On the Complexity of Worst Case and Expected Time in a Circuit

The computational delay of a circuit can be described by the natural concept of time Jakoby et al. STOC94]. We show that for a given input x and circuit C the computation of timeC(x) is P-complete. Moreover, we show that it is NP-complete to decide whether there exists an input x such that timeC(x) t for a given time bound t. We introduce the notion of worst time of a circuit and show that to decide whether a given time bound is the worst time of a circuit is BH2-complete. We also prove that the computation of an arbitrary worst case input is FP NP tt-hard, whereas the search of the lexicographically minimal worst case input is FP NP-complete and of the lex. middle worst case input is FP #P-complete. Computation of the expected time E D (timeC) of a circuit C with respect to a distribution D generated by circuit D is #P-complete under metric reducibility. Nevertheless we show that a polynomial time bounded probabilistic Turing machine approximates E D (timeC) up to an arbitrary additive constant with high probability.