Universal Computation Is 'Almost Surely' Chaotic (1111.4949v9)

Abstract: Fixed point iterations are known to generate chaos, for some values in their parameter range. It is an established fact that Turing Machines are fixed point iterations. However, as these Machines operate in integer space, the standard notions of a chaotic system is not readily applicable for them. Changing the state space of Turing Machines from integer to rational space, the condition for chaotic dynamics can be suitably established, as presented in the current paper. Further it is deduced that, given random input, computation performed by a Universal Turing Machine would be 'almost surely' chaotic.