On read-once threshold formulae and their randomized decision tree complexity

TC⁰ is the class functions computable by polynomial-size, constant-depth formulae with threshold gates. Read-once TC⁰ (RO-TC⁰) is the subclass of TC⁰ which restricts every variable to occur exactly once in the formula. Our main result is a (tight) linear lower bound on the randomized decision tree complexity of any function in RO-TC⁰. This relationship between threshold circuits and decision trees bears significance on both models of computation. Regarding decision trees, this is the first class of functions for which such a strong bound is known. Regarding threshold circuits, it may be considered as a possible first step towards proving TC⁰ ≠ NC¹; generalizing our lower bound to all functions in TC⁰ would establish this separation. Another structural result we obtain is that a read-once threshold formula uniquely represents the function it computes.