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LL(1) Parsing

Learn about LL parsing techniques, including first and follow sets computation, prediction table creation, left factoring, recursion removal, and LL(k) grammars. Examples and guidance provided to enhance understanding.

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LL(1) Parsing

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  1. LL(1) Parsing CPSC 388 Ellen Walker Hiram College

  2. Adding Prediction • No lookahead: case statement based on first token (e.g. “exp” in book) • Adding lookahead: consider two sets • FIRST(S): All possible first tokens of S (can include ) • FOLLOW (S): All possible tokens that can follow S (can include $, for end of string)

  3. Finding First(X) • If there is a rule X->x…, where x is a terminal, then x is in First(X) • If there is a rule X-> , or, if S can derive  through a sequence of rules, then  is in First(S) • If there is a rule X->A…, where A is a non-terminal, then everything in First(A) except  is in First(S) • If there is a rule X->AB… and  is in First(A), then everything in First(B) except  is in First(X)

  4. Finding Follow(X) • $ is in Follow(S), where S is the start non-terminal. • If there is a rule A->…Xx… , where x is a terminal, then x is in Follow(X) • If there is a rule A->…XB…, where B is a non-terminal, then everything in First(B) except  is in Follow(X) • If there is a rule A-> …X, then everything in Follow(A) is in Follow(X) • If there is a rule A-> …XB… and First(B) is , then re-evaluate the rule as if B weren’t there

  5. S->abSXbY |  X-> cX |  Y -> aY |  First(S) = First(X) = First(Y) = Follow(S) = Follow(X) = Follow(Y) = First & Follow Example

  6. Using First & Follow Sets • Use an explicit stack as in the CFL to PDA algorithm. • If the current character is in First (S), then it’s OK to pop S and push its rule beginning with this character • If the current character is in Follow (S) then it’s OK to pop S without pushing (i.e. use S-> e rule)

  7. LL(1) Example • Grammar: S-> aSb | e • First(S): {a, e} • Follow(S): {b, $}

  8. LL Parse Table • Table of Non-Terminals vs. Terminals • If S->rhs and a is in FIRST(S) and a is in FIRST(rhs) put S->rhs in M[S,a] • If S->e and a is in FOLLOW(S), put S-> e in M[S,a] • If there is only one entry in each cell, it is LL(1)

  9. LL(1) Table Example

  10. Left Factoring • Two rules start with the same RHS • Make a new rule to distinguish them • S-> if ( exp ) S • S-> if ( exp ) S else S becomes • S-> if ( exp ) S S’ • S’ -> else S | e

  11. Left Recursion Removal • Rewrite rules to avoid left recursion S->Sx | y becomes S-> yS’ and S’ -> xS’ | e • (Note: x and y can be arbitrary strings)

  12. Example: • E -> E + T | T • T -> T * F | F • F -> n | (E) n stands for “number” • Left-recursion removal? • LL(1) table?

  13. LL(k) Grammars • Look at k terminals instead of 1 terminal • First(S) is all sequences of k terminals that can begin S • Follow(S) is all sequences of k terminals that can follow S • Col. headers of table are sequences of k terminals instead of single terminals • First & Follow computations get messy!

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