Pixel Matrices and Other Compositional Analyses of Interconnected Systems
Abstract
The new world of interconnected-everything brings new challenges to those who wish to understand it and keep society safe from unintended and unimagined consequences. With constant communication and feedback loops being the norm, the space of behaviors is too large to analyze by simulation alone. Today's nearly unlimited computational power must be used more wisely, so that our knowledge of a system can evolve along with the system itself. New mathematical techniques are needed to provide the algebraic formulas for \emph{combining our insights}, just as we combine components, allowing us to anticipate the behavior of an assembled system. Category theory is the mathematics of combination and compositionality, so it is well-suited as a foundation for such work. We propose to investigate compositional techniques for analyzing systems of all sorts. At the mathematical center of many disciplines, one needs to solve a system of simultaneous equations. As mundane, abstract, and worked-over as this may seem, a new elementary technique was recently discovered with the potential to change how we approach such problems. This technique is highly compositional---the solutions to subsystems can be combined to form a solution of the whole---and it emerged out of a similarly compositional approach to understanding the behavior of networked machines. Just as circuits can be combined to form computers, machines of all scales can be interconnected to form more complex machines. The common theme is compositionality: whether combining the constraints and requirements necessary to design a robot, or combining the equations that describe its function, our goal is to find analyses that are scalable and reusable, so that the knowledge we gain today is efficiently utilized in the networks of tomorrow.
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 11, 2024
- Accession Number
- AD1230383
Entities
People
- David Spivak
Organizations
- Massachusetts Institute of Technology