We consider nondeterministic finite automata (NFA) and nondeterministic Buchi automata (NBA), which describe regular languages and omega-regular languages, respectively. Central problems for these are computationally hard (PSPACE-complete), e.g., checking language inclusion, equivalence and universality, as well as finding automata of minimal size for a given language.
In spite of the high worst-case complexity, recent algorithms and software-tools can solve many instances of nontrivial size. Here we give an overview over these techniques. They include antichain techniques exploiting logical subsumption, the construction of congruence bases, and automata minimization methods based on transition pruning and state-space quotienting w.r.t. generalized simulation preorders. In particular, multipebble simulations and the more practical lookahead simulations can in polynomial time compute very good under-approximations of the PSPACE-complete relations of (trace-)language inclusion. A collection of current papers and tools is available at www.languageinclusion.org.