Abstract
We answer two open questions by (Gruber, Holzer, Kutrib, 2009) on the state-complexity of representing sub- or superword closures of context-free grammars (CFGs): (1) We prove a (tight) upper bound of $$2^{\mathcal {O}(n)}$$ 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 $$2^{2^{\mathcal {O}(n)}}$$ following from (1). Furthermore, we prove that the inequivalence problem for NFAs representing sub- or 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.
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