LR(0) Parsing Summary. LR(0) Parsing Table. LR(0) Limitations. A NonLR(0) Grammar. LR(0) Parsing Table CS412/CS413


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1 LR(0) Parsing ummary C412/C41 Introduction to Compilers Tim Teitelbaum Lecture 10: LR Parsing February 12, 2007 LR(0) item = a production with a dot in RH LR(0) state = set of LR(0) items valid for viable prefixes Compute LR(0) states and build DFA: tart state: V(ε) = { [. ] } * Other states: V(αX) = V(α) x * Build the LR(0) parsing table from the DFA Use the LR(0) parsing table to determine whether to reduce or to shift C 412/41 pring 2007 Introduction to Compilers 1 C 412/41 pring 2007 Introduction to Compilers 2 LR(0) Limitations An LR(0) machine only works if each state with a reduce action has only one possible reduce action and no shift action With some grammars, construction gives states with shift/reduce or reduce/reduce conflicts Need to use lookahead to choose ok [ L L,. ] shift /reduce [ L L,. ] [.,L ] reduce / reduce [L,L. ] [L. ] LR(0) Parsing Table ( ) id, ε L 1 s s2 g4 2 id id id id id s s2 g7 g5 4 accept 5 s6 s8 6 (L) (L) (L) (L) (L) 7 L L L L L 8 s s2 g9 9 L L, L L, L L, L L, L L, C 412/41 pring 2007 Introduction to Compilers C 412/41 pring 2007 Introduction to Compilers 4 A NonLR(0) Grammar Grammar for addition of bers: Leftassociative version is LR(0) Rightassociative version is not LR(0) [. ] [. ] [. ] [. ] LR(0) Parsing Table 1 2 What to do in state 2: shift or reduce? [. ] [. ] [. ] [. ] [. ] [. ] ε 2 s/ 4 [. ] 6 5 [. ] [. ] C 412/41 pring 2007 Introduction to Compilers 5 C 412/41 pring 2007 Introduction to Compilers 6 1
2 LR(1) Parsing LR Parsing = easy extension of LR(0) For each reduction A β, look at the next symbol c Apply reduction only if c is in FOLLOW(A), or c= ε and * γa LR parsing table eliminates some conflicts ame as LR(0) table except reduction rows Adds reductions A βonly in the columns of symbols in FOLLOW(A) xample: FOLLOW()={} but * γ ε 2 s LR Parsing Table Reductions do not fill entire rows Otherwise, same as LR(0) ε 2 s s4 g2 g s7 7 accept C 412/41 pring 2007 Introduction to Compilers 7 C 412/41 pring 2007 Introduction to Compilers 8 LR(k) Use the LR(0) machine states as rows of table Let Q be a state and u be a lookahead string Action(Q,u) = shift Goto(Q,b) if Q contains an item of the form [A β 1.bβ ], with u FIRT k (bβ FOLLOW k (A)) Action(Q,u) = accept if Q = { [. ] } and u=ε Action(Q,u) = reduce i if Q contains the item [A β.], where A βis the ith production of G and u FOLLOW k (A), or u=ε and * γa Action(Q,u) = error otherwise G is LR(k) iff the Action function given above is singlevalued for all Q and u, i.e, there are no shiftreduce or reducereduce conflicts. C 412/41 pring 2007 Introduction to Compilers 9 LR(1) Parsing Get as much power as possible out of 1 lookahead symbol parsing table LR(1) grammar = recognizable by a shift/reduce parser with 1symbol lookahead LR(1) parsing uses similar concepts as LR(0) Parser states = sets of items LR(1) item = LR(0) item lookahead symbol following the production LR(0) item : [. ] LR(1) item : [. ] C 412/41 pring 2007 Introduction to Compilers 10 LR(1) tates LR(1) state = set of LR(1) items LR(1) item = [ A α.β b], where b in Σ {ε} Meaning: α already matched at top of the stack; next expect to see βb horthand notation [ A α. B b 1,, b n ] means: [ A α. β b 1 ] [ A α. β b n ] xtend closure and goto operations [.,ε] [. ] LR(1) Closure LR(1) closure operation on set of items For each item in : [A α.bβ b] and for each production B γ, add the following item to : [B.γ FIRT(βb)], or [B.γ ε] if FIRT(βb)={} Repeat until nothing changes imilar to LR(0) closure, but also keeps track of the lookahead symbol C 412/41 pring 2007 Introduction to Compilers 11 C 412/41 pring 2007 Introduction to Compilers 12 2
