Need to create CFG that requires sum of other letters

Question

I have a homework assignment that requires me to create CFG $G$ for

$$L = \{a^i b^{i+j+k} c^j d^k\}$$

so that it can accept words like ab, aaabbbbd, abbbcd, but it should not accept abba, aabbbbbc, or abc since the number of b must be the number of a+c+d.

I need CFG of L(G)

• I tried using chatgpt but the answers are not correct as I cross checked them with both gpt and myself.
• I tried to cross check this pdf but they are not as complex as my question.
• There is also a solution on chegg I can't see the whole solution as I don't have a subscription, but the final solution is also wrong as I checked it with both chatgpt and myself.
• Also researched multiple sources but did not found the answer.

The best one i came up with this grammer but it doesn't feel right since it does not have corelation between b c and d.

1. attempt: 

S → AB 
A → aAb | ε
B -> bCEdD
C -> bB | ε
D -> dD | ε
E -> bEc | ε

2. attempt : 

S → AB 
A → aA | ε
B → bBC 
C → cC | Cd | ε 
D → dD | ε 

Is it possible to do it with cfg and how?

Answer 1

Use the nesting structure of the language to obtain a context-free grammar.

$a^i\, b^{i+j+k}\, c^j d^k = a^i b^i\, b^k b^j c^j d^k $

We can see that the language is the concatenation of strings of the form $a^i\, b^i $ and strings of the form $b^{j+k}\, c^j d^k$ which each separately form context-free languages.

Answer 2

$S\rightarrow AD$
$A\rightarrow aAb,\varepsilon$
$D\rightarrow bDd, C$
$C\rightarrow bCc, \varepsilon$.

In this solution I divided the word into a prefix of $a^ib^i$ and a suffix of $b^{j+k}c^jd^k$. The latter part is generated by building $b^kCd^k$ and then replacing $C$ with $b^jc^j$.