Title: Normalizing resistor circuits
Abstract: A classical Electrical Engineering problem is to determine whether two networks of resistors are equivalent. The standard solution is to use a process of eliminating internal nodes (the star/mesh transformation) which may be seen as a rewriting procedure. In the EE literature this is organized into a matrix problem which can be solved by a Gaussian elimination procedure. This approach essentially works for resistances taken from the positive reals while secretly using negatives. However, it cannot handle negative resistances. Negative resistances, while counter-intuitive, arise in stabilizer quantum mechanics where networks of “resistors” over finite fields occur. The rewriting approach as compared to the matrix approach is more general and works for any positive division rig. We shall discuss how the rewriting theory can be modified to handle negatives.
In this talk we shall start by introducing categories of resistors. These are “hypergraph categories” whose maps are generated by resistors. Resistors are then self-dual maps satisfying certain formal properties (including an infinite family of star-mesh equalities). The problem of network equivalence is, thus, the decision problem for maps in these categories. The fact that there is a terminating and confluent rewriting system not only solves the decision problem (up to certain equalities) but also has the effect of guaranteeing, very basically, the associativity of the composition.
Work with: Robin Cockett and Amolak Kalra
Abstract courtesy: Robin (at FMCS ’23)