3 LR(1) tart tate Initial state: start with [. apply the closure operation xample: sum grammar ['. ε] closure ε], then ['. ε] [. ε] [.,ε] LR(1) Goto Operation LR(1) goto operation = describes transitions between LR(1) states Algorithm: for a state and a symbol Y ={[A αy.β b] [A α.yβ b] } Goto(, Y) = Closure( ) 1 [. ε] [. ε] Goto( 1, ) 2 Closure({ [. ε] }) C 412/41 pring 2007 Introduction to Compilers 1 C 412/41 pring 2007 Introduction to Compilers 14 LR(1) DFA Construction If = Goto(,X) then add an edge labeled X from to [ '. ε] [. ε] [.,ε] [. ε] [. ε] [.,ε] [. ε] [.,ε] [. ε] [. ε] [.,ε] LR(1) Reductions Reductions correspond to LR(1) items of the form [A β. x] [ '. ε] [. ε] [.,ε] [. ε] [. ε] [.,ε] [. ε] [.,ε] [. ε] [. ε] [.,ε] C 412/41 pring 2007 Introduction to Compilers 15 C 412/41 pring 2007 Introduction to Compilers 16 LR(1) Parsing Table Construction LR(1) Parsing Table xample ame as construction of LR(0) parsing table, except for reductions If [A β. b] state Q, then: Action(Q,b) is Reduce(A β) 1 2 [. ε] [. ε] [.,ε] [. ε] [. ε] [. ε] [. ε] [.,ε] Fragment of the Parsing table: ε s C 412/41 pring 2007 Introduction to Compilers 17 C 412/41 pring 2007 Introduction to Compilers 18
4 LR(1) but not LR(1) Let G have productions aab Ac A a ε V(a) = { [ a.ab ] [ A a. ] [ A.a ] [ A. ] } FOLLOW(A) = {b,c} reducereduce conflict C 412/41 pring 2007 Introduction to Compilers 19 LALR(1) Grammars Problem with LR(1): too many states LALR(1) Parsing (LookAhead LR) Construct LR(1) DFA and then merge any two LR(1) states whose items are identical except lookahead Results in smaller parser tables Theoretically less powerful than LR(1) [ id. ] [ id. ε] =? [. ε] [. ] LALR(1) Grammar = a grammar whose LALR(1) parsing table has no conflicts C 412/41 pring 2007 Introduction to Compilers 20 Classification of Grammars LL(1) LR(1) LALR(1) LR(1) LR(0) LR(k) LR(k1) LL(k) LL(k1) LL(k) LR(k) LR(0) LR(1) LALR(1) LR(1) C 412/41 pring 2007 Introduction to Compilers 21 Automate the Parsing Process Can automate: The construction of LR parsing tables The construction of shiftreduce parsers based on these parsing tables Automatic parser generators: yacc, bison, CUP LALR(1) parser generators Not much difference compared to LR(1) in practice maller parsing tables than LR(1) Augment LALR(1) grammar specification with declarations of precedence, associativity output: LALR(1) parser program C 412/41 pring 2007 Introduction to Compilers 22 Associativity hift/reduce Conflict [. ] [.,ε] What happens if we run this grammar through LALR construction? shift/reduce conflict shift: 1(2) reduce: (12) 12 ^ C 412/41 pring 2007 Introduction to Compilers 2 C 412/41 pring 2007 Introduction to Compilers 24 4
5 Grammar in CUP nonterminal ; terminal PLU, LPARN... precedence left PLU; when shifting a conflicts with reducing a production, choose reduce ::= PLU LPARN RPARN NUMBR ; Precedence CUP can also handle operator precedence T T T T ( ) ( ) C 412/41 pring 2007 Introduction to Compilers 25 C 412/41 pring 2007 Introduction to Compilers 26 Conflicts without Precedence ( ) [. ] [. ] [. ] [. ] C 412/41 pring 2007 Introduction to Compilers 27 Predecence in CUP precedence left PLU; precedence left TIM; // TIM > PLU ::= PLU TIM... RUL: in conflict, choose reduce if last terminal of production has higher precedence than symbol to be shifted; choose shift if viceversa. In tie, use associativity (left or right) given by precedence rule [. ] [. ] [. ] [. ] reduce hift C 412/41 pring 2007 Introduction to Compilers 28 ummary Lookahead information makes LR(1), LALR(1), LR(1) grammars expressive Automatic parser generators support LALR(1) grammars Precedence, associativity declarations simplify grammar writing C 412/41 pring 2007 Introduction to Compilers 29 5
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