Some new results on decidability for elementary algebra and geometry

Robert M. Solovay, R.D. Arthan, John Harrison. ArXiV preprint 0904.3482, submitted to Annals of Pure and Applied Logic, 2009. This is superseded by the corresponding published paper

Abstract:

Abstract We carry out a systematic study of decidability for theories of (a) real vector spaces, inner product spaces, and Hilbert spaces and (b) normed spaces, Banach spaces and metric spaces, all formalised using a 2-sorted first-order language. The theories for list (a) turn out to be decidable while the theories for list (b) are not even arithmetical: the theory of 2-dimensional Banach spaces, for example, has the same many-one degree as the set of truths of second-order arithmetic. We find that the purely universal and purely existential fragments of the theory of normed spaces are decidable, as is the ∀∃ fragment of the theory of metric spaces. These results are sharp of their type: reductions of Hilberts 10th problem show that the ∃∀ fragments for metric and normed spaces and the ∀∃ fragment for normed spaces are all undecidable.

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Bibtex entry:

@UNPUBLISHED{solovay-arthan-harrison,
        author          = "Robert M. Solovay and R.D. Arthan and
                           John Harrison",
        title           = "Some new results on decidability for
                           elementary algebra and geometry",
        note            = "ArXiV preprint 0904.3482; submitted to
                           Annals of Pure and Applied Logic. Available at
 {\url{http://arxiv.org/PS_cache/arxiv/pdf/0904/0904.3482v1.pdf}}",
        year            = 2009}