Topological computing
Encyclopedia
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
.
. He showed that the logical units can be associated with the topologically different geometrical figures. 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.
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
. Other aspects of topological theory of electromagnetic field and applications of topology in physics and electromagnetism can be found from.
, Boolean
, reconfigurable
and pseudo-quantum logics.
The first predicate logic processor based on the idea of topological computing was designed and tested by FPGA modeling in 2007.
The first quantum topological computer and its theory were proposed by A. Kitaev et al. in 2002.
A theory of a particular case of topological processors has been proposed by Ryabov and Serov (2007) which is for parallel operating of elementary image voxels.
Relations of topology, physics and computations are considered by Baez and Stay (2009).
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
.
Science and theory
The first who bound up the topology of figures and the logical systems was Alfred TarskiAlfred Tarski
Alfred Tarski was a Polish logician and mathematician. Educated at the University of Warsaw and a member of the Lwow-Warsaw School of Logic and the Warsaw School of Mathematics and philosophy, he emigrated to the USA in 1939, and taught and carried out research in mathematics at the University of...
. He showed that the logical units can be associated with the topologically different geometrical figures. At the end of the 1980s, the topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
of the electromagnetic field
Electromagnetic field
An electromagnetic field is a physical field produced by moving electrically charged objects. It affects the behavior of charged objects in the vicinity of the field. The electromagnetic field extends indefinitely throughout space and describes the electromagnetic interaction...
was studied in detail and a topological theory of guided waves and components was proposed and applied to many microwave integrated components.
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
Maxwell's equations
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electrodynamics, classical optics, and electric circuits. These fields in turn underlie modern electrical and communications technologies.Maxwell's equations...
. Other aspects of topological theory of electromagnetic field and applications of topology in physics and electromagnetism can be found from.
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. They excel in increased noise immunity due to their topological nature. The signals carry digital information by their skeletons and magnitudes of impulses and they can model the predicatePredicate logic
In mathematical logic, predicate logic is the generic term for symbolic formal systems like first-order logic, second-order logic, many-sorted logic or infinitary logic. This formal system is distinguished from other systems in that its formulae contain variables which can be quantified...
, Boolean
Boolean logic
Boolean algebra is a logical calculus of truth values, developed by George Boole in the 1840s. It resembles the algebra of real numbers, but with the numeric operations of multiplication xy, addition x + y, and negation −x replaced by the respective logical operations of...
, reconfigurable
Reconfigurable computing
Reconfigurable computing is a computer architecture combining some of the flexibility of software with the high performance of hardware by processing with very flexible high speed computing fabrics like field-programmable gate arrays...
and pseudo-quantum logics.
Topological processors and computers
A computing unit performing the operations with topologically modulated impulses is a topological processor proposed in 1991–1992. 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.
The first quantum topological computer and its theory were proposed by A. Kitaev et al. in 2002.
A theory of a particular case of topological processors has been proposed by Ryabov and Serov (2007) which is for parallel operating of elementary image voxels.
Relations of topology, physics and computations are considered by Baez and Stay (2009).