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Something to keep in mind at all times: '''never eliminate the rules corresponding to the initial symbol''' | Something to keep in mind at all times: '''never eliminate the rules corresponding to the initial symbol''' | ||
− | + | <!-- | |
− | There are two ways of handling the mutual recursion: either | + | There are two ways of handling the mutual recursion: either expanding F in G or G in F. |
== Expanding G in F == | == Expanding G in F == | ||
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F' -> c b F' | (eps) | F' -> c b F' | (eps) | ||
F" -> d b F' | a c F" e F' | b F' e F' | F" -> d b F' | a c F" e F' | b F' e F' | ||
− | + | --> | |
− | == Expanding F in G == | + | === Elimination of Mutual Recursion: Expanding F in G === |
Initial grammar: | Initial grammar: | ||
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F" -> c G' b | (eps) | F" -> c G' b | (eps) | ||
− | Factoring F' prefixes: | + | Factoring F' prefixes, we get to the final version: |
G -> a c G" | b c G' | (eps) | G -> a c G" | b c G' | (eps) | ||
G' -> b c G' | (eps) | G' -> b c G' | (eps) | ||
G" -> a c F' e c G' | b F" e c G' | d G' | G" -> a c F' e c G' | b F" e c G' | d G' | ||
− | F' -> a c F' e F" | b F" e F" | + | F' -> a c F' e F" | b F" e F" | d G' b |
F" -> c G' b | (eps) | F" -> c G' b | (eps) | ||
− | [[category: | + | === FIRST & FOLLOW sets === |
− | [[category: | + | |
+ | FIRST(G) = { a, b, (eps) } FOLLOW(G) = { $ } | ||
+ | FIRST(G') = { b, (eps) } FOLLOW(G') = { $, b } | ||
+ | FIRST(G") = { a, b, d } FOLLOW(G") = { $ } | ||
+ | FIRST(F') = { a, b, d } FOLLOW(F') = { e } | ||
+ | FIRST(F") = { c, (eps) } FOLLOW(F") = { e } | ||
+ | |||
+ | |||
+ | [[category:Compiladores]] | ||
+ | [[category:Ensino]] |
Consider the following grammar, where G is the initial symbol and {a,b,c,d,e} is the set of terminal symbols:
O -> a G -> F c | O c d | (eps) F -> G b | O c F e
Something to keep in mind at all times: never eliminate the rules corresponding to the initial symbol
Initial grammar:
O -> a G -> F c | O c d | (eps) F -> G b | O c F e
Eliminating the singularity (O->a):
G -> F c | a c d | (eps) F -> G b | a c F e
Expanding F in G:
G -> G b c | a c F e c | a c d | (eps) F -> G b | a c F e
Eliminating left recursion in G:
G -> a c F e c G' | a c d G' | G' G' -> b c G' | (eps) F -> G b | a c F e
Factoring G prefixes:
G -> a c G" | b c G' | (eps) G' -> b c G' | (eps) G" -> F e c G' | d G' F -> a c G" b | b c G' b | b | a c F e
Factoring F prefixes:
G -> a c G" | b c G' | (eps) G' -> b c G' | (eps) G" -> F e c G' | d G' F -> a c F' | b F" F' -> G" b | F e F" -> c G' b | (eps)
Eliminating non-terminal left corners (note that F becomes unreachable):
G -> a c G" | b c G' | (eps) G' -> b c G' | (eps) G" -> a c F' e c G' | b F" e c G' | d G' F' -> a c F' e c G' b | b F" e c G' b | d G' b | a c F' e | b F" e F" -> c G' b | (eps)
Factoring F' prefixes, we get to the final version:
G -> a c G" | b c G' | (eps) G' -> b c G' | (eps) G" -> a c F' e c G' | b F" e c G' | d G' F' -> a c F' e F" | b F" e F" | d G' b F" -> c G' b | (eps)
FIRST(G) = { a, b, (eps) } FOLLOW(G) = { $ } FIRST(G') = { b, (eps) } FOLLOW(G') = { $, b } FIRST(G") = { a, b, d } FOLLOW(G") = { $ } FIRST(F') = { a, b, d } FOLLOW(F') = { e } FIRST(F") = { c, (eps) } FOLLOW(F") = { e }