Empirical logic of finite automata: microstatements versus macrostatements

We compare the two approaches to the empirical logic of automata. The first, called partition logic (logic of microstatements), refers to experiments on individual automata. The second one, the logic of simulation (logic of macrostatements), deals with ensembles of automata.Comment: late

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

A novel approach to self-organizing, highly-complex systems (HCS), such as living organisms and artificial intelligent systems (AIs), is presented which is relevant to Cognition, Medical Bioinformatics and Computational Neuroscience. Quantum Automata (QAs) were defined in our previous work as generalized, probabilistic automata with quantum state spaces (Baianu, 1971). Their next-state functions operate through transitions between quantum states defined by the quantum equations of motion in the Schroedinger representation, with both initial and boundary conditions in space-time. Such quantum automata operate with a quantum logic, or Q-logic, significantly different from either Boolean or Łukasiewicz 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 with defined biological relations that represent physiological functions of primordial organisms, single cells and higher organisms

Finite automata models of quantized systems: conceptual status and outlook

Since Edward Moore, finite automata theory has been inspired by physics, in particular by quantum complementarity. We review automaton complementarity, reversible automata and the connections to generalized urn models. Recent developments in quantum information theory may have appropriate formalizations in the automaton context.Comment: 12 pages, prepared for the Sixth International Conference on Developments in Language Theory, Kyoto, Japan, September 18-21, 200

Logical equivalence between generalized urn models and finite automata

To every generalized urn model there exists a finite (Mealy) automaton with identical propositional calculus. The converse is true as well.Comment: 9 pages, minor change

Extending Kolmogorov's axioms for a generalized probability theory on collections of contexts

Kolmogorov's axioms of probability theory are extended to conditional probabilities among distinct (and sometimes intertwining) contexts. Formally, this amounts to row stochastic matrices whose entries characterize the conditional probability to find some observable (postselection) in one context, given an observable (preselection) in another context. As the respective probabilities need not (but, depending on the physical/model realization, can) be of the Born rule type, this generalizes approaches to quantum probabilities by Auff\'eves and Grangier, which in turn are inspired by Gleason's theorem.Comment: 18 pages, 3 figures, greatly revise

Generalized event structures and probabilities

For the classical mind, quantum mechanics is boggling enough; nevertheless more bizarre behavior could be imagined, thereby concentrating on propositional structures (empirical logics) that transcend the quantum domain. One can also consistently suppose predictions and probabilities which are neither classical nor quantum, but which are subject to subclassicality; that is, the additivity of probabilities for mutually exclusive, co-measurable observables, as formalized by admissibility rules and frame functions.Comment: 10 pages, 10 figures in Information and Complexity, World Scientific Series in Information Studies: Volume 6, ed. by Mark Burgin and Cristian S Calude, Chapter 11, pp. 276-300, (World Scientific, Singapore, 2016