Grant: Discovery Grants Program - Individual

Understanding and predictively manipulating an important class of models, finitary discrete-event dynamical systems, still lacks adequate predictive frameworks for knowledge users and scientists (in contrast to continuous dynamical systems where calculus, linear algebra, superposition principles and well-developed engineering computational tools are available and have been developed over the last several centuries). Groundwork for deep understanding of finite discrete-event dynamical systems exists in its basic mathematical underpinnings, but this needs to be nurtured into a paradigm of practice with an applicable body of methodological, computational and mathematical tools (1) to be widely exploited throughout Science and Engineering, and (2) to augment AI and human cognitive capabilities. Finite discrete-event dynamical systems models are ubiquitous throughout natural science and engineering: Common modelling techniques include the use of finite automata, Petri nets, reaction networks in biology and chemistry, genetic regulatory networks, Boolean and multivalued logic networks, cortical column neural models, graph-theoretically described systems with different transition types and time-scales, and dynamic networks (whose state-spaces may grow or shrink adaptively, while remaining finite). All are increasingly prevalent computational models that transform in response to external events and internal transitions. This research will create 21st century computational tools rooted in rigorous 19th and 20th century theorems from the mathematical theory of symmetry and algebraic theory of automata that guarantee it is always possible to create a global, hierarchically organized system of coordinates for a given finite (and often infinite) discrete-event dynamical system. These computational coordinate systems will allow one to 'navigate' a given discrete event system, to know the 'latitude and longitude' of a system state, and how to transform it by a sequence of steps into another desired state. A special case is what the decimal expansion does for arithmetic: one can calculate in a positional notation, following highly restricted 'carry laws' that relate change in values along the hierarchy of positions, where change of value is computed in a simple way, e.g. adding single digits, one-by-one. Such an abstract representation of global system state in a coarse-to-fine manner is possible (not just for numbers), but for any finite discrete-event system, and could facilitate easy manipulation in sample applications such as how to: -let AI automatically integrate diverse information to recommend strategies and discrete moves to achieve a desired goal -in a strategic situation / tactical game, concisely represent state of play in compact informative form for a human operator to move effectively to a desired configuration -manipulate a cancerous cell to induce it to approach to a non-malignant state by systematically 'adjusting' its state coordinates.