {d} ^{2}F&+&\mathrm {r} ^{2}F\\4.&\ldots \end{array}}$

In our analysis of the transformation $F,$ we carried out Step 1 in the more familiar form $\mathrm {E} F=F+\mathrm {D} F$ and we have just reached Step 2 in the form $\mathrm {D} F=\mathrm {d} F+\mathrm {r} F,$ where $\mathrm {r} F$ is the residual term remaining for us to examine next.

Note.  I'm am trying to give quick overview here, and this forces me to omit many picky details.  The picky reader may wish to consult the more detailed presentation of this material at the following locations:

• Differential Logic and Dynamic Systems

• The Secant Operator
• Higher Dimensional Targets

Let's push on with the analysis of the transformation:

${\begin{matrix}F&:&(u,v)&\mapsto &(f(u,v),g(u,v))&=&(~~{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}~~,~~{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}~~)\end{matrix}}$

For ease of comparison and computation, I will collect the Figures we need for the remainder of the work together on one page.

Computation Summary for Logical Disjunction

Figure 1.1 shows the expansion of $f={\texttt {((}}u{\texttt {)(}}v{\texttt {))}}$ over $[u,v]$ to produce the expression:

${\begin{matrix}uv&+&u{\texttt {(}}v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\end{matrix}}$

Figure 1.2 shows the expansion of $\mathrm {E} f={\texttt {((}}u+\mathrm {d} u{\texttt {)(}}v+\mathrm {d} v{\texttt {))}}$ over $[u,v]$ to produce the expression:

${\begin{matrix}uv\cdot {\texttt {(}}\mathrm {d} u~\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {))}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {((}}\mathrm {d} u{\texttt {)}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}\end{matrix}}$

In general, $\mathrm {E} f$ tells you what you would have to do, from wherever you are in the universe $[u,v],$ if you want to end up in a place where $f$ is true.  In this case, where the prevailing proposition $f$ is ${\texttt {((}}u{\texttt {)(}}v{\texttt {))}},$ the indication $uv\cdot {\texttt {(}}\mathrm {d} u~\mathrm {d} v{\texttt {)}}$ of $\mathrm {E} f$ tells you this:  If $u$ and $v$ are both true where you are, then just don't change both $u$ and $v$ at once, and you will end up in a place where ${\texttt {((}}u{\texttt {)(}}v{\texttt {))}}$ is true.

Figure 1.3 shows the expansion of $\mathrm {D} f$ over $[u,v]$ to produce the expression:

${\begin{matrix}uv\cdot \mathrm {d} u~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u{\texttt {(}}\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {)}}\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {))}}\end{matrix}}$

In general, ${\mathrm {D} f}$ tells you what you would have to do, from wherever you are in the universe $[u,v],$ if you want to bring about a change in the value of $f,$ that is, if you want to get to a place where the value of $f$ is different from what it is where you are.  In the present case, where the reigning proposition $f$ is ${\texttt {((}}u{\texttt {)(}}v{\texttt {))}},$ the term $uv\cdot \mathrm {d} u~\mathrm {d} v$ of ${\mathrm {D} f}$ tells you this:  If $u$ and $v$ are both true where you are, then you would have to change both $u$ and $v$ in order to reach a place where the value of $f$ is different from what it is where you are.

Figure 1.4 approximates ${\mathrm {D} f}$ by the linear form $\mathrm {d} f$ that expands over $[u,v]$ as follows:

${\begin{matrix}\mathrm {d} f&=&uv\cdot 0&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}\\[8pt]&=&&&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}\end{matrix}}$

Figure 1.5 shows what remains of the difference map ${\mathrm {D} f}$ when the first order linear contribution $\mathrm {d} f$ is removed, namely:

