We investigate the complexity of several predicates on regular sets. In particular, we show: 1) the equivalence and emptiness problem for regular expressions using only the operators -, $\cup$, ., and $\cap$ are p-complete. 2) the emptiness problem for regular expressions using the operators -, $\cup$, ., $\cap$ and * is tape-hard; 3) the emptiness problem for regular expressions using the operators -, $\cup$, ., $\cap$ and 2 is tape-hard; 4) the equivalence problem for regular expressions using the operators -, $\cup$, ., $\cap$ and * is not polynomial in tape; and 5) the equivalence problem for regular expressions using the operators -, $\cup$, ., $\cap$ and 2 requires exponential time.