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

TC⁰ is the class of 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 exactly one occurrence in the formula. The 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 toward proving TC⁰≠NC¹; generalizing the lower bound to all functions in TC⁰ will establish this separation. Another structural result is that a read-once threshold formula uniquely represents the function it computes