The Inclusion Problem for Forest Languages under Substitutions

We consider algorithms and lower bounds for various problems over forest languages; as input models we allow forest algebras, deterministic forest automata and nondeterministic forest automata. For the equivalence problem, we give an almost-linear-time algorithm for both forest algebras and deterministic forest automata; this is complemented by a polynomial time hardness result. The emptiness problem is complete for polynomial time over each of the three models. Additionally, we consider the emptiness of intersection problem for forest algebras and deterministic forest automata; this problem turns out to be complete for exponential time. It is well-known that the corresponding problems for word languages are complete for nondeterministic logarithmic space and for polynomial space, respectively. Equipped with this toolbox of algorithms and lower bounds, we consider various inclusion problems for regular forest languages under substitutions. The substitutions in this paper replace leaf variables by forest languages. Depending on the direction of the inclusion, the problem for a given substitution is either complete for polynomial time or for exponential time; in particular, the equivalence problem under substitutions is complete for exponential time and, hence, more difficult than the equivalence problem for forest languages without substitutions. If we ask whether there exists a substitution such that a given inclusion holds, then this problem is either complete for NP or exponential time, depending on whether we consider inclusion or equivalence; moreover, the problem is undecidable if the substitution is applied on both sides.