Computation is employed to incredible success in a massive variety of applications, and yet it is difficult to formally state what our computations are. Finding a way to model computations is not only valuable to understanding them, but central to automatic manipulations and formal verification. Often the most interesting computations are not functions with inputs and outputs, but ongoing systems that continuously react to user input. In the automata-theoretic approach, computations are modeled as words, a sequence of letters representing a trace of a computation. Each automaton accepts a set of words, called its language. To model reactive computation, we use Büchi automata: automata that operate over infinite words. Although the computations we are modeling are not infinite, they are unbounded, and we are interested in their ongoing properties. For thirty years, Büchi automata have been recognized as the right model for reactive computations.

In order to formally verify computations, however, we must also be able to create specifications that embody the properties we want to prove these systems possess. To date, challenging algorithmic problems have prevented Büchi automata from being used as specifications. I address two challenges to the use of Buechi automata as specifications in formal verification. The first, complementation, is required to check program adherence to a specification. The second, determination, is used in domains such as synthesis, probabilistic verification, and module checking. I present both empirical analysis of existing complementation constructions, and a new theoretical contribution that provides more deterministic complementation and a full determination construction.