Forecasting Black Holes in Abstract Geometrical Computation is Highly Unpredictable

In Abstract geometrical computation for black hole computation (MCU ’04, LNCS 3354), the author provides a setting based on rational numbers, abstract geometrical computation, with super-Turing capability: any recursively enumerable set can be decided in finite time. To achieve this, a Zeno-like construction is used to provide an accumulation similar in effect to the black holes of the black hole model. We prove here that forecasting an accumulation is Σ 2 -complete (in the arithmetical hierarchy) even if only energy conserving signal machines are addressed (as in the cited paper). The Σ 2 -hardness is achieved by reducing the problem of deciding whether a recursive function (represented by a 2-counter automaton) is strictly partial. The Σ 2-membership is proved with a logical characterization. Key-words. Abstract geometrical computation, Accumulation forecasting, Arithmetical hierarchy, Black hole model, Energy conservation, SuperTuring computation, Turing universality, Zeno phenomena.