Prerequisite – Undecidability, Decidable and undecidable problems

Identifying languages (or problems*) as decidable, undecidable or partially decidable is a very common question in GATE. With correct knowledge and ample experience, this question becomes very easy to solve.

A language is undecidable if it is not decidable. An undecidable language maybe a partially decidable language or something else but not decidable. If a language is not even partially decidable , then there exists no Turing machine for that language.

In this topic you will see Decidability table and shortcut to learn them.

Intersection of 2 recursively enumerable languages is recursively enumerable, so it is decidable.

Problems | RL | DCFL | CFL | CSL | Rec.L | REL |
---|---|---|---|---|---|---|

Does ‘w’ belongs to language L? (i.e, membership problem, where ‘w’ is any string) | D | D | D | D | D | UD |

Is L= null? (i.e, emptiness problem) | D | D | D | UD | UD | UD |

Is L= E^{*} ? (i.e, completeness problem.where, E^{*} is set of all languages possible over given alphabet) |
D | D | UD | UD | UD | UD |

Is L1= L2 ? (i.e, equality problem. L1 and L2 are languages of same type.) | D | D | UD | UD | UD | UD |

Is L1 subset of L2 ? (i.e, subset problem) | D | UD | UD | UD | UD | UD |

Is L1 intersection of L2= null? | D | UD | UD | UD | UD | UD |

Is ‘L’ finite or not? (i.e, finiteness problem) | D | D | D | UD | UD | UD |

Is compliment of ‘L’ a language of same type or not? | D | D | UD | D | D | UD |

Is intersection of two languages of same type or not? | D | UD | UD | D | D | D |

Is ‘L’ regular language or not? (‘L’ is any language.) | D | D | UD | UD | UD | UD |

In the above table,

'RL' implies Regular language. 'CFL' implies Context free language. 'DCFL' implies deterministic context free language. 'CSL' implies Context sensitive language. 'REC.L' implies Recursive language. 'REL' implies Recursive enumerable language. 'D' implies that the problem is decidable. 'UD' implies that the problem is undecidable.

**Note:**

**Regular language:**It Decidable for all problems.**CFL:**It is decidable for emptiness problem, finiteness problem, and membership problem.**CSL and REC.L:**Both are decidable for membership problem, Is compliment of ‘L’ a language of same type or not?, and (Is intersection of two languages of same type or not?.**REC:**It is decidable for (Is intersection of two languages of same type or not?)**DCFL**It is decidable for everything decidable in**CFL**plus (Is compliment of ‘L’ a language of same type or not?), (Is ‘L’ regular language?).

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