Independence Criteria for Implementations

As explained in the previous post Counting Implementations: The Problem of Size, the measure of a conscious computation within a system will be assumed to be proportional to the number of independent implementations of that computation. Independence Criteria for Implementations (ICI) must be chosen appropriately, and all possible mappings must be considered so as to maximize the number of simultaneous independent implementations for all conscious computations.

Intuitively as well as based on human observations, measure should not be a super-strong function of brain size or flexibility, so criteria that lead to exponential dependence on size are unlikely to be correct. In particular, mappings that include a functional channel plus an arbitrary combination of additional channels (any of which could have been a functional one given different initial conditions) would not be independent (intuitively) as the functional channel does all of the explanatory work about what computations are occurring, and as (problematically) the number of possible combinations grows exponentially with the number of channels. However, the Born Rule suggests that the size (squared amplitude) of terms in the wavefunction is indeed a factor, either directly or indirectly.


ICI proposal 1: Substate-style Criteria

For substates within a computation, the criterion for their independence from each other is roughly speaking that each substate should depend (in whole or in part) on a different physical variable (which may be either actual variables much like particle positions in classical configuration space, or an index which a group of variables depends on much like the quantum wavefunction depends on directions in particle-configuration space). Could a similar idea hold for independence of implementations?

Consider a digital variable that can take on 4 possible values. It can correspond to a pair of bits with possible values 00, 01, 10, 11. Within a computation, these can not be considered as independent substate bits because allowing that sort of thing would open the door to clock-and-dial-style false implementations. But for two different implementations (whether of the same computation or of different ones), each of which is by itself an allowed implementation, there seems to be no problem with allowing each of the above bits to play a role in a different one of the implementations. So while it is possible to use similar independence criteria across implementations as within them, that seems unnecessarily restrictive, and I do not find it intuitively appealing.

But that's not the only problem. Consider a system with N channels each of which can perform the same kinds of computations, similar to the earlier example of the Problem of Size, but for this example, suppose that it is a system with linear dynamics. It could for example be a quantum system, although any linear system would do.

For the initial conditions suppose that channel #1 is set up to perform a computation of interest, while the other channels are in their static conditions. For a quantum system, that means that the wavefunction is zero in the other channels, and remains so.

Consider mappings which are based on sums over various combinations of the channels. For example, a mapping M1 maps values of sums in corresponding substates of channels #1, 3, 5, and 8 to computational states. Another mapping M2 uses different channels, such as #1, 2, 3, 4, and 5.

How many implementations of the computation are performed by this system? If the Born Rule can be derived for this system, that depends on the amplitude of the wavefunction, but that's not what I want to focus on yet. In any case, the substate-style criterion leaves no direct role for the amplitude, though see below for how it might play an indirect role based on overcoming 'noise' in a many-particle world.

The question now is: How does the number of implementations depend on the number of channels, N? Intuitively, because there is no activity in channels other than #1, the number of channels should not affect the number of implementations. But mappings that combine different channels linearly, such as M1 and M2 do, satisfy the substate-style independence criteria. If each new mapping can either include or reject each channel (other than channel #1, which must be included in order to implement the computation), there are 2^(N-1) such mappings. This is actually an overestimate, because for any pair of mappings there must be at least one channel that the first includes but the second does not, and also vice versa. For example, the mapping that includes all odd # channels is not independent of the one that includes all channels. Still, this is not the proper dependence on N, which should be none.

It can be concluded that the substate-style criteria are not appropriate.


ICI proposal 2: Full Independence

An alternative is to allow any set of mappings such that being given each implementation mapping and the corresponding computational state places no restrictions on what computational states any of the other mappings correspond to. For the case of the classical ball system considered for the Problem of Size this rules out the problematic arbitrary combinations of channels, because all of the mappings using the same functional channel would have to be in the same computational state (or in one which can be mapped to it using similar substates and transition rules). It also allows the digital variable with 4 possible values to play a role in two different implementations.

For the linear system with N channels, this alone is not sufficient, because the same problem that the substate-style criteria encountered applies here. An added restriction can be placed as follows:

Change in Private:
If there is a region either of physical variables, physical indices, or physical values which is not shared by the mappings:
Both for initial and subsequent time steps, a difference in any computational sub-state that the mapping uses the non-shared region to determine must require a corresponding difference in the non-shared region, so that if there is no such difference there, no computational state is mapped to. For a given mapping, the shared region may or may not have to change also (not restricted by this condition).

This solves the N-channel problem because the non-shared channels don't meet the above conditions if the mappings give computational states that are determined by the state of channel #1 when the other channels remain at zero wavefunction.

The reference to a nonshared region of "physical values" for a particular physical variable is something that wouldn't make sense for substate independence within a single mapping, since all substates must be mapped to simultaneously within a single mapping. For different implementations, there is no such restriction.

What does it imply for quantum mechanics? That question will be studied in more detail in later posts, but note that scaling up wavefunction amplitudes does not appear to open up any possibilities for additional implementations under these criteria, for either of the above ICI proposals. As will be explored, the only way to produce the appearance of the Born Rule under such circumstances would be if a fixed 'noise level' in the wavefunction were somehow competing with the 'signal' and if increasing the amplitude allows more implementations indirectly because it allows a smaller volume of configuration space to support an implementation capable of beating the noise. This idea is interesting but has its own problems.


ICI proposal 3: Partial Independence (Note: This is a work in progress)

Knowing that the Born Rule is the goal, perhaps independence criteria could be found that satisfy the restrictions on independence suggested by the Problem of Size, while also opening the door to a more direct relationship between wavefunction amplitude and number of implementations. Increased complexity would argue against it, but if the problems of the competing-noise approach can be avoided that would be a point in its favor. Can such criteria be formulated, and if so, are they at all an intuitively reasonable possibility?