Creating Grammar for specific strings

Question

Develop a grammar for the language that consists of all strings formed from the characters a and b that have the same number of a as b. For example, the strings aabb, baba, and aabbababba are part of this language, but not the strings abb, bba, and aabbabb. Explain why your grammar "does the job". If possible, is there a way without epsilon?

Answer 1

The grammar provided indeed does the job of generating all strings formed from the characters 'a' and 'b' that have the same number of 'a's as 'b's.

S --> ε

S --> aSbS

S --> bSaS

S --> aS

S --> bS

S --> a

S --> b

here is the explanation for the above

1. S is the start symbol of the grammar.
2. The rules S --> aSbS and S --> bSaS ensure that for each 'a' generated, there is a corresponding 'b' generated, maintaining the same number of 'a's and 'b's.
3. The rules S --> aS and S --> bS allow for the addition of single 'a's or 'b's, respectively, to maintain the balance.
4. The rules S --> a and S --> b generate single 'a's and 'b's, respectively.
5. The rule S --> ε allows for the generation of an empty string, which is valid for this language as it has an equal number of 'a's and 'b's (0 of each).

This grammar covers all possible cases of strings with equal numbers of 'a's and 'b's, including those with more complex arrangements such as aSbS, where 'a's and 'b's are interspersed.

Regarding your question about whether it's possible to construct the grammar without using epsilon productions:

It's indeed possible to construct a grammar without epsilon productions, but it would require more complex rules and might not be as intuitive. The grammar you provided is clear and concise, making it easy to understand and use. Therefore, while it's possible to construct such a grammar without epsilon productions, it might not be as straightforward or easy to interpret. The grammar you've provided strikes a good balance between simplicity and completeness for generating strings with equal numbers of 'a's and 'b's.