Talk:PlanetPhysics/Categories of Quantum Automata and Quantum Computers

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\section{Categories of Quantum Automata, \\ N-- \L ukasiewicz Algebras and Quantum Computers}

\htmladdnormallink{Quantum automata}{http://planetphysics.us/encyclopedia/QuantumComputers.html} were defined (in ref.\cite{IB71}) as generalized, probabilistic automata with \htmladdnormallink{quantum state spaces}{http://planetphysics.us/encyclopedia/NonAbelianQuantumAlgebraicTopology3.html}. Their next-state \htmladdnormallink{functions}{http://planetphysics.us/encyclopedia/Bijective.html} operate through transitions between quantum states defined by the quantum equations of \htmladdnormallink{motions}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} in the Schr\"{o}dinger \htmladdnormallink{representation}{http://planetphysics.us/encyclopedia/CategoricalGroupRepresentation.html}, with both initial and \htmladdnormallink{boundary}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html} conditions in \htmladdnormallink{space-time}{http://planetphysics.us/encyclopedia/SR.html}. A new \htmladdnormallink{theorem}{http://planetphysics.us/encyclopedia/Formula.html} is proven which states that the \emph{\htmladdnormallink{category of quantum automata}{http://planetphysics.us/encyclopedia/CategoryOfQuantumAutomata.html} and automata--homomorphisms has both limits and colimits.} Therefore, both categories of quantum automata and classical automata (\htmladdnormallink{sequential machines}{http://planetphysics.us/encyclopedia/AAT.html}) are \emph{bicomplete.} A second new theorem establishes that the standard automata \htmladdnormallink{category}{http://planetphysics.us/encyclopedia/Cod.html} is a subcategory of the quantum automata category. The quantum automata category has a faithful representation in the category of Generalized (\textbf{M,R})--Systems which are open, \htmladdnormallink{dynamic}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html} bio-networks (\cite{ICB87}) with defined biological \htmladdnormallink{relations}{http://planetphysics.us/encyclopedia/Bijective.html} that represent physiological functions of primordial(s), single cells and the simpler organisms. A new \emph{category of quantum computers} is also defined in terms of \emph{reversible} quantum automata with quantum state spaces represented by \htmladdnormallink{topological groupoids}{http://planetphysics.us/encyclopedia/GroupoidHomomorphism2.html} that admit a local characterization through unique 'quantum' \emph{\htmladdnormallink{Lie algebroids}{http://planetphysics.us/encyclopedia/LieAlgebroids.html}}. On the other hand, the category of n-- \textsl{\L}ukasiewicz algebras has a subcategory of \emph{centered} n-- \textsl{\L}ukasiewicz algebras (ref. \cite{GGV70}) which can be employed to design and construct subcategories of quantum automata based on n--\textsl{\L}ukasiewicz \htmladdnormallink{diagrams}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html} of existing VLSI. Furthermore, as shown in ref.(\cite{GGV70}) the category of centered n--\textsl{\L}ukasiewicz algebras and the category of Boolean algebras are naturally equivalent. A `no-go' conjecture is also proposed which states that Generalized (\textbf{M,R})--Systems \htmladdnormallink{complexity}{http://planetphysics.us/encyclopedia/Complexity.html} prevents their complete computability (\cite{ICB87,BGGB2k7}) by either standard or quantum automata.

\begin{thebibliography}{99}

\bibitem{IB71} Baianu, I.1971.``Organismic Supercategories and Qualitative Dynamics of Systems." \emph{Bull. Math.Biophysics}., 33, 339-353.

\bibitem{GGV70} Georgescu, G. and C. Vraciu 1970. ``On the Characterization of \L ukasiewicz Algebras." \emph{J. Algebra}, 16 (4), 486-495.

\bibitem{ICB77} Baianu, I.C. 1977. ``A Logical Model of Genetic Activities in \L ukasiewicz Algebras: The Non-linear Theory." \emph{Bulletin of Mathematical Biology}, 39:249-258 (1977).

\bibitem{ICB87} Baianu, I.C. 1987. ``Computer Models and Automata Theory in Biology and Medicine" (A Review). In: \emph{"Mathematical Models in Medicine.}",vol.7., M. Witten, Ed., Pergamon Press: New York, pp.1513-1577.

\bibitem{BGGB2k7} Baianu, I.C., J. Glazebrook, G. Georgescu and R.Brown. 2007. ``A Novel Approach to Complex Systems Biology based on Categories, Higher Dimensional Algebra and A Generalized \L ukasiewicz Topos. ", \emph{Axiomathes},vol.17,(in press): 46 pp.

\end{thebibliography}

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