FIRST Set in Syntax Analysis

FIRST(X) for a grammar symbol X is the set of terminals that begin the strings derivable from X. 

Rules to compute FIRST set: 

1. If x is a terminal, then FIRST(x) = { ‘x’ }
2. If x-> Є, is a production rule, then add Є to FIRST(x).
3. If X->Y1 Y2 Y3….Yn is a production, 
   1. FIRST(X) = FIRST(Y1)
   2. If FIRST(Y1) contains Є then FIRST(X) = { FIRST(Y1) – Є } U { FIRST(Y2) }
   3. If FIRST (Yi) contains Є for all i = 1 to n, then add Є to FIRST(X).

Example 1: 

Production Rules of Grammar
E  -> TE’
E’ -> +T E’|Є
T  -> F T’
T’ -> *F T’ | Є
F  -> (E) | id

FIRST sets
FIRST(E) = FIRST(T) = { ( , id }
FIRST(E’) = { +, Є }
FIRST(T) = FIRST(F) = { ( , id }
FIRST(T’) = { *, Є }
FIRST(F) = { ( , id }

Example 2: 

Production Rules of Grammar
S -> ACB | Cbb | Ba
A -> da | BC
B -> g | Є
C -> h | Є

FIRST sets
FIRST(S) = FIRST(A) U FIRST(B) U FIRST(C)
         = { d, g, h, Є}
FIRST(A) = { d } U FIRST(B) = { d, g , Є }
FIRST(B) = { g , Є }
FIRST(C) = { h , Є }

Notes: 

1. The grammar used above is Context-Free Grammar (CFG). Syntax of most of the programming language can be specified using CFG.
2. CFG is of the form A -> B , where A is a single Non-Terminal, and B can be a set of grammar symbols ( i.e. Terminals as well as Non-Terminals)

In the next article “FOLLOW sets in Compiler Design” we will see how to compute Follow sets. 

This article is compiled by Vaibhav Bajpai. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

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