Emptiness problem

In theoretical computer science and formal language theory, a formal language is empty if its set of valid sentences is the empty set. The emptiness problem is the question of determining whether a language is empty given some representation of it, such as a finite-state automaton.^[1] For an automaton having ${\displaystyle n}$ states, this is a decision problem that can be solved in ${\displaystyle O(n^{2})}$ time,^[2] or in time ${\displaystyle O(n+m)}$ if the automaton has n states and m transitions. However, variants of that question, such as the emptiness problem for non-erasing stack automata, are PSPACE-complete.^[3]

The emptiness problem is undecidable for context-sensitive grammars, a fact that follows from the undecidability of the halting problem. It is, however, decidable for context-free grammars.^[3]