Ãlukasiewicz - Moisil Many - Valued Logic Algebra of Highly - Complex Systems

The fundamentals of à Lukasiewicz-Moisil logic algebras and their applications to complex genetic network dynamics and highly complex systems are presented in the context of a categorical ontology theory of levels, Medical Bioinformatics and self-organizing, highly complex systems. Quantum Automata were defined in refs.[2] and [3] as generalized, probabilistic automata with quantum state spaces [1]. 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 biosystem 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– à Lukasiewicz algebras has a subcategory of centered n– à Lukasiewicz algebras (as proven in ref. [2]) which can be employed to design and construct subcategories of quantum automata based on n–à Lukasiewicz diagrams of existing VLSI. Furthermore, as shown in ref.[2] the category of