Implicit complexity over an arbitrary structure: Quantifier alternations

We provide machine-independent characterizations of some complexityclasses, over an arbitrarystructure, 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 safe recursion with predicative substitutions and that digital polynomial alternating time corresponds to safe recursion with digital predicative substitutions. © 2005 Elsevier Inc. All rights reserved.