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* '''Scholium.'''  The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, § 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic.  And of course there is the strange but true story of how the Turin machines of the 1840's prefigured the Turing machines of the 1940's [Men, 225-297].  At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].

* '''Scholium.'''  The fact that a difference calculus can be developed for boolean functions is well known [Fuji], [Koh, § 8-4] and was probably familiar to Boole, who was an expert in difference equations before he turned to logic.  And of course there is the strange but true story of how the Turin machines of the 1840's prefigured the Turing machines of the 1940's [Men, 225-297].  At the very outset of general purpose, mechanized computing we find that the motive power driving the Analytical Engine of Babbage, the kernel of an idea behind all of his wheels, was exactly his notion that difference operations, suitably trained, can serve as universal joints for any conceivable computation [M&M], [Mel, ch. 4].

Given this language, the particular Example that I take up here can be described as the family of 4^th gear curves through E⁴''X'' = 〈''A'', d''A'', d²''A'', d³''A'', d⁴''A''〉.  These are the trajectories generated subject to the dynamic law d⁴''A'' = 1, where it is understood in such a statement that all higher order differences are equal to 0.  Since d⁴''A'' and all higher d^''k''''A'' are fixed, the temporal or transitional conditions (initial, mediate, terminal - transient or stable states) vary only with respect to their projections as points of E³''X'' = 〈''A'', d''A'', d²''A'', d³''A''〉.  Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of E³''X''.  It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figures 16-a and 16-b.  (NB.  I leave it as an exercise for the reader to connect the dots in the second figure.)

Given this language, the particular Example that I take up here can be described as the family of 4^th gear curves through E⁴''X'' = 〈''A'', d''A'', d²''A'', d³''A'', d⁴''A''〉.  These are the trajectories generated subject to the dynamic law d⁴''A'' = 1, where it is understood in such a statement that all higher order differences are equal to 0.  Since d⁴''A'' and all higher d^''k''''A'' are fixed, the temporal or transitional conditions (initial, mediate, terminal - transient or stable states) vary only with respect to their projections as points of E³''X'' = 〈''A'', d''A'', d²''A'', d³''A''〉.  Thus, there is just enough space in a planar venn diagram to plot all of these orbits and to show how they partition the points of E³''X''.  It turns out that there are exactly two possible orbits, of eight points each, as illustrated in Figure 16.

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<p>[[Image:Diff Log Dyn Sys -- Figure 16 -- A Couple of Fourth Gear Orbits.gif|center]]</p>

<p><center><font size="+1">'''Figure 16.  A Couple of Fourth Gear Orbits'''</font></center></p>

|    o    4<---|----/----|----3    o        o    |

Figure 16-a.  A Couple of Fourth Gear Orbits:  1

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|    |    o        o    3    o    1   o    |    |

Figure 16-b.  A Couple of Fourth Gear Orbits:  2

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With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation.  With this end in sight, let us index the states ''q'' in E^''m''''X'' with the dyadic rationals (or the binary fractions) in the half-open interval [0,&nbsp;2).  Formally and canonically, a state ''q''_''r'' is indexed by a fraction ''r''&nbsp;=&nbsp;''s''/''t'' whose denominator is the power of two ''t''&nbsp;=&nbsp;2^''m'' and whose numerator is a binary numeral that is formed from the coefficients of state in a manner to be described next.  The ''differential coefficients'' of the state ''q'' are just the values d^''k''''A''(''q''), for ''k''&nbsp;=&nbsp;0&nbsp;to&nbsp;''m'', where d⁰''A'' is defined as being identical to ''A''.  To form the binary index d₀'''.'''d₁&hellip;d_''m'' of the state ''q'' the coefficient d^''k''''A''(''q'') is read off as the binary digit ''d''_''k'' associated with the place value 2^&ndash;''k''.  Expressed by way of algebraic formulas, the rational index ''r'' of the state ''q'' can be given by the following equivalent formulations:

With a little thought it is possible to devise an indexing scheme for the general run of dynamic states that allows for comparing universes of discourse that weigh in on different scales of observation.  With this end in sight, let us index the states ''q'' in E^''m''''X'' with the dyadic rationals (or the binary fractions) in the half-open interval [0,&nbsp;2).  Formally and canonically, a state ''q''_''r'' is indexed by a fraction ''r''&nbsp;=&nbsp;''s''/''t'' whose denominator is the power of two ''t''&nbsp;=&nbsp;2^''m'' and whose numerator is a binary numeral that is formed from the coefficients of state in a manner to be described next.  The ''differential coefficients'' of the state ''q'' are just the values d^''k''''A''(''q''), for ''k''&nbsp;=&nbsp;0&nbsp;to&nbsp;''m'', where d⁰''A'' is defined as being identical to ''A''.  To form the binary index d₀'''.'''d₁&hellip;d_''m'' of the state ''q'' the coefficient d^''k''''A''(''q'') is read off as the binary digit ''d''_''k'' associated with the place value 2^&ndash;''k''.  Expressed by way of algebraic formulas, the rational index ''r'' of the state ''q'' can be given by the following equivalent formulations: