Implicit complexity over an arbitrary structure: Quantifier alternations

We provide machine-independent characterizations of some complexity classes, over an arbitrary structure, in the model of computation proposed by L. Blum, M. Shub and S. Smale. We show that the levels of the polynomial hierarchy correspond to safe recursion with predicative minimization and the levels of the digital polynomial hierarchy to safe recursion with digital predicative minimization. Also, we show that polynomial alternating time corresponds to ∗This author has been supported by French Ministry of Research through ANR Project SOGEA. †This author has been supported City University of Hong Kong SRG grant 7001290. ‡This author has also been supported City University of Hong Kong SRG grant 7001290.