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Article type: Research Article
Authors: Kreczmar, Antonia
Affiliations: [a] Institute of Informatics, University of Warsaw
Abstract: In the present paper we investigate algorithmic properties of fields. We prove that axioms of formally real fields for the field R of reals and axioms of fields of characteristic zero for the field C of complex numbers, give the complete characterization of algorithmic properties. By Kfoury’s theorem programs which define total functions over R or C are effectively equivalent to loop-free programs. Examples of programmable and nonprogrammable functions and relations over R and C are given. In the case of ordered reals the axioms of Archimedean ordered fields completely characterize algorithmic properties. We show how to use the equivalent version of Archimed’s axiom (the exhaustion rule) in order to prove formally the correctness of some iterative numerical algorithms.
Keywords: programs and programmability, algorithmic properties, programmability in fields, axioms for algorithmic properties of reals, ordered reals and complex numbers
DOI: 10.3233/FI-1977-1113
Journal: Fundamenta Informaticae, vol. 1, no. 1, pp. 195-230, 1977
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