CategoriesOfQuantumAutomataAndQuantumComputers 2008-09-28 15:14:48 2008-09-28 15:17:24 Topic \\ N-- \L ukasiewicz Algebras and Quantum Computers} quantum automata categories; limits and colimits; bicomplete categories centered n--\textsl{\L}ukasiewicz algebras % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \usepackage{graphicx} \newtheorem{thm}{Theorem}[section] \newtheorem{cor}[thm]{Corollary} \numberwithin{equation}{section} \newcommand{\norm}[1]{\left\Vert#1\right\Vert} \newcommand{\abs}[1]{\left\vert#1\right\vert} \newcommand{\set}[1]{\left\{#1\right\}} \newcommand{\Real}{\mathbb R} \newcommand{\eps}{\varepsilon} \newcommand{\To}{\longrightarrow} \newcommand{\BX}{\mathbf{B}(X)} \newcommand{\A}{\mathcal{A}} \section{Categories of Quantum Automata, N- \L ukasiewicz Algebras and Quantum Computers} Quantum Automata were defined (in ref.\cite{IB71}) as generalized, probabilistic automata with quantum state spaces. Their next-state functions operate through transitions between quantum states defined by the quantum equations of motions in the Schr\"{o}dinger representation, with both initial and boundary conditions in space-time. A new theorem is proven which states that the \emph{category of quantum automata and automata--homomorphisms has both limits and colimits.} Therefore, both categories of quantum automata and classical automata (sequential machines) are \emph{bicomplete.} A second new theorem establishes that the standard automata category 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, dynamic bio-networks (\cite{ICB87}) with defined biological relations 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 topological groupoids that admit a local characterization through unique 'quantum' \emph{Lie algebroids}. 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 diagrams 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 complexity 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}