Parallel Real-Time Complexity Theory

We present a new complexity theoretic approach to real-time computations. We define timed ω-languages as a new formal model for such computations, that we believe to allow a unified treatment of all variants of real-time computations that are meaningful in practice. To our knowledge, such a practically meaningful formal definition does not exist at this time. In order to support our claim that timed ω-languages capture all the real-time characteristics that are important in practice, we use this formalism to model the two most important features of real-time algorithms, namely the presence of deadlines and the realtime arrival of input data. We emphasize the expressive power of our model by using it to formalize aspects from the areas of real-time database systems and ad hoc networks. We also offer a complexity theoretic characterization of parallel real-time computations. First, we define complexity classes that capture the intuitive notion of resource requirements for real-time computations in a parallel environment. Then, we show that real-time algorithms form an infinite hierarchy with respect to the number of processors used, and that all the problems solvable in nondeterministic logarithmic space (NLOGSPACE) can be solved in real time by a parallel machine, no matter how tight the real-time constraints are. As well, we show that, once real-time constraints are dropped, several other real-time problems are in effect in NLOGSPACE. Therefore, we conjecture that NLOGSPACE contains exactly all the computations that admit feasible (poly(n) processors) real-time parallel implementations.