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Given languages L₁ L ₂⊆Σ ∗ , define

L₁ ⋄L₂={xy:x∈L 1,y∈L 2and ∣x∣=∣y∣}.

Show that if L₁ and L₂ are regular languages, then L₁ ⋄L₂is context-free.

Answer :

Final answer:

The student question refers to formal language theory regarding regular and context-free languages. A language generated from two regular languages with matched string lengths (L₁ ⋄L₂) can be demonstrated to be a context-free language using a non-deterministic pushdown automaton.

Explanation:

The given question pertains to formal language theory, dealing with regular languages and context-free languages. L₁ and L₂ are regular languages that are subsets of Σ*. The operation ⋄ generates a new language, where representation is a string xy such that x is a part of L₁ and y of L₂, with the additional requirement that the length of x matches that of y.

Regular languages are recognized by finite automata, which is a type of computational model with limited memory. Conversely, context-free languages are recognized by pushdown automata, which have more memory in the form of a stack. This additional memory allows context-free languages to recognize more complex patterns such as balanced parentheses.

To prove that L₁ ⋄L₂ is a context-free language, consider a non-deterministic pushdown automaton (NPDA). As each string is read in parallel from L₁ and L₂, we push a symbol onto the stack for each symbol read from L₁ and concurrently pop a symbol from the stack for each symbol read from L₂. If, at the end of the reading, the stack is empty (indicating that we have read equal lengths of symbols from L₁ and L₂), we enter an accept state. Thus, we can construct an NPDA that accepts L₁ ⋄L₂, demonstrating that L₁ ⋄L₂ is indeed a context-free language.

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