Decision Problems of CFG

In this chapter, we will cover some interesting decision problems related to Context-Free Grammars (CFGs). We will see whether these problems are solvable in CFG or not. The answer will be Yes or No. In other words, they help us understand what kind of questions we can answer about CFGs using algorithms. Let us see the four main problems for a better understanding of the concept.

Problem 1: String Generation Problem

The problem states, imagine we have a CFG and a string. We want to check if the grammar can produce that exact string or not?

Solution

• First, we change the grammar into a special form called Chomsky Normal Form. This makes things easier to work with.
• Then, we check the string length. If the length is say n.
• We create all possible ways to generate strings using the grammar, but we only use 2n-1 steps. Because in Chomsky Normal Form, that's the maximum number of steps needed to create a string of length n.
• Finally, we check whether any of these ways produce our string. If yes, the answer will be positive.

This method always gives us an answer like yes or no, so we call it "decidable."

Problem 2: Emptiness Problem

The Problem states, consider we have a CFG, and we want to check if it can generate any strings at all? Or the string is empty of not

Solution

Here is a simple way to solve it out −

• We start by marking all the terminal symbols in the grammar. These are like the building blocks.
• Then, we look at each variable (non-terminal) in the grammar. If it can produce only marked symbols, we mark it too.
• We keep doing this until we can't mark any more variables.
• At the end, we check if the start symbol (the main variable) is marked. If it's not marked, the language is empty!

This method also always gives us a clear answer in yes or no, so it's decidable too.

Problem 3: Decidability of Context-Free Language?

The Problem states if we have a context-free language (CFL) and a string. We want to check whether the string belongs to that language or not?

Solution

We can always answer this question. In the following steps −

• We take the grammar that generates our CFL.
• We use the method from Problem 1 to check if this grammar can generate our string.
• If it can, the string is in the language. If not, it's not in the language.

This shows that every context-free language is decidable.

Problem 4: Equivalence of Context Free Grammar

The Problem states, if we have two CFGs and we want to know whether they generate the same language or not?

Solution

We can't always answer this question. So this problem can be said as "undecidable". There's no algorithm that can always tell us if two CFGs generate the same language.

This is because context-free languages have some limitations −

• They are not closed under intersection (when you combine two CFLs, you might not get a CFL).
• They are not closed under complementation (when you flip a CFL, you might not get a CFL).

These properties make it impossible to create a general method to compare two CFLs.

Conclusion

In automata theory, we use several classes of problems. The most interesting class of problems are decidable problems. It is nothing but checking whether that can be solvable or not.

In this chapter, we highlighted a set of decision problems corresponding to contextfree grammars. There are four problems that we covered here. Among them, three are decidable and the last equivalence is not decidable.