---
title: "Emptiness problem"
chunk: 1/1
source: "https://en.wikipedia.org/wiki/Emptiness_problem"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T11:33:06.962515+00:00"
instance: "kb-cron"
---

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.


== See also ==
Intersection non-emptiness problem


== References ==