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Decidability Table in Theory of Computation
  • Difficulty Level : Hard
  • Last Updated : 28 Oct, 2020

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.

 



ProblemsRLDCFLCFLCSLRec.LREL
Does ‘w’ belongs to language L? (i.e, membership problem, where ‘w’ is any string)DDDDDUD
Is L= null? (i.e, emptiness problem)DDDUDUDUD
Is L= E* ? (i.e, completeness problem.where, E* is set of all languages possible over given alphabet)DDUDUDUDUD
Is L1= L2 ? (i.e, equality problem. L1 and L2 are languages of same type.)DDUDUDUDUD
Is L1 subset of L2 ? (i.e, subset problem)DUDUDUDUDUD
Is L1 intersection of L2= null?DUDUDUDUDUD
Is ‘L’ finite or not? (i.e, finiteness problem)DDDUDUDUD
Is compliment of ‘L’ a language of same type or not?DDUDDDUD
Is intersection of two languages of same type or not?DUDUDDDD
Is ‘L’ regular language or not? (‘L’ is any language.)DDUDUDUDUD

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: 
 

  1. Regular language: It Decidable for all problems.
  2. CFL: It is decidable for emptiness problem, finiteness problem, and membership problem.
  3. 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?.
  4. REC: It is decidable for (Is intersection of two languages of same type or not?)
  5. 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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