Can the intersection of two context-free languages be a context-free language? Provide an example to support your answer.
The intersection of two context-free languages can indeed be a context-free language. To understand why, we need to consider the properties of context-free languages and their intersection.
A context-free language is a language that can be generated by a context-free grammar. A context-free grammar consists of a set of production rules that define how to generate strings in the language. These rules typically have a single non-terminal symbol on the left-hand side and a sequence of terminals and non-terminals on the right-hand side. The language generated by a context-free grammar is the set of all strings that can be derived from the start symbol using these production rules.
The intersection of two languages is the set of strings that belong to both languages. In the case of context-free languages, the intersection can be a context-free language if certain conditions are met.
One condition is that the two context-free grammars generating the languages must have disjoint sets of non-terminal symbols. This means that the non-terminal symbols used in one grammar should not appear in the other grammar. If this condition is satisfied, we can construct a new context-free grammar for the intersection language by combining the production rules of the two grammars.
Let's consider an example to illustrate this. Suppose we have two context-free languages L1 and L2, generated by the context-free grammars G1 and G2, respectively. The non-terminal symbols in G1 are A, B, and C, while G2 uses non-terminal symbols X, Y, and Z. If we want to find the intersection of L1 and L2, we can create a new context-free grammar G3 with non-terminal symbols A', B', C', X', Y', and Z', where the primes denote the non-terminal symbols in G3. We can then define the production rules for G3 by combining the production rules of G1 and G2, while replacing the non-terminal symbols with their primed counterparts.
By constructing G3 in this way, we can generate a new context-free language that is the intersection of L1 and L2. This demonstrates that the intersection of two context-free languages can indeed be a context-free language.
The intersection of two context-free languages can be a context-free language if the two context-free grammars generating the languages have disjoint sets of non-terminal symbols. By constructing a new context-free grammar that combines the production rules of the original grammars, we can generate the intersection language.
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