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G is a given CFG, is L(G) regular? It's an undecidable problem.

But My argument is, Language is given and if i can do any of the following things then it will be regular else non-regular:

  • Creation of DFA/NFA
  • Writing Left Linear or Right Linear Grammar
  • Writing Regular Expression

Please tell me why is it undecidable?

Manu Thakur
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1 Answers1

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"Undecidable" means there is no algorithm that can decide the problem. Let's delve into what these terms mean.

An algorithm is anything that you can code into a Turing machine. Turing machines are not creative, they do not think, they cannot get lucky or learn new things to try. They are coded one way and then have to work the same on all inputs without any possibility of changing. Change its behavior and you have a new Turing machine.

To decide in this context means to correctly determine for each problem instance whether the answer is yes or no. You have to be able to say yes or no for each instance with 100% certainty in finite time; it's not enough to be able to say "yes" only, or "no" only, or even both only most of the time.

To answer your question then:

  • You are (presumably) not a Turing machine
  • Nothing stops you (or a Turing machine) for answering the problem for vast numbers of real problem instances of interest
  • It is currently unknown whether this problem is undecidable for human beings; we can only prove it's undecidable for Turing machines (and equivalent systems of computation).
  • The Church-Turing thesis conjectures that human beings' computational facilities are not in excess of those of Turing machines, but this is not proven
Patrick87
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