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{{DISPLAYTITLE:Differential Analytic Turing Automata}}

{{DISPLAYTITLE:Differential Analytic Turing Automata}}

My aim is to chart a course from general ideas about ''transformational equivalence classes of graphs'' to a notion of ''differential analytic turing automata'' (DATA).  It may be a while before we get within sight of that goal, but it will provide some measure of motivation to name the thread after the envisioned end rather than the more homely starting place.

My aim is to chart a course from general ideas about ''transformational equivalence classes of graphs'' to a notion of ''differential analytic turing automata'' (DATA).  It may be a while before we get within sight of that goal, but it will provide some measure of motivation to name the thread after the envisioned end rather than the more homely starting place.

That should be enough to get started.

That should be enough to get started.

==Note 2==

==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:

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:

:* [http://stderr.org/pipermail/inquiry/2004-February/thread.html#1160 Differential Logic B]

:* [http://stderr.org/pipermail/inquiry/2004-February/thread.html#1160 Differential Logic B]

I am currently rewriting these presentations in hopes of making them as clear as they can be, so please let me know if you have any questions.

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.

 

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

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

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

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

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

 

 

I am preparing a more fleshed-out 1-variable example, but in the mean time, for anybody who's finished all that other reading, there is a more detailed account of differential logic in the following paper:

:* [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0|Differential Logic and Dynamic Systems]]

:* [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0|Differential Logic and Dynamic Systems]]

:* [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0#Tables_of_Propositional_Forms|Tables of Propositional Forms]]

:* [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0#Tables_of_Propositional_Forms|Tables of Propositional Forms]]

==Note 5==

==Example 2==

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

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

:* [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0#Example_1._A_Square_Rigging|Example 1. A Square Rigging]]

:* [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0#Example_1._A_Square_Rigging|Example 1. A Square Rigging]]

==Note 6==

==Example 3==

One more example may serve to suggest just how much dynamic complexity can be built on a universe of discourse that has but a single logical feature at its base.

One more example may serve to suggest just how much dynamic complexity can be built on a universe of discourse that has but a single logical feature at its base.

:* [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0#Example_2._Drives_and_Their_Vicissitudes|Example 2. Drives and Their Vicissitudes]]

:* [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0#Example_2._Drives_and_Their_Vicissitudes|Example 2. Drives and Their Vicissitudes]]

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

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

:* [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0#Example_2._Drives_and_Their_Vicissitudes|Example 2. Drives and Their Vicissitudes]]

:* [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0#Example_2._Drives_and_Their_Vicissitudes|Example 2. Drives and Their Vicissitudes]]

==Note 8==

==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.

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.

:* [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0#Transformations_of_Type_B2_.E2.86.92_B2|Transformations of Type '''B'''² &rarr; '''B'''²]]

:* [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0#Transformations_of_Type_B2_.E2.86.92_B2|Transformations of Type '''B'''² &rarr; '''B'''²]]

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

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

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

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

==Note 10==

==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.

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.

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

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

By way of getting our feet back on solid ground, let's crank up our current case of a transformation of discourse, <math>F : U^\circ \to X^\circ,</math> with concrete type <math>[u, v] \to [x, y]</math> or abstract type <math>\mathbb{B}^2 \to \mathbb{B}^2,</math> 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 <math>F : [u, v] \to [u', v'].</math>

By way of getting our feet back on solid ground, let's crank up our current case of a transformation of discourse, <math>F : U^\circ \to X^\circ,</math> with concrete type <math>[u, v] \to [x, y]</math> or abstract type <math>\mathbb{B}^2 \to \mathbb{B}^2,</math> 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 <math>F : [u, v] \to [u', v'].</math>

In the upshot there are two basins of attraction, the state <math>(1, 1)\!</math> and the state <math>(1, 0),\!</math> with the orbit <math>(1, 1)\!</math> making up an isolated basin and the orbit <math>(0, 0), (0, 1), (1, 0)\!</math> leading to the basin <math>(1, 0).\!</math>

In the upshot there are two basins of attraction, the state <math>(1, 1)\!</math> and the state <math>(1, 0),\!</math> with the orbit <math>(1, 1)\!</math> making up an isolated basin and the orbit <math>(0, 0), (0, 1), (1, 0)\!</math> leading to the basin <math>(1, 0).\!</math>

On first examination of our present example we made a likely guess at a form of rule that

On first examination of our present example we made a likely guess at a form of rule that

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 <math>(1, 1)\!</math> is no problem, as the rule <math>\texttt{du~=~dv~=~0}</math> describes it pithily enough.  When it comes to the other orbit, the first thing that comes to mind is to write out the law <math>\texttt{du~=~v}, ~\texttt{dv~=~(u)}.</math>

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 <math>(1, 1)\!</math> is no problem, as the rule <math>\texttt{du~=~dv~=~0}</math> describes it pithily enough.  When it comes to the other orbit, the first thing that comes to mind is to write out the law <math>\texttt{du~=~v}, ~\texttt{dv~=~(u)}.</math>

==Note 13==

==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.

