Mathematical Characterization of System-of-Systems Attributes

The practice of complex system engineering has been hampered by a lack of rigorous and consistent language across disciplines. Terms such as “complexity” and “system of systems” have been subject to various verbal definitions that are inconsistent from author to author, and from field to field. Rich literatures in philosophy, management and engineering have the promise of mutual reinforcement and support, but the lack of consistency makes correlation difficult. In this Chapter, we provide a comprehensive review of the literature around the questions of system complexity and systems of systems (SoS). Extending previous work in this area, we offer a package of four set theoretic equations expressing the essential attributes used in constructing taxonomies of system types. They are consistent with the set of attributes evolved by system engineering researchers up to this point, and together they provide a complete description of the taxonomy of systems up to but not including complex SoS. Finally, we examine the attribute of emergence. It remains an open question whether emergence can actually exist in a system made up entirely of engineered components. This has eloquently been posed as the question of emergence versus ignorance—is an unpredictable result truly a feature of the complex system, or merely an artifact of our lack of understanding? We conclude by offering several mathematical approaches that may have promise in extending this system of definitions to include emergence as well.