Question

Provide  context-free grammars for the following languages :

(a) {(a^n)(b^m) | n ≥ (2^m)}.

(b) {(a^i)(b^j)(c^k)(d^l)| i + j = k + l}.

Answer 1

(a). {(a^n)(b^m) | n ≥ (2^m)}

S -> aaSb | aS | λ

(b) If this solution seems strange, it may seem less so if you tackle this problem first. We introduce two non-terminals, L1 and L2 - the first will produce all legal strings with more a's than d's, and the second will produce more d's than a's. For the first we give the production rules

L1→aL1d

L1→M1

M1→aM1c

M1→N1

N1→bN1c

Similarly,

L2→aL2d

L2→M2

M2→bM2d

M2→N2

N2→bN2c

Clearly distinguishing between N1 and N2 is unnecessary, but I think it makes the presentation more readable. Finally, to formally put this into a proper CFG we add in

S→L1

S→L2

N1→ε

N2→ε

where S is the start symbol, and εε is the empty string