For it implies that at least across the kinds of systems that we considered in that chapter there is in fact an upper limit on the sophistication of computations that can be done.

And one of the consequences of this is that it implies that most systems whose behavior seems complex should be universal.

For what this phenomenon implies is that even though a system may follow definite underlying laws its overall behavior can still have aspects that fundamentally cannot be described by reasonable laws.

So what this implies is that to answer questions about the t -step behavior of a multiway system can take any ordinary Turing machine a number of steps that increases faster than any power of t .

Computational irreducibility implies that it can be arbitrarily difficult to find minimal or optimal rules.

For in particular it ends up being almost impossible to distinguish whatever genuine instability and apparent randomness may be implied by the Navier–Stokes equations from artifacts that get introduced through the discretization procedure used in solving the equations on a computer. … Indeed, there is even some evidence that singularities might almost inevitably form, which would imply a breakdown of the equations, and perhaps a need to account for underlying molecular processes.

But this number grows faster than r d for any d —implying that the network has no correspondence to ordinary space in any finite number of dimensions.

In the case of substitution systems for strings, locality of underlying replacement rules immediately implies overall locality of effects in the system.

But the Principle of Computational Equivalence implies that many ordinary physical processes are computationally just as sophisticated as human thinking.

This program always halts, yet it does not correspond to any possible value of i —even though universality implies that any program should be encodable by a single integer i .