suggests that in our actual universe there are limits on the sizes and densities of components that we can ever expect to manipulate.
In present-day physics the standard mathematical formalism of quantum mechanics is often interpreted as suggesting that quantum systems work like multiway systems, potentially following many paths in parallel. And indeed within the usual formalism one can construct quantum computers that may be able to solve at least a few specific problems exponentially faster than ordinary Turing machines.
But particularly after my discoveries in Chapter 9, I strongly suspect that even if this is formally the case, it will still not turn out to be a true representation of ultimate physical reality, but will instead just be found to reflect various idealizations made in the models used so far.
And so in the end it seems likely that there really can in some fundamental sense be an almost exponential difference in the amount of computational effort needed to find the behavior of a system with given particular initial conditions, and to solve the inverse problem of determining which if any initial conditions yield particular behavior.
In fact, my suspicion is that such a difference will exist in almost any system whose behavior seems to us complex. And among other things this then implies many fundamental limits on the processes of perception and analysis that we discussed in Chapter 10.
Such limits can ultimately be viewed as being consequences of the phenomenon of computational irreducibility. But a much more direct consequence is one that we have discussed before: that even given a particular initial condition it can require an irreducible amount of computational work to find the outcome after a given number of steps of evolution.
One can specify the number of steps t that one wants by giving the sequence of digits in t. And for systems with sufficiently simple behavior—say repetitive or nested—the pictures on page 744 indicate that one can typically determine the outcome with an amount of effort that is essentially proportional to the length of this digit sequence.
But the point is that when computational irreducibility is present, one may in effect explicitly have to follow each of the t steps of evolution—again requiring exponentially more computational work.