Topological computing

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Topological computing is the designing and building of hardware and software based on the processing of topologically modulated signals or objects which differ from each other in their spatio-time topology.

Contents 1 Science and theory 2 Topologically modulated signals 3 Topological processors and computers 4 Related results 5 References

[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.^[2][3][4] The topological theory, nonlocal by its nature, describes the electric and magnetic fields by their topological schemes or skeletons 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.

[edit] Topologically modulated signals

In transmission lines, the electromagnetic impulses can be generated with the topologically different spatio-time content; these are the topologically modulated signals.^[3][5] They excel in increased noise immunity due to their topological nature.^[6] The signals carry digital information by their skeletons and magnitudes of impulses and they can model the predicate, Boolean, reconfigurable and pseudo-quantum logics.^{[5][7][8][9][10]} Additionally, the signals can be used to imitate the spatio-time pattern activity of the brain.^[9]

[edit] Topological processors and computers

A computing unit performing the operations with topologically modulated impulses is a topological processor.^[5][7] This processor united with other necessary conventional digital units makes up the topological computer. The first predicate logic topological processor was created in 2007. It is designed for the fast operating with predicated data, spatio-temporal information, etc.^[10]

[edit] Related results

1. Logic of topology is a particular case of the spatial one.^[11] Some information on the digital topology which is the basic for topological computing is in ^[12][13].

2. A theory of a particular case of topological processors has been proposed recently in ^[14] which is for parallel operating of elementary image voxels.

3. The idea of increased noise immunity of topological computing (1992-1993) is used in quantum topological computations (1997).

Spatial computing and spatial artificial intelligence; Hypercube nano-computing; Computing of patterned signals; Topological quantum computers for increased noise-immune computations

[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. ^ V.I. Gvozdev anf G.A. Kouzaev, Physics and the Field Topology of Three-dimensional Microwave Integrated Circuits, Soviet Microelectronics, Vol. 21, pp.1-17, Jan.1992.
5. ^ ^{a b c} V.I. Gvozdev anf G.A. Kouzaev, Microwave Flip-flop, Patent of Russian Federation, #2054794, 05.26.1992.
6. ^ 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.
7. ^ ^{a b} G.A. Kouzaev, “Topological Computing”, WSEAS Trans., Computers, Vol. 5, pp. 1247-1250, 2006, http://www.worldses.org/online/download.htm
8. ^ G.A. Kouzaev, “Spatio-Temporal Electromagnetic Field Shapes and their Logical Processing”, http://arxiv.org/abs/physics/0701081
9. ^ ^{a b} G.A. Kouzaev, “Logic for Electromagnetic Field Patterns”, http://aps.arxiv.org/abs/0805.4600
10. ^ ^{a b} A.N. Kostadinov and G.A. Kouzaev, “Predicate Logic Processor of Spatially Patterned Signals”, In: Recent Advances in Systems Engineering and Applied Mathematics, WSEAS Publ., pp. 94-96, 2008, http://www.worldses.org/online/download.htm
11. ^ Handbook of Spatial Logic, M. Allielo, I. Pratt-Hartmann, and J. van Benthem (Eds), Springer, 2007.
12. ^ V. Kovalevsky, Geometry of Locally Finite Spaces, 2008.
13. ^ Digital and Image Geometry, G. Bertrand, A. Imiya and R. Klette (Eds), LNCS-2243, Springer, 2001.
14. ^ 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.ru/files/PRIP2007.pdf

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