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] 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]

References

1. Sipser, Michael (2012). Introduction to the Theory of Computation. Cengage Learning. ISBN 9781285401065.
2. "Lecture 6: Properties of Regular Languages - II". COMS W3261 CS Theory. Department of Computer Science, Columbia University. Retrieved 2019-08-22.
3. Hopcroft, J. E.; Ullman, J. D (1979). Introduction to Automata Theory, Languages, and Computation (first ed.). Addison-Wesley. ISBN 81-7808-347-7.