Implicit complexity over an arbitrary structure: Quantifier alternations

Authors:
Olivier Bournez;Felipe Cucker;Paulin Jacobé de Naurois;Jean-Yves Marion
Affiliations:
LORIA, 615 rue du Jardin Botanique, BP 101, 54602 Villers-lès-Nancy Cedex, Nancy, France;Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong;LORIA, 615 rue du Jardin Botanique, BP 101, 54602 Villers-lès-Nancy Cedex, Nancy, France;LORIA, 615 rue du Jardin Botanique, BP 101, 54602 Villers-lès-Nancy Cedex, Nancy, France
Venue:
Information and Computation
Year:
2006

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Abstract

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 safe recursion with predicative substitutions and that digital polynomial alternating time corresponds to safe recursion with digital predicative substitutions.