The fundamentals of Lukasiewicz-Moisil logic algebras [1],[15]--[16], [31],
[35], [37]--[39] are presented in the context of their applications to complex genetic network dynamics, highly complex systems, quantum automata [2]--[3] and quantum supercomputers. Our novel approach to the Categorical Ontology Theory of Levels impacts on Medical Bioinformatics and self-organizing, Highly Complex Systems (HCS), such as living organisms and artificial intelligent systems (AIs). Quantum Automata (QAs) were defined in refs.[2] and [3] as generalized, probabilistic automata mwith quantum state spaces [24]. Their next--state functions operate through transitions between quantum states defined by the quantum equations of motion in the Schrodinger representation, with both initial and boundary conditions in spacetime. Such quantum automata operate with a quantum logic, or Q--logic, significantly different from either Boolean or Lukasiewicz many--valued logic. A new theorem is proposed 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 biosystem networks [4]--[5] with defined biological relations that represent physiological functions of primordial(s), single cells and higher organisms [4]--[5],[8]. A new category of quantum computers is here 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--Lukasiewicz algebras has a subcategory of centered
n--Lukasiewicz algebras (as proven in ref.[15]) which can be employed to design and construct subcategories,of quantum automata based on
n--Lukasiewicz diagrams of existing VLSI. Furthermore, as shown in ref.[16] the category of
centered n--Lukasiewicz algebras and the category of Boolean algebras are naturally
equivalent. A `no--go' conjecture is also proposed here which states that Generalized (M,R)--Systems complexity prevents their complete computability (as defined in refs.[4]--[5]) by either standard, or quantum, automata [7]--[9]. Last-but-not-least, a homotopy category and Homotopy Theory are suggested for generalised, dynamic realisations of extended (M,R)--systems endowed with a topological structure, leading to a Higher Dimensional Algebra
of generalised (M,R)-systems and their dynamic
realisations.
