Context-Free Language Closure Properties: Homomorphism and Reversal

This question asks about the closure properties of the family of context-free languages (CFLs). Specifically, it examines whether CFLs are closed under homomorphism and reversal.

Analyzing Statement I: Closure under Homomorphism

Statement I claims that the family of context free languages is closed under homomorphism. A homomorphism is a function that maps symbols from one alphabet to strings over another alphabet. This mapping is then extended to strings and languages.

If \(L\) is a context-free language over an alphabet \(\Sigma\), and \(h\) is a homomorphism from \(\Sigma^*\) to \(\Delta^*\), then \(h(L) = \{h(w) \mid w \in L\}\) is the language obtained by applying the homomorphism \(h\) to every string in \(L\). The statement asserts that \(h(L)\) is also a context-free language.

This property is indeed true. If \(L\) is generated by a Context-Free Grammar (CFG) \(G = (V, \Sigma, P, S)\), a CFG \(G' = (V, \Delta, P', S)\) for \(h(L)\) can be constructed. The production rules \(P'\) are derived from \(P\) by replacing each terminal symbol \(a \in \Sigma\) on the right-hand side of every production with its homomorphic image \(h(a)\). Non-terminal symbols and the structure of the grammar remain the same.

For example, if a production in \(G\) is \(A \to aBb\), and the homomorphism is \(h(a) = 01\), \(h(b) = 1\), the corresponding production in \(G'\) would be \(A \to 01B1\).

Therefore, Statement I is true.

Analyzing Statement II: Closure under Reversal

Statement II claims that the family of context free languages is closed under reversal. The reversal of a string \(w = w_1 w_2 \dots w_n\) (where \(w_i\) are symbols) is \(w^R = w_n \dots w_2 w_1\). The reversal of a language \(L\), denoted \(L^R\), is the set of reversals of all strings in \(L\): \(L^R = \{w^R \mid w \in L\}\).

The statement asserts that if \(L\) is a context-free language, then \(L^R\) is also a context-free language.

This property is also true. If \(L\) is generated by a CFG \(G = (V, \Sigma, P, S)\), a CFG \(G^R = (V, \Sigma, P^R, S)\) for \(L^R\) can be constructed. The production rules \(P^R\) are obtained by reversing the right-hand side of every production rule in \(P\). That is, if \(A \to \alpha\) is a production in \(P\), then \(A \to \alpha^R\) is a production in \(P^R\).

For example, if a production in \(G\) is \(A \to aBb\), the corresponding production in \(G^R\) would be \(A \to bBa\).

Therefore, Statement II is true.

Conclusion

Both Statement I (closure under homomorphism) and Statement II (closure under reversal) are true properties of context-free languages.

Based on the analysis:

• Statement I is true.
• Statement II is true.

Thus, both statements are true.

Revision Table: Context-Free Language Properties

Here's a quick summary of the key properties discussed:

• Homomorphism: Applying a symbol-to-string mapping to every string in a CFL results in another CFL.
• Reversal: Reversing every string in a CFL results in another CFL.

Additional Information: More on CFLs and Closure

Context-Free Languages are a fundamental class of languages in theoretical computer science, often used to describe the syntax of programming languages. Their closure properties are important for understanding their capabilities and limitations.

• Other Closure Properties: CFLs are also closed under union, concatenation, Kleene star, and intersection with regular languages.
• Non-Closure Properties: However, CFLs are notably *not* closed under intersection and complementation. This means that the intersection of two CFLs is not necessarily a CFL, and the complement of a CFL is not necessarily a CFL.
• Context-Free Grammars (CFGs): CFLs are precisely the languages that can be generated by Context-Free Grammars or recognized by Pushdown Automata. The closure properties are often proven by showing how to construct a new CFG or Pushdown Automaton for the resulting language.