Time and Space Complexity of Inside-Out Macro Languages

Starting from Fischer's IO Standard Form Theorem we show that for each inside-out (or IO-) macro language $L$, there is a $\lambda$-free IO-macro grammar with the following property: for each $x$ in $L$, there is a derivation of $x$ of length at most linear in the length of $x$. Then we construct a nondeterministic log-space bounded auxiliary pushdown automaton which accepts $L$ in polynomial time. Therefore the IO-macro languages are (many-one) log-space reducible to the context-free languages. Consequently, the membership problem for IO-macro languages can be solved deterministically in polynomial time and in space $(\log n)^2$.