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.

I'll break this here in case anyone wants to try and do the work for <math>g\!</math> on their own.

I'll break this here in case anyone wants to try and do the work for <math>g\!</math> on their own.

===Computation Summary : <math>g(u, v) = \texttt{((u,~v))}</math>===

===Computation Summary : <math>g(u, v) = \texttt{((u,~v))}</math>===

</pre>

</pre>

==Note 15==

==Differential : Locally Linear Approximation==

{| cellpadding="2" cellspacing="2" width="100%"

{| cellpadding="2" cellspacing="2" width="100%"

As it turns out, there are just four linear propositions in the associated ''differential universe'' <math>\operatorname{d}U^\circ = [du, dv],</math> and these are the propositions that are commonly denoted:  <math>\texttt{0}, \texttt{du}, \texttt{dv}, \texttt{du + dv},</math> in other words, <math>\texttt{()}, \texttt{du}, \texttt{dv}, \texttt{(du, dv)}.</math>

As it turns out, there are just four linear propositions in the associated ''differential universe'' <math>\operatorname{d}U^\circ = [du, dv],</math> and these are the propositions that are commonly denoted:  <math>\texttt{0}, \texttt{du}, \texttt{dv}, \texttt{du + dv},</math> in other words, <math>\texttt{()}, \texttt{du}, \texttt{dv}, \texttt{(du, dv)}.</math>

==Note 16==

==Notions of Approximation==

{| cellpadding="2" cellspacing="2" width="100%"

{| cellpadding="2" cellspacing="2" width="100%"

Well, <math>g,\!</math> that was easy, seeing as how <math>\operatorname{D}g</math> is already linear at each locus, <math>\operatorname{d}g = \operatorname{D}g.</math>

Well, <math>g,\!</math> that was easy, seeing as how <math>\operatorname{D}g</math> is already linear at each locus, <math>\operatorname{d}g = \operatorname{D}g.</math>

==Note 17==

==Analytic Series==

We have been conducting the differential analysis of the logical transformation <math>F : [u, v] \mapsto [u, v]</math> defined as <math>F : (u, v) \mapsto ( ~\texttt{((u)(v))}~, ~\texttt{((u, v))}~ ),</math> and this means starting with the extended transformation <math>\operatorname{E}F : [u, v, du, dv] \to [u, v, du, dv]</math> and breaking it into an analytic series, <math>\operatorname{E}F = F + \operatorname{d}F + \operatorname{d}^2 F + \ldots,</math> and

We have been conducting the differential analysis of the logical transformation <math>F : [u, v] \mapsto [u, v]</math> defined as <math>F : (u, v) \mapsto ( ~\texttt{((u)(v))}~, ~\texttt{((u, v))}~ ),</math> and this means starting with the extended transformation <math>\operatorname{E}F : [u, v, du, dv] \to [u, v, du, dv]</math> and breaking it into an analytic series, <math>\operatorname{E}F = F + \operatorname{d}F + \operatorname{d}^2 F + \ldots,</math> and

: [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0#The_Secant_Operator_:_E|The Secant Operator]]

: [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0#The_Secant_Operator_:_E|The Secant Operator]]

: [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0#Taking_Aim_at_Higher_Dimensional_Targets|Higher Dimensional Targets]]

: [[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0#Taking_Aim_at_Higher_Dimensional_Targets|Higher Dimensional Targets]]

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

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

Figure 1.5.  Remainder rf = Df + df

Figure 1.5.  Remainder rf = Df + df

</pre>

</pre>

===Computation Summary : <math>g(u, v) = \texttt{((u, v))}</math>===

===Computation Summary : <math>g(u, v) = \texttt{((u, v))}</math>===

</pre>

</pre>

==Note 20==

==Visualization==

In my work on "[[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0|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 <math>F : [u, v] \to [u, v],\!</math> we've been making use of what I call the ''areal view'' of the extended universe of discourse, <math>[u, v, du, dv],\!</math> 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 that we have gathered about <math>F\!</math> from several different angles, and see if it can be fitted into a coherent picture of the transformation <math>F : (u, v) \mapsto ( ~\texttt{((u)(v))}~, ~\texttt{((u, v))}~ ).</math>

In my work on "[[Directory:Jon_Awbrey/Papers/Differential_Logic_and_Dynamic_Systems_2.0|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 <math>F : [u, v] \to [u, v],\!</math> we've been making use of what I call the ''areal view'' of the extended universe of discourse, <math>[u, v, du, dv],\!</math> 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 that we have gathered about <math>F\!</math> from several different angles, and see if it can be fitted into a coherent picture of the transformation <math>F : (u, v) \mapsto ( ~\texttt{((u)(v))}~, ~\texttt{((u, v))}~ ).</math>