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Recursive Unsolvability of Group Theoretic Problems

Michael O. Rabin

Journal of Symbolic Logic 23 (1):55-56 (1958)

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Detecting properties from descriptions of groups.Iva Bilanovic, Jennifer Chubb & Sam Roven - 2020 - Archive for Mathematical Logic 59 (3-4):293-312.details We consider whether given a simple, finite description of a group in the form of an algorithm, it is possible to algorithmically determine if the corresponding group has some specified property or not. When there is such an algorithm, we say the property is recursively recognizable within some class of descriptions. When there is not, we ask how difficult it is to detect the property in an algorithmic sense. We consider descriptions of two sorts: first, recursive presentations in terms of (...) Download Export citation Bookmark
Unrecognizability of manifolds.A. V. Chernavsky & V. P. Leksine - 2006 - Annals of Pure and Applied Logic 141 (3):325-335.details We present a modernized proof, with a modification by M.A. Shtan’ko, of the Markov theorem on the unsolvability of the homeomorphy problem for manifolds. We then discuss a proof of the S.P. Novikov theorem on the unrecognizability of spheres for n≥5, from which we obtain a corollary about unrecognizability of all manifolds of dimension at least five. An analogous argument then proves the unrecognizability of stabilizations of all four-dimensional manifolds. We also give a brief overview of known results concerning algorithmic (...) Download Export citation Bookmark 1 citation

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