Q3Compiler Design

In the realm of compiler design, specifically for Top-Down Predictive Parsers (like the LL(1) parser), the parser must decide exactly which grammatical production rule to execute just by looking at the current input token. To achieve this deterministic, non-backtracking behavior, the compiler engineers mathematically compute two critical sets for the Context-Free Grammar: the FIRST set and the FOLLOW set.

Defining the FIRST Set

The FIRST(X) set is mathematically defined as the absolute collection of all terminal symbols that can begin any string derived from the symbol X. If X can derive the empty string $ϵ$, then $ϵ$ is violently injected into FIRST(X). It tells the parser: "If I am trying to build X, what are the possible first tokens I should expect to see?"

Defining the FOLLOW Set

The FOLLOW(A) set is strictly defined for non-terminals. It is the collection of all terminal symbols that can physically appear immediately to the right of A in any valid derivation. Crucially, the $ϵ$ symbol is absolutely never allowed in a FOLLOW set. If A is the rightmost symbol in a derivation, the end-of-input marker $ is injected.

A Rigorous Example

Consider the following Context-Free Grammar: 1. $SightarrowAB$ 2. $Aightarrowa|ϵ$ 3. $Bightarrowb|c$

• Calculating FIRST sets: - FIRST(A): The rules are $Aightarrowa$ and $A\to ϵ$. Therefore, FIRST(A) = {a, epsilon}. - FIRST(B): The rules are $Bightarrowb$ and $Bightarrowc$. Therefore, FIRST(B) = {b, c}. - FIRST(S): $S$ starts with $A$. So we include FIRST(A) (which is a). But because $A$ can become $ϵ$, $S$ can also start with $B$. So we must also include FIRST(B). Therefore, FIRST(S) = {a, b, c}.
• Calculating FOLLOW sets: - FOLLOW(S): Because $S$ is the start symbol, we immediately inject the end marker $. FOLLOW(S) = {$}. - FOLLOW(A): Look at $SightarrowAB$. What directly follows $A$? It is $B$. Therefore, FOLLOW(A) mathematically inherits FIRST(B). FOLLOW(A) = {b, c}. - FOLLOW(B): Look at $SightarrowAB$. $B$ is at the absolute end of the string, meaning whatever follows $S$ must follow $B$. Therefore, FOLLOW(B) = FOLLOW(S) = {$}.

By computing these sets, the parser can perfectly populate its predictive parsing table, completely eliminating catastrophic trial-and-error backtracking.