Answer :
To prove that l1 l2 is context-free, we can construct a context-free grammar (CFG) that generates the language, Let G1 be a CFG for l1 and G2 be a CFG for l2. We can then construct a new CFG G for l1 l2 as follows:
S -> AB, A -> x, B -> y.
where x is any string in l1 of length n, y is any string in l2 of length n, and n is a non-negative integer, This CFG generates strings of the form xy where x is in l1 and y is in l2, and |x| = |y|. Since l1 and l2 are regular languages, they can be recognized by finite automata, which in turn can be converted into a CFG. Therefore, G1 and G2 exist and we can construct G as described above.
Let's start by constructing a CFG for l1 l2.
1. Assume that l1 and l2 have the deterministic finite automata (DFA) A1 and A2, respectively.
2. Let's denote the state sets for A1 and A2 as Q1 and Q2, respectively.
3. Create a new set of non-terminal symbols N = {A_q1q2 | q1 ∈ Q1, q2 ∈ Q2}.
4. Create a new start symbol S.
5. Add the following rules for the start symbol S: - For each pair of states (q1, q2) ∈ Q1 × Q2, add a rule S -> A_q1q2.
6. For each non-terminal symbol A_q1q2 ∈ N, add the following rules:
- For each input symbol a ∈ Σ, add rules A_q1q2 -> aA_q1'a_q2' if δ1(q1, a) = q1' and δ2(q2, a) = q2'.
- If both q1 and q2 are accepting states in A1 and A2, respectively, add a rule A_q1q2 -> ε.
The new CFG generates the language l1 l2 because it essentially simulates the DFAs A1 and A2 in parallel, with the constraint that the length of x and y must be the same.
Since we can construct a context-free grammar that generates l1 l2, we can conclude that if l1 and l2 are regular languages, then l1 l2 is context-free.
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Final answer:
The concatenation of two regular languages l1 and l2 where the lengths of the concatenated words are equal forms a context-free language. This is because a context-free grammar, which can recognize this type of structures, can be created by combining the regular expressions of the two initial regular languages.
Explanation:
The question is asking about language concatenation in formal language theory, hence the product l1 l2, which is a set of all words produced by joining a word from l1 with a word from l2 such that their lengths are equal. Regular languages are the most simple type of formal language, while context-free languages are more complex and can express more grammatical rules.
Given l1 and l2 are regular languages, they are associated with regular expressions and can be accepted by finite automata. However, to prove that the resulting product language, l1 l2, is context-free, we need to demonstrate that it can be processed with a pushdown automaton or produced by a context-free grammar, which has the ability to handle balanced set of parentheses, which is the case for the condition |x| = |y|.
To illustrate, suppose you have a context-free grammar that generates l1 and another one that generates l2. If you combine these two grammars, you can create a new context-free grammar that generates the concatenation of l1 and l2 where |x| = |y|. Therefore, if l1 and l2 are regular, their concatenation l1 l2 where |x| = |y| is context-free.
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