Comment on Open Thread, July 1-15, 2013

Fri, 12 Jul 2013 22:38:32 GMT

Uncountability doesn't seem like a big deal to me. Just give the Turing machine an auxiliary tape containing the real parameter.

A related question is whether a Turing machine can fully simulate the dynamical system, that is, whether it can compute the state at any future time to any precision using only finitely many bits from the starting parameters.

I think the answer to that differential equations with initial conditions a computable real number can evolve to an uncomputable real number. But if the initial random number is not arbitrary, but guaranteed random, maybe it is computable. (That is, maybe the inputs with computable futures have full measure.)

Discuss