Context-Free Constraints for Language $L = \{a^ib^jc^kd^\ell\}$

The language is defined as $L = \{a^ib^jc^kd^\ell \mid i, j, k, \ell \ge 0\}$. We need to identify which constraints on the exponents $i, j, k, \ell$ make $L$ a context-free language (CFL).

A language is context-free if it can be recognized by a pushdown automaton (PDA). We examine the constraints provided in the options that are confirmed to result in CFLs:

Option A: Constraint $i + k = j + \ell$

This condition relates the sum of counts of characters $a$ and $c$ to the sum of counts of characters $b$ and $d$. The language $L_A = \{a^ib^jc^kd^\ell \mid i+k = j+\ell\}$ is context-free. A pushdown automaton can be constructed to verify this equality by managing counts possibly using its stack and states.

Option C: Constraints $i = \ell$ and $j = k$

This condition requires matching pairs: the count of 'a's ($i$) equals the count of 'd's ($\ell$), and the count of 'b's ($j$) equals the count of 'c's ($k$), specifically forming strings of the structure $a^i b^j c^j d^i$.

The language $L_C = \{a^i b^j c^j d^i \mid i, j \ge 0\}$ is context-free. It can be generated by the context-free grammar:

• $S \to a S d \mid T$
• $T \to b T c \mid \epsilon$

This grammar generates strings adhering to the specified constraints.

Option D: Constraint $i + j = k + \ell$

This condition requires the sum of the counts of 'a' and 'b' characters to equal the sum of the counts of 'c' and 'd' characters.

The language $L_D = \{a^ib^jc^kd^\ell \mid i+j = k+\ell\}$ is context-free. A pushdown automaton can recognize it. The PDA pushes a marker onto the stack for each character 'a' or 'b'. For each character 'c' or 'd', it pops a marker. If the stack is empty at the end of the input, the condition $i+j = k+\ell$ is met.

Thus, the constraints ensuring the language $L$ is context-free are $i+k=j+\ell$, $i=\ell$ and $j=k$, and $i+j=k+\ell$.