Topological computing

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Topological computing is the designing and building of hardware and software based on the processing of topologically modulated electromagnetic impulses that differ from each other in their spatio-temporal topology.

Contents

[edit] Science and theory

The first who bound up the topology of figures and the logical systems was Alfred Tarski. He showed that the logical units can be associated with the topologically different geometrical figures.[1] At the end of the 1980s, the topology of the electromagnetic field was studied in detail and a topological theory of guided waves and components was proposed and applied to many microwave integrated components.[2][3][4][5]

The topological theory, nonlocal by its nature, describes the electric and magnetic fields by their topological schemes or skeletons[clarification needed] composed of the field-force map separatrices and field-equilibrium manifolds. These skeletons are coupled to each other through the topological analogs of the Maxwell's equations[4][clarification needed]. Other aspects of topological theory of electromagnetic field and applications of topology in physics and electromagnetism can be found from[6][7][8][9][10][11][12].

[edit] Topologically modulated signals

In transmission lines, the electromagnetic impulses can be generated with the topologically different spatio-temporal content; these are the topologically modulated signals.[3][13] They excel in increased noise immunity due to their topological nature.[14][15] The signals carry digital information by their skeletons and magnitudes of impulses and they can model the predicate, Boolean, reconfigurable and pseudo-quantum logics.[13][15][16][17][18][19][20][21]

[edit] Topological processors and computers

A computing unit performing the operations with topologically modulated impulses is a topological processor proposed in 1991–1992.[13][21] This processor united with other necessary conventional digital units makes up the topological computer.

The first predicate logic processor based on the idea of topological computing was designed and tested by FPGA modeling in 2007.[19][20]

The first quantum topological computer and its theory were proposed by A. Kitaev et al. in 2002.[22]

A theory of a particular case of topological processors has been proposed by Ryabov and Serov (2007)[23] which is for parallel operating of elementary image voxels.

Relations of topology, physics and computations are considered by Baez and Stay (2009).[24]

[edit] Related results

1. Logic of topology is a particular case of the spatial one.[25][26] Some information on the digital topology which is the basic for topological computing is in [27][28].

[edit] References

  1. ^ J.C.C. McKinsey and A. Tarski, The algebra of topology, Annals of Mathematics, Vol. 45, No. 1, pp. 141–191, Jan. 1944. http://www.jstor.org/stable/1969080
  2. ^ V.I. Gvozdev and G.A. Kouzaev, "A field approach to the CAD of microwave three-dimensional integrated circuits", Proc. Conf. Microwave Three-Dimensional Integrated Circuits, Tbilisi, USSR, pp. 67–73, 1988
  3. ^ a b G.A. Kouzaev, Mathematical fundamentals of topological electrodynamics and the three-dimensional microwave integrated circuits’ simulation. In: Electrodynamics and Techniques of Microwaves and EHF, Moscow. MIEM Publ., pp. 37–44, 1991.
  4. ^ a b V.I. Gvozdev and G.A. Kouzaev, Physics and the field topology of three-dimensional microwave integrated circuits", Soviet Microelectronics, Vol. 21, pp. 1–17, Jan.1992.
  5. ^ G.A., Kouzaev, M.J. Deen, N.K. Nikolova, and A. Rahal, Cavity models of planar components grounded by via-holes and their experimental verification. IEEE Trans., Microwave Theory and Techniques, Vol. 54, pp. 1033–1042, 2006.
  6. ^ F. Ranada , A topological theory of the electromagnetic field, Lett Math. Phys., Vol. 18, pp. 97–106, 1989.
  7. ^ T. W. Barret, Topological Foundations of Electromagnetism, World Sci., 2008.
  8. ^ P.W. Gross and P.R. Cotiuga, Electromagnetic Theory and Computations: A Topological Approach, Cambridge University Press, 2004.
  9. ^ H. Eschrig, Topology and Geometry for Physics, Dresden University of Technology.
  10. ^ G. W. Afanasief, Topological Effects in Quantum Mechanics, Kluwer Acad. Publ., 1999.
  11. ^ C. Nash, Topology and physics – a historical essay, arXiv:hep-th\9709135v4, 31 Dec. 1997.
  12. ^ L. Boi, Geometrical and topological foundations of theoretical physics: From gauge theories to string program, IJMMS, Vol. 34, pp. 1777–1836, 2004.
  13. ^ a b c V.I. Gvozdev and G.A. Kouzaev, Microwave flip-flop, Patent of Russian Federation, #2054794, 05.26.1992.
  14. ^ D.V. Bykov, V.I. Gvozdev, and G.A. Kouzaev, Contribution to the theory of topological modulation of electromagnetic field”, Russian Physics Doklady, Vol. 38, pp. 512–514, 1993.
  15. ^ a b G.A. Kouzaev, Communications by vector manifolds, In: Proc. European Computing Conf., Vol. 1, Series: Lecture Notes in Electrical Engineering, Vol. 27, Springer Verlag, 2009, pp. 617–624.
  16. ^ G.A. Kouzaev, Logic for electromagnetic field patterns, http://aps.arxiv.org/abs/0805.4600
  17. ^ G.A. Kouzaev, Spatio-temporal electromagnetic field shapes and their logical processing”, http://arxiv.org/abs/physics/0701081
  18. ^ G.A. Kouzaev, Qubit logic modeling by electronic circuits and electromagnetic signals, http://arxiv.org/abs/quant-ph/0108012
  19. ^ a b G.A. Kouzaev and A.N. Kostadinov, Predicate gates, components, and a processor for spatial logic, J. Circuits, Systems, and Computers, Vol. 19, No. 7 (2010), pp. 1–25.
  20. ^ a b G.A. Kouzaev and A.N. Kostadinov, Predicate logic processor, Materials of Innovation Forum’08, June 5, 2008, Toronto, Canada, Enterprise Toronto Publ., 2008.
  21. ^ a b V. I. Gvozdev and G. A. Kouzaev, "Topological computer", Computers and People, 1, pp. 2–5, 1992.
  22. ^ Michael H. Freedman, Alexei Kitaev, Michael J. Larsen, and Zhenghan Wang, Bull. Amer. Math. Soc., 40, 31 (2003), "Topological quantum computation"
  23. ^ G.G. Ryabov and V.A. Serov, Simplicial-lattice model and metric-topological constructions, Proc. 9th Int. Conf. Pattern Recognition and Information Processing, PRIP’2007, 22–24 May 2007, Minsk, Belarus, Vol. 2, pp. 135–140, 2007. http://www.vizcom.srcc.msu.ru/files/PRIP2007.pdf
  24. ^ J.C. Baez and M. Stay, Physics, topology, logic and computation: A Rosetta stone", 2009, http://arxiv.org/abs/0903.0340, http://math.ucr.edu/home/baez/rosetta.pdf
  25. ^ Handbook of Spatial Logic, M. Allielo, I. Pratt-Hartmann, and J. van Benthem (Eds), Springer, 2007.
  26. ^ A.G. Cohn and D. Mark, Spatial Information Theory, Berlin Heidelberg, Springer Verlag, 2005.
  27. ^ V. Kovalevsky, Geometry of Locally Finite Spaces, 2008.
  28. ^ Digital and Image Geometry, G. Bertrand, A. Imiya and R. Klette (Eds), LNCS-2243, Springer, 2001.
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