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How to construct an LL(1) parsing table ?

How to construct an LL(1) parsing table ?. S  A S b S  C A  a C  c C C  . LL(1) Parsing Table. The idea is easy:. What is the first terminal symbols you will see from deriving AS b , C , a , … and so on?. First Sets. N : the set of non-terminal symbols

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How to construct an LL(1) parsing table ?

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  1. How to construct an LL(1) parsing table? • S A S b • S  C • A  a • C  c C • C   LL(1) Parsing Table The idea is easy: What is the first terminal symbols you will see from deriving ASb,C, a, … and so on? IT 327

  2. First Sets N : the set of non-terminal symbols T : the set of terminal symbols Let ω ( N  T )*. Definition: First set of ω, First(ω), is defined as follows: First(ω) = The set of every first (the left most) terminal symbol that appears in any possible string derived from ω. If ω derives , then let   First(ω); otherwise   First(ω); . * * First(ω) = { a | (a  T and ω a ω’ ) or ( ω a and a = )} IT 327

  3. S A S b • S  C • A  a • C  c C • C   First-set Examples • First() = {} • First(c C ) = {c} • First(a) = {a} • First(C)= First(c C )  First() = {c, } • First(ASb) = First(aSb) = {a} • First(S) = First(ASb)  First(C)= {a , c, } • First(Sb) = (First(S)- {})  First(b) = {a , b , c} IT 327

  4. N : the set of non-terminal symbols T : the set of terminal symbols Let a T, A N, ωi ( N  T )* Algorithm to find First Sets • First() = {} • First(a ω) = {a} • If A ω1 |ω2 ......... |ωn • First(A) = First(ω1 )  First(ω2)  .....  First(ωn) • Assume ω : •  First(A), • then First(Aω) = First(A) •  First(A), • then First(Aω) = (First(A) - {})  First(ω) IT 327

  5. N : the set of non-terminal symbols T : the set of terminal symbols More (?) algorithmic procedure • First(x): • if (x = ) return {}; • if (x = a ωand a  T) return {a}; • if (x = A , A N and A ω1 |ω2 ......... |ωn) • return First(ω1 )  First(ω2)  .....  First(ωn); • if (x = Aωand ω ) • if ( First(A)) • return First(A); • else • return (First(A) - {})  First(ω); IT 327

  6. Follow Sets N : the set of non-terminal symbols T : the set of terminal symbols let A, S  N and S is the start symbol Definition: Follow(A), the follow set of A, is defined as follows: Follow (A) = The set of every terminal symbol (including $) that can appear to the right of A in any possible sentential form derived from the start symbol. Follow(A) = { a | S ωAa ω’ and a  T} * IT 327

  7. Rules (recursive) for finding Follow Sets S …………Aa... Let a T, A, B N, and ,  ( N  T )* S$ • If S is the start symbol, then let $ Follow (S). • If A B, then let Follow (A)  Follow (B). • If A B  •  First () , then letFirst () Follow (B). •  First () , then let • (First () - {})  Follow (A)  Follow (B). …Ax…  … Bx… …Ax…  … Bx… IT 327

  8. S A S b • S  C • A  a • C  c C • C   Follow-set Examples • Follow (S) = {b, $} • S is the start symbol, $  Follow (S) • S A S b,b  Follow (S) • Follow (A ) = {a, b, c} • S A S b, First (S b) Follow (A). • First (S b) = (First (S)- {})  First (b) = {a , b , c} • Follow (C) = {b, $} • S  C, Follow (S) Follow (C) • C  c C, Follow (C) Follow (C) IT 327

  9. Convert rules(definitions) of follow sets into an algorithm. How? Let a T, A, B N, and ,  ( N  T )* S …………Aa... • If S is the start symbol, then let $ Follow (S). • If A B, then let Follow (A)  Follow (B). • If A B  •  First () , then letFirst () Follow (B). •  First () , then let • (First () - {})  Follow (A)  Follow (B). Check every rule and work on every Non-terminal symbols appearing in the right hand side of the production rules. IT 327

  10. Principle: repeatedly updating until nothing will further change. • Initialize Follow(X) =  for every X N; • Let Follow (S) = {$}. • Repeat the following procedure until every follow-set stops changing: Check every rule and work on every non-terminal symbol appeared in the right hand side of the rule. • If A B, and B N , then • add every element in Follow (A) into Follow (B). • If A B , and B N , then • if  First (), then add every element inFirst () into Follow (B). • if  First (), then add every terminal symbol in First () and Follow(A) into Follow (B).(note:   T) IT 327

  11. Construction of LL(1) parsing talbes For ever production rule A ω, do the following : • For every a First (ω) and a , put A ω into T [A, a] • If  First (ω) , then for every a Follow (A), • put A ω into T [A, a] First (ASb) = {a}, First (C) = {c, }, • S A S b • S  C • A  a • C  c C • C   1 2 2 2 Follow (S) = {b, $}, 3 First (a) = {a}, First (cC) = {c}, First () = {}, 5 4 5 Follow (C) = {b, $} IT 327

  12. Condition for being an LL(1) grammar: For every A N with A , A   and    • First ()  First () = . • If  First (), then First ()  Follow (A) = . Q: Can an LL(1) grammar be ambiguous? A: No. IT 327

  13. Another Example: • E T E’ • E’ + T E’ • E’  • T  F T’ • T’  * F T’ • T’  • F  ( E ) • F  id • E E + T • ET • T  T * F • T  F • F  ( E ) • F  id IT 327

  14. First and Follow Sets First (F) = { (, id } First (T’) = { *,  } First (T) = First (FT’) = { (, id } First (E’) = { +,  } First (E) = First (TE’) = {(, id } • E T E’ • E’ + T E’ • E’  • T  F T’ • T’  * F T’ • T’  • F  ( E ) • F  id Follow (E) Follow (E’) Follow (T) Follow (T’) Follow (F) {$ } {} {} {} {} {), $ } {} {+} {} {*} { ), $ } {), $ } { +,), $ } { +} { +,*} { ), $ } {), $ } { +,), $ } {+,), $ } {+ ,*, ), $ } initialization E to E’ and T T to T’ and F check those caused by first sets check those caused by the follow sets from the previous stage IT 327

  15. First Follow LL(1) Parsing Table E E’ T T’ F { (, id } {+,  } { (, id } {*,  } {(, id } { ), $ } {), $ } { +,), $ } {+,), $ } {+ ,*, ), $ } • E T E’ • E’ + T E’ • E’  • T  F T’ • T’  * F T’ • T’  • F  ( E ) • F  id 1 1 2 3 3 4 4 6 5 6 6 8 7 IT 327

  16. id*(id+id) The entry [E, id] in the Parsing table: LL(1)_PT[E, id] = 1 • E •  T E’ • F T’ E’ • idT’ E’ • id *F T’ E’ • id * (E )T’ E’ • id * (T E’ ) T’ E’ • id * (F T’ E’)T’ E’ • id * ( idT’ E’)T’ E’ • id * ( idE’)T’ E’ • id * ( id+TE’)T’ E’ • id * ( id +F T’ E’)T’ E’ • id * ( id + idT’ E’)T’ E’ • id * ( id + id E’)T’ E’ • id * ( id + id)T’ E’ • id * ( id + id)E’ • id * ( id + id) • E T E’ • E’ + T E’ • E’  • T  F T’ • T’  * F T’ • T’  • F  ( E ) • F  id left-most derivation IT 327

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