Logspace Hierarchies, Polynomial Time and the Complexity of Fairness Problems Concerning $\omega $-Machines

Abstract

In this paper, we define a restricted logspace oracle hierarchy which turns out to be equivalent to the logspace alternation hierarchy (of Chandra, Kozen and Stockmeyer) and thus is contained within the second level of the logspace oracle heirarchy (of Ruzzo, Simon and Tompa). We then examine problems concerning various types of “fair” computations with respect to $\omega $-Finite State Machines ($\omega $-FSM’s) and $\omega $-One Counter Machines ($\omega $-1CM’s). For example, we consider the nonemptiness problem for $\omega $-FSM’s and $\omega $-1CM’s where acceptance is defined in the usual fashion, but with a fairness constraint imposed on accepting computations. Our results yield problems that are complete not only for LOGSPACE and PTIME but the second and third levels of the restricted logspace oracle hierarchy as well. As far as we know, these are the first natural problems shown to be complete for various levels of the logspace alternation hierarchy. The problems are also of independent interest. In fact, the nonemptiness problem (with fairness constraints) for w-machines has been shown to have immediate applications to the verification of concurrent finite state programs. Furthermore, the results can be used to strengthen known results concerning some related fairness problems that involve temporal logic (e.g. model checking).