LUKASIEWICZ-MOISIL MANY-VALUED LOGIC ALGEBRA OF HIGHLY-COMPLEX SYSTEMS

The fundamentals of ÃLukasiewicz-Moisil logic algebras and their applica-tions to complex genetic network dynamics and highly complex systems arepresented in the context of a categorical ontology theory of levels, MedicalBioinformatics and self-organizing, highly complex systems. Quantum Au-tomata were de¯ned in refs.[2] and [3] as generalized, probabilistic automatawith quantum state spaces [1]. Their next-state functions operate throughtransitions between quantum states de¯ned by the quantum equations of mo-tions in the SchrÄodinger representation, with both initial and boundary condi-tions in space-time. A new theorem is proven which states that thecategory ofquantum automata and automata{homomorphisms has both limits and colim-its. Therefore, both categories of quantum automata and classical automata(sequential machines) arebicomplete. A second new theorem establishes thatthe standard automata category is a subcategory of the quantum automatacategory. The quantum automata category has a faithful representation in thecategory of Generalized (M,R){Systems which are open, dynamic biosystemnetworks [4] with de¯ned biological relations that represent physiological func-tions of primordial(s), single cells and the simpler organisms. A newcategoryof quantum computersis also de¯ned in terms of reversible quantum automatawith quantum state spaces represented by topological groupoids that admit alocal characterization through unique, quantum Lie algebroids. On the otherhand, the category of n{ÃL ukasiewicz algebras has a subcategory ofcenteredn{ÃL ukasiewicz algebras (as proven in ref. [2]) which can be employed to designand construct subcategories of quantum automata based on n{ÃLukasiewicz diagrams of existing VLSI. Furthermore, as shown in ref.[2] the category of centered n{ÃL ukasiewicz algebras and the category of Boolean algebras are nat-urally equivalent. A `no-goʹ conjecture is also proposed which states that Gen-eralized (M,R){Systems complexity prevents their complete computability (asshown in refs. [5]{[6]) by either standard, or quantum, automata.