${\begin{array}{*{9}{l}}\mathrm {r} f&=&uv\cdot \mathrm {d} u~\mathrm {d} v&+&u{\texttt {(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)}}v\cdot \mathrm {d} u~\mathrm {d} v&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot \mathrm {d} u~\mathrm {d} v\\[8pt]&=&\mathrm {d} u~\mathrm {d} v\end{array}}$

o---------------------------------------o | | | o | | /%\ | | /%%%\ | | /%%%%%\ | | o%%%%%%%o | | /%\%%%%%/%\ | | /%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o | | /%\%%%%%/%\%%%%%/%\ | | /%%%\%%%/%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o%%%%%%%o | | /%\%%%%%/%\%%%%%/%\%%%%%/%\ | | /%%%\%%%/%%%\%%%/%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\%/%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o%%%%%%%o%%%%%%%o | | |\%%%%%/%\%%%%%/ \%%%%%/%\%%%%%/| | | | \%%%/%%%\%%%/ \%%%/%%%\%%%/ | | | | \%/%%%%%\%/ \%/%%%%%\%/ | | | | o%%%%%%%o o%%%%%%%o | | | | |\%%%%%/ \ / \%%%%%/| | | | | | \%%%/ \ / \%%%/ | | | | | u | \%/ \ / \%/ | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 1.1. f = ((u)(v))

o---------------------------------------o | | | o | | /%\ | | /%%%\ | | /%%%%%\ | | o%%%%%%%o | | /%\%%%%%/%\ | | /%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o | | /%\%%%%%/ \%%%%%/%\ | | /%%%\%%%/ \%%%/%%%\ | | /%%%%%\%/ \%/%%%%%\ | | o%%%%%%%o o%%%%%%%o | | /%\%%%%%/%\ /%\%%%%%/%\ | | /%%%\%%%/%%%\ /%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\ /%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o%%%%%%%o%%%%%%%o | | |\%%%%%/ \%%%%%/%\%%%%%/ \%%%%%/| | | | \%%%/ \%%%/%%%\%%%/ \%%%/ | | | | \%/ \%/%%%%%\%/ \%/ | | | | o o%%%%%%%o o | | | | |\ /%\%%%%%/%\ /| | | | | | \ /%%%\%%%/%%%\ / | | | | | u | \ /%%%%%\%/%%%%%\ / | v | | | o---+---o%%%%%%%o%%%%%%%o---+---o | | | \%%%%%/ \%%%%%/ | | | | \%%%/ \%%%/ | | | | du \%/ \%/ dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 1.2. Ef = ((u + du)(v + dv))

o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | o o o | | / \ /%\ / \ | | / \ /%%%\ / \ | | / \ /%%%%%\ / \ | | o o%%%%%%%o o | | / \ / \%%%%%/ \ / \ | | / \ / \%%%/ \ / \ | | / \ / \%/ \ / \ | | o o o o o | | |\ /%\ /%\ /%\ /| | | | \ /%%%\ /%%%\ /%%%\ / | | | | \ /%%%%%\ /%%%%%\ /%%%%%\ / | | | | o%%%%%%%o%%%%%%%o%%%%%%%o | | | | |\%%%%%/%\%%%%%/%\%%%%%/| | | | | | \%%%/%%%\%%%/%%%\%%%/ | | | | | u | \%/%%%%%\%/%%%%%\%/ | v | | | o---+---o%%%%%%%o%%%%%%%o---+---o | | | \%%%%%/ \%%%%%/ | | | | \%%%/ \%%%/ | | | | du \%/ \%/ dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 1.3. Difference Map Df = f + Ef

o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | o o o | | / \ / \ / \ | | / \ / \ / \ | | / \ / \ / \ | | o o o o | | / \ /%\ /%\ / \ | | / \ /%%%\ /%%%\ / \ | | / \ /%%%%%\ /%%%%%\ / \ | | o o%%%%%%%o%%%%%%%o o | | |\ /%\%%%%%/ \%%%%%/%\ /| | | | \ /%%%\%%%/ \%%%/%%%\ / | | | | \ /%%%%%\%/ \%/%%%%%\ / | | | | o%%%%%%%o o%%%%%%%o | | | | |\%%%%%/%\ /%\%%%%%/| | | | | | \%%%/%%%\ /%%%\%%%/ | | | | | u | \%/%%%%%\ /%%%%%\%/ | v | | | o---+---o%%%%%%%o%%%%%%%o---+---o | | | \%%%%%/ \%%%%%/ | | | | \%%%/ \%%%/ | | | | du \%/ \%/ dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 1.4. Linear Proxy df for Df

