(a) What difficulties would arise if we allowed the empty string as the left side of a production in an unrestricted grammar? Explain with the help of an example.

Introduction

Unrestricted grammar, also known as Type-0 grammar, is the most general form of grammar in the Chomsky hierarchy. It allows any string to be transformed into any other string, given that the length of the output string is not less than the input string. In the context of unrestricted grammar, the empty string, denoted by ε, is a special string with no characters.

Difficulties with Empty String as the Left Side of a Production

Allowing the empty string as the left side of a production in an unrestricted grammar can lead to several difficulties:

1. Non-termination and Infinite Loops: If ε is allowed on the left side of a production, it means that we can replace ε with any string on the right side of the production, even when ε is not present in the string. This can lead to non-termination or infinite loops in the derivation process. For instance, if we have a production rule ε → AB, we can keep applying this rule indefinitely, leading to an infinite loop.

2. Ambiguity and Non-determinism: If ε is allowed on the left side of a production, it can lead to ambiguity in the grammar. Ambiguity in a grammar means that there can be more than one leftmost derivation for a single string in the language. This can lead to non-determinism, where it is not clear which production rules to apply to derive a string.

Mathematical Formulas and Reasoning

Let's denote a production rule with ε on the left side as ε → α, where α is any string of terminals and non-terminals.

1. Non-termination and Infinite Loops: Since ε is always a substring of any string, we can always apply the production rule ε → α, regardless of the current string. This means that we can keep applying this rule indefinitely, leading to an infinite loop.

2. Ambiguity and Non-determinism: If we have two production rules ε → α and ε → β, and both α and β can derive a string γ, then there is ambiguity in the grammar. This is because there are two different sequences of production rules that can derive γ.

Example

Let's consider an unrestricted grammar G with the following production rules:

1. ε → AB
2. A → a
3. B → b

We can derive the string "ab" in the following way:

1. ε → AB (using rule 1)
2. AB → aB (using rule 2)
3. aB → ab (using rule 3)

However, we can keep applying rule 1 indefinitely, leading to an infinite loop:

1. ε → AB (using rule 1)
2. AB → AAB (using rule 1)
3. AAB → AAAB (using rule 1)
4. ...

Technical Properties

The example grammar G is an unrestricted grammar, as it allows any string to be transformed into any other string. The language generated by G is {ab}, which consists of a single string "ab". The derivations in G are non-terminating due to the presence of the production rule ε → AB.

Conclusion

Allowing the empty string as the left side of a production in an unrestricted grammar can lead to non-termination, infinite loops, ambiguity, and non-determinism. These difficulties can make the grammar and its language harder to analyze and understand. To avoid these difficulties, it is generally recommended not to allow ε on the left side of a production in an unrestricted grammar.

Summary

Allowing the empty string as the left side of a production in an unrestricted grammar can lead to difficulties such as non-termination, infinite loops, ambiguity, and non-determinism. This can make the grammar and its language harder to analyze and understand. It is generally recommended not to allow ε on the left side of a production in an unrestricted grammar.

Analogy

Imagine a production line where the left side is an empty conveyor belt. If we allow the empty string as the left side of a production in an unrestricted grammar, it's like allowing the conveyor belt to transport any item, even when there is nothing on it. This can cause the production line to run indefinitely, leading to confusion and inefficiency.