Upper bounds for integrated information

Originally developed as a theory of consciousness, integrated information theory provides a mathematical framework to quantify the causal irreducibility of systems and subsets of units in the system. Specifically, mechanism integrated information quantifies how much of the causal powers of a subset of units in a state, also referred to as a mechanism, cannot be accounted for by its parts. If the causal powers of the mechanism can be fully explained by its parts, it is reducible and its integrated information is zero. Here, we study the upper bound of this measure and how it is achieved. We study mechanisms in isolation, groups of mechanisms, and groups of causal relations among mechanisms. We put forward new theoretical results that show mechanisms that share parts with each other cannot all achieve their maximum. We also introduce techniques to design systems that can maximize the integrated information of a subset of their mechanisms or relations. Our results can potentially be used to exploit the symmetries and constraints to reduce the computations significantly and to compare different connectivity profiles in terms of their maximal achievable integrated information.Author summary: Integrated Information Theory (IIT) offers a theoretical framework to quantify the causal irreducibilty of a system, subsets of the units in a system, and the causal relations among the subsets. For example, mechanism integrated information quantifies how much of the causal powers of a subset of units in a state cannot be accounted for by its parts. Here, we provide theoretical results on the upper bounds for this measure, how it is achieved, and why mechanisms with overlapping parts cannot all be maximally integrated. We also study the upper bounds for integrated information of causal relations among the mechanisms. The ideas introduced here can potentially pave the way to design systems with optimal causal irreducibility and to develop computationally lightweight exact or approximate measures for integrated information.