I show really be done with Integrated
Information Theory (IIT), in Aaronson’s simplified
formulation, but I noticed a rather interesting difficult.

In my previous
post on the subject, I noticed that a double grid system where there
are two grids stacked on top of one another, with the bottom grid
consisting of inputs and the upper grid of outputs, and each upper value
being the logical OR of the (up to) five neighboring input values will
be conscious according to IIT if all the values are zero and the grid is
large enough.

In this post, I am going to give some conjectures about the
mathematics rather than even a proof sketch. But I think the conjectures
are pretty plausible and, if true, it shows something fishy about IIT’s
measure of integrated information.

Consider our dual grid system, except now the grids are with some
exceptions rectangular, with a length of *M* along the *x*-axis and a width of *N* along the *y*-axis (and the stacking along the
*z*-axis). But there are the
following exceptions to the rectangularity:

In other words, at two *x*-coordinate areas, the grids have
bottlenecks, of slightly different sizes. We suppose *M* is significantly larger than *N*, and *N* is very, very large (say, 10^{15}).

Let *A*_{k}
be the components on the grids with *x*-coordinates less than *k* and let *B*_{k} be the
remaining components. I suspect (with a lot of confidence) that the
optimal choice for a partition {*A*, *B*} that minimizes the
“modified *Φ* value” *Φ*(*A*,*B*)/min (|*A*|,|*B*|)
will be pretty close to {*A*_{k}, *B*_{k}}
where *k* is in one of the
bottlenecks. Thus to estimate *Φ*, we need only look at the *Φ* and modified *Φ* values for {*A*_{M/4}, *B*_{M/4}}
and {*A*_{M/2}, *B*_{M/2}}.
Note that if *k* is *M*/4 or *M*/2, then min (|*A*|,|*B*|) is
approximately 2*M**N*/4
and 2*M**N*/2,
respectively, since there are two grids of components.

I suspect (again with a lot of confidence) that *Φ*(*A*_{k},*B*_{k})
will be approximately proportional to the width of the grid around
coordinate *k*. Thus, *Φ*(*A*_{M/4},*B*_{M/4})/min (*A*_{M/4},*B*_{M/4})
will be approximately proportional to (*N*/8)/(2*N**M*/4) = 0.25/*M*
while *Φ*(*A*_{M/2},*B*_{M/2})/min (*A*_{M/2},*B*_{M/2})
will be approximately proportional to (*N*/10)/(2*N**M*/2) = 0.1/*M*.

Moreover, I conjecture that the optimal partition will be close to
{*A*_{k}, *B*_{k}}
for some *k* in one of the
bottlenecks. If so, then our best choice will be close to {*A*_{M/2}, *B*_{M/2}},
and it will yield a *Φ* value
approximately proportional to *N*/10.

Now modify the system by taking each output component at an *x*-coordinate less than *M*/4 and putting four more output
components besides the original output component, and with the very same
value as the original output component.

I strongly suspect that the
optimal partition will again be obtained by cutting the system at one of
the two bottlenecks. The *Φ*
values of at the *M*/4 and *M*/2 bottlenecks will be
unchanged—mere duplication of outputs does not affect information
content—but the modified *Φ*
values (obtained by dividing *Φ*(*A*,*B*) by min (|*A*|,|*B*|)) will be
(*N*/8)/(6*N**M*/4) = 0.083/*M*
and (*N*/10)/(2*N**M*/2) = 0.1/*M*.
Thus the optimal choice will be to partition the system at the *M*/4 bottleneck. This will yield a
*Φ* value approximately
proportional to *N*/8. Which is
bigger than *N*/10.

For concreteness, let’s now imagine that each output is an LED. We
now see that if we replace some of the LEDs by five LEDs (namely, the
ones in the left-hand quarter of the system), we increase the amount of
integrated information from *N*/10 to *N*/8. This has got to be wrong.
Simply by duplicating LEDs we don’t add anything to the information
content. And we certainly don’t make a system more conscious just by
lighting up a portion of it with additional LEDs.

Notice, too, that IIT has a special proviso: if one system is a part
of another with a higher degree of consciousness, the part system has
*no* consciousness. So now imagine that a *Φ* value proportional to *N*/10 is sufficiently large for
significant consciousness, so our original system, without extra output
LEDs, is conscious. Now, besides the left quarter of the LEDs, add the
quadruples of new LEDs that simply duplicate the original LED values
(they might not even be electrically connected to the original system:
they might sense whether the original LED is on, and light up if so).
According to IIT, then, the new system is more conscious than the
old—and the old system has had its consciousness destroyed, simply by
adding enough duplicates of its LEDs. This seems wrong.

Of course, my conjectures and back-of-the-evelope calculations could be false.