1. S L $
   L L ; E
   L E
   E x

2. S L $
   L E ; L
   L E
   E x

3. S L $
   L L ; L
   L E
   E x

4. S L $
   L x T
   T
   T ; L

(Note: The in grammar (d) is meant to represent the empty string. )

Taking into account the fact that we are only interested in right-most derivations, an apparently obvious alternate definition intended to capture what we have in mind might be:

Def.

We say that a string (V_n U V_t)^* is a feasible prefix for G if is different from S and there exists a rightmost derivation

S

for some string V_t^*.

The definition used in class, by contrast is:

Def.

We say that a string (V_n U V_t)^* is a viable prefix for G if there exists a rightmost derivation

S N _{1 2}

for some N V_n, ₁, ₂ (V_n U V_t)^* and V_t^* such that = ₁.

1. Does being a feasible prefix imply that a string must be a viable prefix?
2. Does being a viable prefix imply that a string must be a feasible prefix? For each of these questions, justify your answer giving proofs or counter examples as approriate.

In your answer, you should assume that the grammars involved contain no useless non-terminals. That is, assume that every non-terminal derives some string of terminal symbols.