Finite Automata for the Sub- and Superword Closure of CFLs: Descriptional and Computational Complexity

We answer two open questions by (Gruber, Holzer, Kutrib, 2009) on the state-complexity of representing subor superword closures of context-free grammars (CFGs): (1) We prove a (tight) upper bound of 2O(n) on the size of nondeterministic finite automata (NFAs) representing the subword closure of a CFG of size n. (2) We present a family of CFGs for which the minimal deterministic finite automata representing their subword closure matches the upper-bound of 2 O(n) following from (1). Furthermore, we prove that the inequivalence problem for NFAs representing subor superword-closed languages is only NP-complete as opposed to PSPACE-complete for general NFAs. Finally, we extend our results into an approximation method to attack inequivalence problems for CFGs.