Mathematical Models of Interactive Computing

Finite computing agents that interact with an environment are shown to be more expressive than Turing machines according to a notion of expressiveness that measures problem-solving ability and is specified by observation equivalence. Sequential interactive models of objects, agents, and embedded systems are shown to be more expressive than algorithms. Multi-agent (distributed) models of coordination, collaboration, and true concurrency are shown to be more expressive than sequential models. The technology shift from algorithms to interaction is expressed by a mathematical paradigm shift that extends inductive definition and reasoning methods for finite agents to coinductive methods of set theory and algebra. An introduction to models of interactive computing is followed by an account of mathematical models of sequential interaction in terms of coinductive methods of non-well-founded set theory, coalgebras, and bisimulation. Models of distributed information flow and multi-agent interaction are developed, and the Turing test is extended to interactive sequential and distributed models of computation. Specification of interactive systems is defined in terms of observable behavior, Godel incompleteness is shown for interaction machines, and explanatory power of physical theories is shown to correspond to expressiveness for models of computation. Our goal is to provide multiple perspective on interactive modeling from the viewpoint of interaction machines and mathematics, to persuade readers that interaction paradigms can play a significant role in narrowing the gap between theoretical models and software practice.