What are the undecidable problems in TOC?

The problems for which we can’t construct an algorithm that can answer the problem correctly in the infinite time are termed as Undecidable Problems in the theory of computation (TOC).

A problem is undecidable if there is no Turing machine that will always halt an infinite amount of time to answer as ‘yes’ or ‘no’.

Examples

The examples of undecidable problems are explained below. Here, CFG refers to Context Free Grammar.

• Whether two CFG L and M equal − Since, we cannot determine all the strings of any CFG, we can predict that two CFG are equal or not.

• Given a context-free language, there is no Turing machine (TM) that will always halt an infinite amount of time and give an answer to whether language is ambiguous or not.

• Given two context-free languages, there is no Turing machine that will always halt an infinite amount of time and give an answer whether two context-free languages are equal or not.

• Whether CFG will generate all possible strings of the input alphabet (∑*) is undecidable.

Halting Problem

The Halting problem is the most famous of the undecidable problems.

Consider the code

num=1;
while(num=0)
{
   num=num+1;
}

It counts up forever since it will never equal 0. This is an example of the halting problem.

Note: Every context-free language is decidable.

Some of the other undecidable problems are:

Totality problem − It decide whether an arbitrary TM halts on all inputs. This is equivalent to the problem of whether a program can ever enter an infinite loop, for any input. It differs from the halting problem, which asks whether it enters an infinite loop for a particular input.

Equivalence problem − It decide whether two TMs accept the same language. This is equivalent to the problem of whether two programs compute the same output for every input.