Unpredictability and Computational Irreducibility

We explore several concepts for analyzing the intuitive notion of computational irreducibility and we propose a robust formal definition, first in the field of cellular automata and then in the general field of any computable function f from N to N. We prove that, through a robust definition of what means "to be unable to compute the n th step without having to follow the same path than simulating the automaton or to be unable to compute f(n) without having to compute f(i) for i=1 to n-1", this implies genuinely, as intuitively expected, that if the behavior of an object is computationally irreducible, no computation of its n th state can be faster than the simulation itself.