Growth Processes on Formulas and Reversible Circuits

Among their many uses, growth processes have been used for constructing reliable networks from unreliable components (Moore and Shannon, 1956) and deriving complexity bounds of various families of functions (Valiant, 1984). Hence, analyzing such processes is an important and challenging problem. In this thesis we parameterize a growth process by its initial conditions and characterize it by the existence and shape of a limiting probability distribution that describes the likelihood of it realizing a particular Boolean function. We identify the limiting distributions of several classes of growth processes on formulas, and derive conditions under which results such as Valiant’s hold. We consider growth processes that use linear, self-dual, and monotone connectives, completely characterizing those processes that use linear or monotone connectives. In the latter case, we derive a novel technique that combines spectral analysis (Savický, 1990) with probabilistic arguments; the technique is also applicable to growth processes that use other connectives. Our characterizations also yields bounds on the convergence of these growth processes. Unfortunately, a comparable definition and characterization of growth processes on general circuits is impractical due to the dependencies between the probabilities associated with various circuit components. However, reversible circuits (Toffoli, 1980) are inherently more structured than general circuits. To study growth processes beyond the formula setting, we propose and characterize growth processes on reversible circuits. Intriguingly, aspects of the characterizations that proved difficult in the former setting—such as proving that the limiting distribution is uniform—turn out to be relatively easy in the latter. In fact, the limiting distribution of a growth process on reversible circuits is characterized completely by its support—the set of functions that the process can generate. Consequently, we also provide bounds on the convergence of these growth processes. Finally, the regular structure of reversible circuits provides ample motivation for considering the reversible circuit complexity of finite Boolean functions—an important issue, since the precursor to such applications as quantum computation is reversible computation. We derive relationships between reversible circuits and other models of computation such as permutation branching programs (Barrington, 1985), based on a new measure that we call bandwidth. By leveraging these relationships, we exhibit a natural gap between two families of reversible circuits that correspond to width-4 and width-5 permutation branching programs. Based on the same measure, we define a hierarchy of families of reversible circuits that corresponds to the SC class hierarchy—a natural circuit-based definition of the class SC. Lastly, we provide constructions for several common Boolean functions and derive sufficient conditions under which a Boolean function has a polynomial-size realization.