o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | o o o | | / \ /%\ / \ | | / \ /%%%\ / \ | | / \ /%%%%%\ / \ | | o o%%%%%%%o o | | / \ /%\%%%%%/%\ / \ | | / \ /%%%\%%%/%%%\ / \ | | / \ /%%%%%\%/%%%%%\ / \ | | o o%%%%%%%o%%%%%%%o o | | |\ / \%%%%%/%\%%%%%/ \ /| | | | \ / \%%%/%%%\%%%/ \ / | | | | \ / \%/%%%%%\%/ \ / | | | | o o%%%%%%%o o | | | | |\ / \%%%%%/ \ /| | | | | | \ / \%%%/ \ / | | | | | u | \ / \%/ \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 1.5. Remainder rf = Df + df

Computation Summary for Logical Equality

Figure 2.1 shows the expansion of $g={\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}$ over $[u,v]$ to produce the expression:

${\begin{matrix}uv&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\end{matrix}}$

Figure 2.2 shows the expansion of $\mathrm {E} g={\texttt {((}}u+\mathrm {d} u{\texttt {,}}~~v+\mathrm {d} v{\texttt {))}}$ over $[u,v]$ to produce the expression:

${\begin{matrix}uv\cdot {\texttt {((}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {))}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {((}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {))}}\end{matrix}}$

In general, $\mathrm {E} g$ tells you what you would have to do, from wherever you are in the universe $[u,v],$ if you want to end up in a place where $g$ is true.  In this case, where the prevailing proposition $g$ is ${\texttt {((}}u{\texttt {,}}~~v{\texttt {))}},$ the component $uv\cdot {\texttt {((}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {))}}$ of $\mathrm {E} g$ tells you this:  If $u$ and $v$ are both true where you are, then change either both or neither of $u$ and $v$ at the same time, and you will attain a place where ${\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}$ is true.

Figure 2.3 shows the expansion of $\mathrm {D} g$ over $[u,v]$ to produce the expression:

${\begin{matrix}uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}\end{matrix}}$

In general, $\mathrm {D} g$ tells you what you would have to do, from wherever you are in the universe $[u,v],$ if you want to bring about a change in the value of $g,$ that is, if you want to get to a place where the value of $g$ is different from what it is where you are.  In the present case, where the ruling proposition $g$ is ${\texttt {((}}u{\texttt {,}}~~v{\texttt {))}},$ the term $uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}$ of $\mathrm {D} g$ tells you this:  If $u$ and $v$ are both true where you are, then you would have to change one or the other but not both $u$ and $v$ in order to reach a place where the value of $g$ is different from what it is where you are.

Figure 2.4 approximates $\mathrm {D} g$ by the linear form ${\mathrm {d} g}$ that expands over $[u,v]$ as follows:

${\begin{array}{*{9}{l}}\mathrm {d} g&=&uv\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&u{\texttt {(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)}}v\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot {\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}\\[8pt]&=&{\texttt {(}}\mathrm {d} u{\texttt {,}}~~\mathrm {d} v{\texttt {)}}\end{array}}$

Figure 2.5 shows what remains of the difference map $\mathrm {D} g$ when the first order linear contribution ${\mathrm {d} g}$ is removed, namely:

${\begin{array}{*{9}{l}}\mathrm {r} g&=&uv\cdot 0&+&u{\texttt {(}}v{\texttt {)}}\cdot 0&+&{\texttt {(}}u{\texttt {)}}v\cdot 0&+&{\texttt {(}}u{\texttt {)(}}v{\texttt {)}}\cdot 0\\[8pt]&=&0\end{array}}$

