[Inquiry] Re: Differential Logic

[Jon Awbrey]

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DLOG.  Note D24

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Example 2.  Drives and Their Vicissitudes (concl.)

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, 2).
Formally and canonically, a state q_r is indexed by a fraction r = s/t
whose denominator is the power of two t = 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 = 0 to m, where d^0.A is defined as being
identical to A.  To form the binary index d_0 . d_1 ... 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^(-k).  Expressed by way of algebraic formulas, the rational
index r of the state q can be given by the following equivalent formulations:

o-------------------------------------------------------------------------------o
|                                                                               |
|  r(q)    =   Sum_k d_k . 2^(-k)          =   Sum_k d^k.A(q) . 2^(-k)          |
|                                                                               |
|  =                                                                            |
|                                                                               |
|  s(q)/t  =  (Sum_k d_k . 2^(m-k)) / 2^m  =  (Sum_k d^k.A(q) . 2^(m-k)) / 2^m  |
|                                                                               |
o-------------------------------------------------------------------------------o

Applied to the example of fourth gear curves, this scheme results in the data
of Tables 17-a and 17-b, which exhibit one period for each orbit.  The states
in each orbit are listed as ordered pairs <p_i, q_j>, where p_i may be read
as a temporal parameter that indicates the present time of the state, and
where j is the decimal equivalent of the binary numeral 's'.  Informally
and more casually, the Tables exhibit the states q_s as subscripted
with the numerators of their rational indices, taking for granted
the constant denominators of 2^m = 2^4 = 16.  Within this set-up,
the temporal successions of states can be reckoned as given by
a kind of "parallel round-up rule".  That is, if <d_k, d_(k+1)>
is any pair of adjacent digits in the state index r, then the
value of d_k in the next state is (d_k)' = d_k + d_(k+1).

Table 17-a.  A Couple of Orbits in Fourth Gear:  Orbit 1
o---------o---------o---------o---------o---------o---------o---------o
| Time    | State   |    A    |   dA    |         |         |         |
|  p_i    |  q_j    |  d^0.A  |  d^1.A  |  d^2.A  |  d^3.A  |  d^4.A  |
o---------o---------o---------o---------o---------o---------o---------o
|         |         |                                                 |
|  p_0    |  q_01   |    0.        0         0         0         1    |
|         |         |                                                 |
|  p_1    |  q_03   |    0.        0         0         1         1    |
|         |         |                                                 |
|  p_2    |  q_05   |    0.        0         1         0         1    |
|         |         |                                                 |
|  p_3    |  q_15   |    0.        1         1         1         1    |
|         |         |                                                 |
|  p_4    |  q_17   |    1.        0         0         0         1    |
|         |         |                                                 |
|  p_5    |  q_19   |    1.        0         0         1         1    |
|         |         |                                                 |
|  p_6    |  q_21   |    1.        0         1         0         1    |
|         |         |                                                 |
|  p_7    |  q_31   |    1.        1         1         1         1    |
|         |         |                                                 |
o---------o---------o---------o---------o---------o---------o---------o

Table 17-b.  A Couple of Orbits in Fourth Gear:  Orbit 2
o---------o---------o---------o---------o---------o---------o---------o
| Time    | State   |    A    |   dA    |         |         |         |
|  p_i    |  q_j    |  d^0.A  |  d^1.A  |  d^2.A  |  d^3.A  |  d^4.A  |
o---------o---------o---------o---------o---------o---------o---------o
|         |         |                                                 |
|  p_0    |  q_25   |    1.        1         0         0         1    |
|         |         |                                                 |
|  p_1    |  q_11   |    0.        1         0         1         1    |
|         |         |                                                 |
|  p_2    |  q_29   |    1.        1         1         0         1    |
|         |         |                                                 |
|  p_3    |  q_07   |    0.        0         1         1         1    |
|         |         |                                                 |
|  p_4    |  q_09   |    0.        1         0         0         1    |
|         |         |                                                 |
|  p_5    |  q_27   |    1.        1         0         1         1    |
|         |         |                                                 |
|  p_6    |  q_13   |    0.        1         1         0         1    |
|         |         |                                                 |
|  p_7    |  q_23   |    1.        0         1         1         1    |
|         |         |                                                 |
o---------o---------o---------o---------o---------o---------o---------o

Jon Awbrey

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