The read once formula of a series-parallel network

For any two-terminal resistive series-parallel network N with n resistors, let R: [0, infinity)](n) -> [0, infinity] denote the resistance function of N. Let R vertical bar{0, infinity}(n) be the {0, infinity}-valued function obtained from R by only allowing the resistors in N to be open/short-circuited. R vertical bar{0, infinity}(n) is a boolean function that can be coded by a boolean formula without repeated variables and no negation symbols-for short, a positive read-once formula. Let both networks N-1 and N-2 have n resistors with resistances r(1,j )= r(2,j) is an element of [0, infinity] for each j = 1, ..., n. We prove that if R-1 vertical bar{0, infinity}(n) = R-2 vertical bar{0, infinity}(n) then R-1 = R-2. We extend this result to the Wheatstone bridge. (C) 2022 Elsevier B.V. All rights reserved.