Hidden information and regularities of information dynamics II

Vladimir S. Lerner 13603 Marina Pointe Drive, Marina Del Rey, CA 90292, USA [email protected] Abstract The introduced entropy functional’s (EF) information measure of a random process integrates all information contribution along the process trajectories, evaluating both the states’ and between states’ bound information connections. By defining the integral information measure on the trajectories of the multi-dimensional Markov stochastic dynamics (as a simple mathematical model of a random nonstationary interactive process), we obtain the formal integral evaluation of the process that includes its multiple long-term correlations. This measure reveals information that is hidden by traditional information measures, which commonly use Shannon’s entropy function for each selected stationary states of the process. Such hidden information could disclose a missing connections during, for example, creation of a human thought, speech, discussion, communication, and so on, being important for evaluation the process’ meaningful information, related to these connections in acceptance, cognition, and perception, which enables producing a logic of the information. The presentation consists of three Parts. In Part1R (revised) we analyze a mechanism of arising information regularities from a stochastic process, measured by EF, independently of the process’ specific source and origin, applying them to a wide class of real material and non material processes, including an intellectual process and a world of virtual reality. Uncovering the process’ regularities leads us to an information law, based on extracting a maximal information from its minimum, which could create these regularities. To formally approach such a law, we solve a variation problem (VP) of converting a stochastic process, measured by EF with its hidden information, to a dynamic process, measured by an information path functional (IPF) on the process’ trajectories. The IPF variation equations determine the information dynamic model, approximating the stochastic process with a maximal functional probability on trajectories and the equations for a joint solution of the identification and optimal control problems, combined with a state’s consolidation. The VP information invariants allow developing a simple computer based procedure for dynamic modeling and prediction of the random diffusion process with following optimal encoding of the identified dynamic model’s operator and control. In Part 2, we study the cooperative processes, arising at the consolidation, as a result of the VP-EF-IPF approach, which is able to produce a multiple cooperative structures, concurrently assembling in a hierarchical information network (IN) and generating the IN’s digital genetic code. We analyze information geometry of the cooperative structures, evaluate a curvature of their geometrical forms and their cooperative information complexities. The dynamic model, which extracts a hidden information of an observed process, does both conversion, cooperation, and then generation of its genetic code, working as an information operating system which creates an information observer. In Part 3 we study the evolutionary information processes and regularities of evolution dynamics, evaluated by entropy functional (EF) of a random field and informational path functional (IPF) of a dynamic space-time process. The VP minimax variation principle, applied to the evolution of both random microprocesses and dynamic macroprocesses, leads to the following evolutionary regularities: creation of the order from stochastics through the evolutionary macrodynamics, described by a gradient of dynamic potential, evolutionary speed and the evolutionary conditions of a fitness