o---------------------------------------o | | | o | | /%\ | | /%%%\ | | /%%%%%\ | | o%%%%%%%o | | /%\%%%%%/%\ | | /%%%\%%%/%%%\ | | /%%%%%\%/%%%%%\ | | o%%%%%%%o%%%%%%%o | | / \%%%%%/%\%%%%%/ \ | | / \%%%/%%%\%%%/ \ | | / \%/%%%%%\%/ \ | | o o%%%%%%%o o | | / \ / \%%%%%/ \ / \ | | / \ / \%%%/ \ / \ | | / \ / \%/ \ / \ | | o o o o o | | |\ / \ /%\ / \ /| | | | \ / \ /%%%\ / \ / | | | | \ / \ /%%%%%\ / \ / | | | | o o%%%%%%%o o | | | | |\ /%\%%%%%/%\ /| | | | | | \ /%%%\%%%/%%%\ / | | | | | u | \ /%%%%%\%/%%%%%\ / | v | | | o---+---o%%%%%%%o%%%%%%%o---+---o | | | \%%%%%/%\%%%%%/ | | | | \%%%/%%%\%%%/ | | | | du \%/%%%%%\%/ dv | | | o-------o%%%%%%%o-------o | | \%%%%%/ | | \%%%/ | | \%/ | | o | | | o---------------------------------------o Figure 2.1. g = ((u, v))

o---------------------------------------o | | | o | | /%\ | | /%%%\ | | /%%%%%\ | | o%%%%%%%o | | / \%%%%%/ \ | | / \%%%/ \ | | / \%/ \ | | o o o | | /%\ /%\ /%\ | | /%%%\ /%%%\ /%%%\ | | /%%%%%\ /%%%%%\ /%%%%%\ | | o%%%%%%%o%%%%%%%o%%%%%%%o | | / \%%%%%/ \%%%%%/ \%%%%%/ \ | | / \%%%/ \%%%/ \%%%/ \ | | / \%/ \%/ \%/ \ | | o o o o o | | |\ /%\ /%\ /%\ /| | | | \ /%%%\ /%%%\ /%%%\ / | | | | \ /%%%%%\ /%%%%%\ /%%%%%\ / | | | | o%%%%%%%o%%%%%%%o%%%%%%%o | | | | |\%%%%%/ \%%%%%/ \%%%%%/| | | | | | \%%%/ \%%%/ \%%%/ | | | | | u | \%/ \%/ \%/ | v | | | o---+---o o o---+---o | | | \ /%\ / | | | | \ /%%%\ / | | | | du \ /%%%%%\ / dv | | | o-------o%%%%%%%o-------o | | \%%%%%/ | | \%%%/ | | \%/ | | o | | | o---------------------------------------o Figure 2.2. Eg = ((u + du, v + dv))

o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | /%\ /%\ | | /%%%\ /%%%\ | | /%%%%%\ /%%%%%\ | | o%%%%%%%o%%%%%%%o | | /%\%%%%%/ \%%%%%/%\ | | /%%%\%%%/ \%%%/%%%\ | | /%%%%%\%/ \%/%%%%%\ | | o%%%%%%%o o%%%%%%%o | | / \%%%%%/ \ / \%%%%%/ \ | | / \%%%/ \ / \%%%/ \ | | / \%/ \ / \%/ \ | | o o o o o | | |\ /%\ / \ /%\ /| | | | \ /%%%\ / \ /%%%\ / | | | | \ /%%%%%\ / \ /%%%%%\ / | | | | o%%%%%%%o o%%%%%%%o | | | | |\%%%%%/%\ /%\%%%%%/| | | | | | \%%%/%%%\ /%%%\%%%/ | | | | | u | \%/%%%%%\ /%%%%%\%/ | v | | | o---+---o%%%%%%%o%%%%%%%o---+---o | | | \%%%%%/ \%%%%%/ | | | | \%%%/ \%%%/ | | | | du \%/ \%/ dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 2.3. Difference Map Dg = g + Eg

o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | /%\ /%\ | | /%%%\ /%%%\ | | /%%%%%\ /%%%%%\ | | o%%%%%%%o%%%%%%%o | | /%\%%%%%/ \%%%%%/%\ | | /%%%\%%%/ \%%%/%%%\ | | /%%%%%\%/ \%/%%%%%\ | | o%%%%%%%o o%%%%%%%o | | / \%%%%%/ \ / \%%%%%/ \ | | / \%%%/ \ / \%%%/ \ | | / \%/ \ / \%/ \ | | o o o o o | | |\ /%\ / \ /%\ /| | | | \ /%%%\ / \ /%%%\ / | | | | \ /%%%%%\ / \ /%%%%%\ / | | | | o%%%%%%%o o%%%%%%%o | | | | |\%%%%%/%\ /%\%%%%%/| | | | | | \%%%/%%%\ /%%%\%%%/ | | | | | u | \%/%%%%%\ /%%%%%\%/ | v | | | o---+---o%%%%%%%o%%%%%%%o---+---o | | | \%%%%%/ \%%%%%/ | | | | \%%%/ \%%%/ | | | | du \%/ \%/ dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 2.4. Linear Proxy dg for Dg

