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Journal of Logic and Computation 2005 15(1):41-58; doi:10.1093/logcom/exh036
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Vol. 15 No. 1, © The Author, 2005. Published by Oxford University Press. All rights reserved.

Implicit Complexity over an Arbitrary Structure: Sequential and Parallel Polynomial Time

Olivier Bournez1, Felipe Cucker2, Paulin Jacobé de Naurois3 and Jean-Yves Marion4

1 INRIA/LORIA, 615 rue du Jardin Botanique, BP 101, 54602 Villers-lès-Nancy Cedex, Nancy, France. E-mail: Olivier.Bournez{at}loria.fr, 2 Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong. E-mail: macucker{at}math.cityu.edu.hk, 3 Nancy II/LORIA, 615 rue du Jardin Botanique, BP 101, 54602 E-mail: Paulin.De-Naurois{at}loria.fr, 4 INPL/LORIA, 615 rue du Jardin Botanique, BP 101, 54602 E-mail: Jean-Yves.Marion{at}loria.fr

We provide several machine-independent characterizations of deterministic complexity classes in the model of computation proposed by L. Blum, M. Shub and S. Smale. We provide a characterization of partial recursive functions over any arbitrary structure. We show that polynomial time over an arbitrary structure can be characterized in terms of safe recursion. We show that polynomial parallel time over an arbitrary structure can be characterized in terms of safe recursion with substitutions.

Keywords: Blum, Shub, Smale model, implicit complexity, complexity theory

Received 4 April 2003.


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