Question

Consider grammar G2. Give an equivalent CFG grammar with no unit productions, lambda productions, or useless productions. You should show the steps that you used to reach your solution.

G2:
a)S →A | CB
b)A →C | D
c)B →bB | b
d)C →cC | c
e)D →dD | d

Answer 1

To create an equivalent CFG without unit, lambda, and useless productions for grammar G2, unit productions were first identified and replaced with the corresponding right-hand sides. Since G2 has no lambda or useless productions, those steps were unnecessary. The resulting CFG is then provided without the unwanted types of productions.

Step-by-step explanation:

Converting Grammar G2 into an Equivalent CFG

To convert the given grammar G2 into an equivalent context-free grammar (CFG) without unit productions, lambda productions, or useless productions, we need to follow certain steps:

1. First, identify any unit productions (productions where a variable is mapped to another variable). In the given grammar, A → C and A → D are unit productions.
2. Replace each unit production by substituting it with the corresponding right-hand side of the variable it points to.
3. For A → C, replace A in S → A to give S → cC | cc (using C → cC | c).
4. For A → D, we add new production alternatives for A which are A → dD | d.
5. The second step involves removing lambda productions (productions that produce an empty string), but since there are none in G2, we can skip this step.
6. Next, handle useless productions (productions that do not produce terminal strings or cannot be reached from the start symbol). Grammar G2 does not contain useless productions, so no changes are needed here.
7. Finally, we rewrite B to remove the unit production B → bB | b.
8. With these changes, our equivalent CFG without unit, lambda, or useless productions is:

• S → cC | cc | dD | d | CB
• B → bB | b
• C → cC | cc
• D → dD | d