o---------------------------------------o | | | o | | / \ | | / \ | | / \ | | o o | | / \ / \ | | / \ / \ | | / \ / \ | | o o o | | / \ / \ / \ | | / \ / \ / \ | | / \ / \ / \ | | o o o o | | / \ / \ / \ / \ | | / \ / \ / \ / \ | | / \ / \ / \ / \ | | o o o o o | | |\ / \ / \ / \ /| | | | \ / \ / \ / \ / | | | | \ / \ / \ / \ / | | | | o o o o | | | | |\ / \ / \ /| | | | | | \ / \ / \ / | | | | | u | \ / \ / \ / | v | | | o---+---o o o---+---o | | | \ / \ / | | | | \ / \ / | | | | du \ / \ / dv | | | o-------o o-------o | | \ / | | \ / | | \ / | | o | | | o---------------------------------------o Figure 2.5. Remainder rg = Dg + dg

| Have I carved enough, my lord --
| Child, you are a bone.
|
| Leonard Cohen, "Teachers" (1967)

Visualization

In my work on Differential Logic and Dynamic Systems, I found it useful to develop several different ways of visualizing logical transformations, indeed, I devised four distinct styles of picture for the job.  Thus far in our work on the mapping $F:[u,v]\to [u,v],$ we've been making use of what I call the areal view of the extended universe of discourse, $[u,v,\mathrm {d} u,\mathrm {d} v],$ but as the number of dimensions climbs beyond four, it's time to bid this genre adieu and look for a style that can scale a little better.  At any rate, before we proceed any further, let's first assemble the information we have gathered about $F$ from several different angles, and see if it can be fitted into a coherent picture of the transformation $F:(u,v)\mapsto (~{\texttt {((}}u{\texttt {)(}}v{\texttt {))}}~,~{\texttt {((}}u{\texttt {,}}~~v{\texttt {))}}~).$

In our first crack at the transformation $F,$ we simply plotted the state transitions and applied the utterly stock technique of calculating the finite differences.

${\text{Orbit 1}}$

${\begin{array}{c|cc|cc|}t&u&v&\mathrm {d} u&\mathrm {d} v\\[8pt]0&1&1&0&0\\1&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }\end{array}}$

A quick inspection of the first Table suggests a rule to cover the case when $u=v=1,$ namely, $\mathrm {d} u=\mathrm {d} v=0.$  To put it another way, the Table characterizes Orbit 1 by means of the data:  $(u,v,\mathrm {d} u,\mathrm {d} v)=(1,1,0,0).$  Another way to convey the same information is by means of the extended proposition:  $uv{\texttt {(}}\mathrm {d} u{\texttt {)(}}\mathrm {d} v{\texttt {)}}.$

${\text{Orbit 2}}$

${\begin{array}{c|cc|cc|cc|}t&u&v&\mathrm {d} u&\mathrm {d} v&\mathrm {d} ^{2}u&\mathrm {d} ^{2}v\\[8pt]0&0&0&0&1&1&0\\1&0&1&1&1&1&1\\2&1&0&0&0&0&0\\3&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }&{}^{\shortparallel }\end{array}}$

A more fine combing of the second Table brings to mind a rule which partly covers the remaining cases, that is, $\mathrm {d} u=v,~\mathrm {d} v={\texttt {(}}u{\texttt {)}}.$  This much information about Orbit 2 is also encapsulated by the extended proposition ${\texttt {(}}uv{\texttt {)((}}\mathrm {d} u{\texttt {,}}~v{\texttt {))(}}\mathrm {d} v{\texttt {,}}~u{\texttt {)}},$ which says that $u$ and $v$ are not both true at the same time, while $\mathrm {d} u$ is equal in value to $v$ and $\mathrm {d} v$ is opposite in value to $u.$

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• Overview • Part 1 • Part 2 • Document History •

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