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.