[Inquiry] Re: Differential Logic A -- Discussion

[Jon Awbrey]

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DLOG A.  Discussion Note 23

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To facilitate the comparison with sign relations, let's reset
our machine table in the form of a relational database for
the 3-adic relation M c X x Y x Z, as shown in Table 2.

Table 2.  Transduction Triples
o---------o---------o---------o
|    X    |    Y    |    Z    |
o---------o---------o---------o
|    1    |    a    |    c    |
|    1    |    b    |    d    |
|    1    |    c    |    d    |
|    1    |    d    |    b    |
o---------o---------o---------o
|    2    |    a    |    b    |
|    2    |    b    |    a    |
|    2    |    c    |    d    |
|    2    |    d    |    c    |
o---------o---------o---------o
|    3    |    a    |    d    |
|    3    |    b    |    c    |
|    3    |    c    |    d    |
|    3    |    d    |    b    |
o---------o---------o---------o

Using the same tactic that we used for the regular representations of groups,
we can express the "meaning" of each input parameter as a logical sum of its
operational effects, where an effect is defined as an ordered pair of states,
let's say in the order <ante-op, post-op>, sometimes written as "ante:post".

   T_1  =  a:c + b:d + c:d + d:b

   T_2  =  a:b + b:a + c:d + d:c

   T_3  =  a:d + b:c + c:d + d:b

In a typical control system scenario, an agent has the job,
given the ability to choose a sequence of input parameters,
to take the system from whatever state it happens to be in
to a specified state, that we may regard as the goal state.

This type of set-up lends itself to a game-theoretic description,
where one player puts the system in any state and another player
has to pick a sequence of input parameters that will bring it to
the designated goal state.  Of course, there are many such games.

Jon Awbrey

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