Abstracts.2018.LinEq

The logic of linear equations, electrical resistance, and statistical noise

by Joel Sjögren

The starting point of this talk is an explicit formula for the elimination of several variables at once in a linear equation. Recalling a correspondence between variable elimination, pure local variables and existential quantification, we will seek to use this formula as the core algorithm of a logic whose propositions are either networks of resistors or collections of statistical information. How many logical operations beyond "there exists" can we define, what laws do they satisfy, and what are their domain-specific interpretations? As an application, I will present the linear Kalman filter as a simple logical formula.

References:

• https://en.wikipedia.org/wiki/Schur_complement
• Tensor Analysis for Networks, by Gabriel Kron, 1939, scanned at https://babel.hathitrust.org/cgi/pt?id=mdp.39015017568331;view=1up
• A New Approach to Linear Filtering and Prediction Problems, by Rudolf Kálmán, 1960, transcribed at https://www.cs.unc.edu/~welch/kalman/media/pdf/Kalman1960.pdf