Functionals Using Bounded Information and the Dynamics of Algorithms

We consider computable functionals mapping the Baire space into the set of integers. By continuity, the value of the functional on a given function depends only on a "critical" finite part of this function. Care: there is in general no way to compute this critical finite part without querying the function on an arbitrarily larger finite part! Nevertheless, things are different in case there is a uniform bound on the size of the domain of this critical finite part. We prove that, modulo a quadratic blow-up of the bound, one can compute the value of the functional by an algorithm which queries the input function on a uniformly bounded finite part. Up to a constant factor, this quadratic blow-up is optimal. We also characterize such functionals in topological terms using uniformities. As an application of these results, we get a topological characterization of the dynamics of algorithms as modeled by Gurevich's Abstract State Machines.