Loop Circuits and Their Relation to Razborov's Approximation Model

Recently, a new technique called the method of approximations has been developed for proving lower bounds on the size of circuits computing certain Boolean functions. To obtain a lower bound on the size complexity size(f) of a certain function f by the method, an appropriate legitimate model M for the function f is chosen, and then a lower bound on the distance ?(f, M) from f to M is derived. The lower bound on ?(f, M) becomes a lower bound on size(f) in view of the fact that size(f) ? ?(f, M). Razborov gave a legitimate monotone model, Mmon(Fmax), and showed that ?(f, Mmon(Fmax)) = ?(size1/3(f)), so there remains a gap between the size size(f) and the distance ?(f, M). Employing his method, the following statements are established: (i) Razborov?s model Mmon(Fmax) is generalized to obtain model M(Fmax), and it is established that ?(f, M(Fmax)) = ?(size12(f)). (ii) Allowing the underlying graphs of circuits to have cycles, a new notion of apparently more powerful circuits, called loop circuits, is introduced, and it is proved that ?(f, M(Fmax)) = ?(sizeloop(f)), where sizeloop(f) denotes the size complexity of f based on loop circuits.