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. For an automaton having n {\displaystyle n} states, this is a decision problem that can be solved in O ( n 2 ) {\displaystyle O(n^{2})} time, or in time O ( n + m ) {\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. The emptiness problem in machine learning and formal languages determines if a model or automaton generates the empty language, which is undecidable for certain alternating multi-head finite automata over single-letter alphabets. 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.

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 n {\displaystyle n} states, this is a decision problem that can be solved in O ( n 2 ) {\displaystyle O(n^{2})} time,^[2] or in time O ( n + m ) {\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 in machine learning and formal languages determines if a model or automaton generates the empty language, which is undecidable for certain alternating multi-head finite automata over single-letter alphabets.^[4]

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]

See also

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• Intersection non-emptiness problem

References

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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. Archived from the original on 2019-10-31. Retrieved 2019-08-22.
3. ^ ^{a b} Hopcroft, J. E.; Ullman, J. D (1979). Introduction to Automata Theory, Languages, and Computation (first ed.). Addison-Wesley. ISBN 81-7808-347-7.
4. ^ Geidmanis, Dainis (1991-03-01). "Unsolvability of the emptiness problem for alternating 1-way multi-head and multi-tape finite automata over single-letter alphabet". Comput. Artif. Intell. 10 (2): 133–141. ISSN 0232-0274.

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