Abstract
In this paper we study the topology of asymptotic cones of groups constructed from \(\mathcal{S}\)-machines running in polynomial time. In particular we directly construct an \(\mathcal{S}\)-machine for an NP-complete problem. Using a part of the machinery shaped by Sapir,Birget and Rips we construct its associated group and we show that every asymptotic cone of this group is not simply connected. The proof is rather geometric and use an argument similar to the one developed by Sapir and Olshanskii. This work aims to give a topological characterization of non-deterministic time complexity class.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Rips, E., Birget, J.C., Olshanskii, A.Y., Sapir, M.V.: Isoperimetric function of groups and computational complexity of the word problem. Annals of Mathematics 156(2), 476–518 (2002)
Gromov, M.: Asymptotics invariants of infinite groups. Geometric Group Theory 2, 1–295 (1993)
Olshanskii, A.Y., Sapir, M.V.: Length and area functions on groups and quasi-isometric higman embeddings. Intern. J. Algebra and Comput. 1, 137–170 (1991)
Papsoglu, P.: On the asymptotic cones of groups satisfying a quadratic isoperimetric inequality. J. Differential Geometry 44, 789–806 (1996)
Birget, J.C., Sapir, M.V., Rips, E.: Isoperimetric and isodiametric functions of groups. Annals of Mathematics 156, 345–466 (2002)
Olshanskii, A.Y., Sapir, M.V.: Groups with non-simply connected asymptotic cones. In: Topology and Asymptotic Aspects of Group Theory. Contemp. Math., vol. 394, pp. 203–208 (2006)
Olshanskii, A.Y., Sapir, M.V.: A finitely presented group with two non-homeomorphic asymptotic cones. Internat. J. Algebra. Comput. 17(2), 421–426 (2007)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Gasperin, A. (2013). Topology of Asymptotic Cones and Non-deterministic Polynomial Time Computations. In: Bonizzoni, P., Brattka, V., Löwe, B. (eds) The Nature of Computation. Logic, Algorithms, Applications. CiE 2013. Lecture Notes in Computer Science, vol 7921. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-39053-1_22
Download citation
DOI: https://doi.org/10.1007/978-3-642-39053-1_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-39052-4
Online ISBN: 978-3-642-39053-1
eBook Packages: Computer ScienceComputer Science (R0)