In the previous post, Restrictions on mappings 1: Independence and Inheritance, the "inheritance" of structured state labels was explained; it allows the same group of underlying variables to be mapped to more than one independent formal variable. In the example a function on a 2-d grid was mapped to a pair of variables.
Transference is something like the reverse process: It allows a set of simpler variables to be mapped to a structured state function on a grid.
This allows ordinary digital computers to implement wave dynamics on a 3-d spaces, which could matter for the question of whether the universe could be ultimately digital. The AdS/CFT correspondence in some models of string theory would need something similar if the bulk model is to be implemented on the boundary in the computational sense.
Transference can be Direct or Indirect. It works like this:
Direct Transference could be used in a mapping by taking the value from a given variable and turning it into a label for structuring a set of new variables.
For example, if there is a single integer variable I(t), we can transfer its value to label to a set of bits B(j) which each only depend on whether I(t) equals the value of its label, e.g.
B(j) = 1 if I = j
B(j) = 0 if I does not = j
These bits can be considered an ordered series of "occupation tests" of the different regions that the underlying variable's value could be in.
Of course, only one of these bits at a time will be nonzero. But they are to be considered independent variables. At this point you might object: If you know the value of the nonzero one, don't you know the other bit values must be zero? But just as Inheritance carved out an exception to the rule for independence, so would Direct Transference carve out an exception to it.
Going the other direction is no problem: If we restrict a mapping such that only one bit in an ordered set B(i) is nonzero, then a new variable I can be constructed such that I has a value equal to the index i of the nonzero bit. Here we are doing the reverse.
We can't double count, though; if we make the new set of variables b(i), we can't make a second independent new set of variables c(i) which gets its label transferred from the same underlying variable I(t) for the same values of I.
If we have two underlying variables I and J, we could similarly use Direct Transference to map them to a 2-d grid of bits, B(i,j), in which only one bit is nonzero.
If we then re-map this grid using inheritance we could arrive back at our original I and J variables. So, basically, what Direct Transference is saying is that these two pictures are really equivalent.
We could also map the two of them to a single 1-d series of variables, e.g. which are the sum of the respective 1-d series of bits. (Since the value of the sum becomes 2 when X=Y, these are trits, not bits.)
Can the variables that were obtained using Direct Transference be used to make a mapping so flexible that it must not be allowed? Something like a clock and dial mapping? The answer to that certainly appears to be, no. And that may be justification enough for allowing it; my philosophy is to be liberal in allowing mappings, as long as those mappings don't allow implementations of arbitrary computations.
Indirect Transference is a little more complicated. Consider a computer simulation of dynamics on a 2-d grid, f(x,y). When the value of f is updated at the pair of parameters x and y, this can be done by setting one variable equal to x, another equal to y, and using them to find the memory location M in the computer at which the corresponding value of f is to be changed. Since updates of f at (x,y) always involve fixed values for each of those parameters, f(x,y) can be labeled by those values. In this way, mapping of the values of f to the actual function of x and y, f(x,y), is considered a valid mapping, even though the computer's memory is not laid out in that matter. This is an example of Indirect Transference. It can be generalized to any case in which a parameter of a function is used.
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