Categories of Quantum Automata,
N– Łukasiewicz Algebras and Quantum Computers

Quantum automata were defined (in ref.[1]) 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 Schrödinger representation, with both initial and boundary conditions in space-time. A new theorem is proven which states that the category of quantum automata and automata–homomorphisms has both limits and colimits. Therefore, both categories of quantum automata and classical automata (sequential machines) are 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 (M,R)–Systems which are open, dynamic bio-networks ([4]) with defined biological relations that represent physiological functions of primordial(s), single cells and the simpler organisms. A new category of quantum computers is also defined in terms of reversible quantum automata with quantum state spaces represented by topological groupoids that admit a local characterization through unique 'quantum' Lie algebroids. On the other hand, the category of n– Łukasiewicz algebras has a subcategory of centered n– Łukasiewicz algebras (ref. [2]) which can be employed to design and construct subcategories of quantum automata based on n–Łukasiewicz diagrams of existing VLSI. Furthermore, as shown in ref.([2]) the category of centered n–Łukasiewicz algebras and the category of Boolean algebras are naturally equivalent. A `no-go' conjecture is also proposed which states that Generalized (M,R)–Systems complexity prevents their complete computability ([4,5]) by either standard or quantum automata.

Bibliography

1

Baianu, I.1971.“Organismic Supercategories and Qualitative Dynamics of Systems." Bull. Math.Biophysics., 33, 339-353.

2

Georgescu, G. and C. Vraciu 1970. “On the Characterization of Łukasiewicz Algebras." J. Algebra, 16 (4), 486-495.

3

Baianu, I.C. 1977. “A Logical Model of Genetic Activities in Łukasiewicz Algebras: The Non-linear Theory." Bulletin of Mathematical Biology, 39:249-258 (1977).

4

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

5

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 Łukasiewicz Topos. " , Axiomathes,vol.17,(in press): 46 pp.