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Notes on First and Follow

Notes on First and Follow. Written by David Walker Edited by Phil Sweany. Computing Nullable Sets. Non-terminal X is Nullable only if the following constraints are satisfied (computed using iterative analysis) base case: if (X := ) then X is Nullable inductive case:

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Notes on First and Follow

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  1. Notes on First and Follow Written by David Walker Edited by Phil Sweany

  2. Computing Nullable Sets • Non-terminal X is Nullable only if the following constraints are satisfied (computed using iterative analysis) • base case: • if (X := ) then X is Nullable • inductive case: • if (X := ABC...) and A, B, C, ... are all Nullable then X is Nullable

  3. Computing First Sets • First(X) is computed iteratively • base case: • if T is a terminal symbol then First (T) = {T} • inductive case: • if X is a non-terminal and (X:= ABC...) then • First (X) = First (X) U First (ABC...) where First(ABC...) = F1 U F2 U F3 U ... and • F1 = First (A) • F2 = First (B), if A is Nullable • F3 = First (C), if A is Nullable & B is Nullable • ...

  4. Computing Follow Sets • Follow(X) is computed iteratively • base case: • initially, we assume nothing in particular follows X • (Follow (X) is initially { }) • inductive case: • if (Y := s1 X s2) for any strings s1, s2 then • Follow (X) = First (s2) UFollow (X) • if (Y := s1 X s2) for any strings s1, s2 then • Follow (X) = Follow(Y) UFollow (X), if s2 is Nullable

  5. building a predictive parser Y ::= c Y ::= X ::= a X ::= b Y e Z ::= X Y Z Z ::= d

  6. building a predictive parser Y ::= c Y ::= X ::= a X ::= b Y e Z ::= X Y Z Z ::= d base case

  7. building a predictive parser Y ::= c Y ::= X ::= a X ::= b Y e Z ::= X Y Z Z ::= d after one round of induction, we realize we have reached a fixed point

  8. building a predictive parser Y ::= c Y ::= X ::= a X ::= b Y e Z ::= X Y Z Z ::= d base case (note, first of Y should include empty string)

  9. building a predictive parser Y ::= c Y ::= X ::= a X ::= b Y e Z ::= X Y Z Z ::= d after one round of induction, no fixed point

  10. building a predictive parser Y ::= c Y ::= X ::= a X ::= b Y e Z ::= X Y Z Z ::= d after two rounds of induction, no more changes ==> fixed point

  11. building a predictive parser Y ::= c Y ::= X ::= a X ::= b Y e Z ::= X Y Z Z ::= d base case

  12. building a predictive parser Y ::= c Y ::= X ::= a X ::= b Y e Z ::= X Y Z Z ::= d after one round of induction, no fixed point

  13. building a predictive parser Y ::= c Y ::= X ::= a X ::= b Y e Z ::= X Y Z Z ::= d after two rounds of induction, fixed point (but notice, computing Follow(X) before Follow (Y) would have required 3rd round)

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