LR Parsing Techniques

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1 LR Parsing Techniques Bottom-Up Parsing - LR: a special form of BU Parser LR Parsing as Handle Pruning Shift-Reduce Parser (LR Implementation) LR(k) Parsing Model - k lookaheads to determine next action Parsing Table Construction: SLR, LR, LLR 1

2 Bottom-Up Parsing bottom-up parser attempts to construct a parse tree for an input string beginning at the leaves (the bottom) and working up towards the root (the top). 2

3 Bottom-Up Parsing: Ex1 BU Parsing: Construct a parse tree from the leaves to the root: left-to-right reduction G: S a B e input: abbcde b c b B d S B B a b b c d e a b b c d e a b b c d e a b b c d e a b b c d e 3

4 Bottom-Up Parsing: Ex2 BU Parsing: Construct a parse tree from the leaves to the root: random reduction G: S a B e input: abbcde b c b B d S B B B a b b c d e a b b c d e a b b c d e a b b c d e a b b c d e 4

5 LR Parsing: BU + Left-to-Right Many ways to construct a parse tree bottom-up Ex1 & Ex2 Prefer a simpler form of parser Left-to-right scanning If scanning strictly Left-to-right Rightmost derivation in reverse (thus the name LR Parser). Why rm.? ( Ex1) Never consider right terminals while reducing left ( N)* Reduce left ( N)* (terminals or non-terminals) as much as possible until no further reduce Shift when no further reduce Reversing the sequence of reduction corresponds to a rightmost derivation LR Parser special form of BU Parser parser with simpler form: left-to-right scan 5

6 LR Parsing: BU + Left-to-Right LR Parsing: Construct a parse tree from the leaves to the root, scanning left-to-right (resulting in rightmost derivation in reverse) a b b c S a B e b c b B d d e a a b b c b b c d e d e a b b c d e input: abbcde a b b c B d e abbcde rm abcde rm ade rm abe rm S a b b S c B d e 6

7 Rightmost Derivation in Reverse E 1 E 5 E 3 E 2 E 3 E 4 E 5 E 4 E 1 E 2 id1 + id2 * id3 id1 + id2 * id3 8

8 LR Parsing The L stands for scanning the input from left to right The R stands for constructing a rightmost derivation in reverse 9

9 LR Parsing LR Parsing =/= Leftmost Reduction The 1 st reducible substring does not always result in successful parse Handle(s): those successfully lead to S Top-Down: Expansion Matching Bottom-Up: Shift/Reduce Locating next handle to reduce [How To??] Handle pruning: hide details below reduced ( N)* a b b c d e a b b c d e 10

10 Handles NOT all (leftmost) reduction ( ) leads to the start symbol S: rm rm (n) rm S Only some handles do handle of a right-sentential form consists of a production a position of where can be replaced by to produce the previous right-sentential form in a rightmost derivation of Right-sent. forms: abbcde rm abcde rm ade rm abe rm S Handles: b b c B d S a B e 11

11 If S * rm rm Handles, then in the position following is a handle of. (The string contains only terminal symbols.) We say a handle rather than the handle since the grammar may be ambiguous. But if the grammar is unambiguous, then every right sentential form has exactly one handle. 12

12 LR Parsing as Handle Pruning rm rm S S Pruning: Find a string that is reducible to S and hide its details by reduction and proceed with the new sentential form. Never consider right terminals while reducing left grammar symbols The string to the right of the handle contains only terminals ( is the rightmost non-terminal) is the leftmost complete interior node with all its children in the tree 15

13 n Example S S S B B B a b b c d e a b c d e a d e S B S a e 16

14 LR Parsing as Handle Pruning (1st reduction sequence) rightmost derivation in reverse can be obtained by handle pruning. Let G = E E+E E*E (E) id (ambiguous!) rm Right-sentential Handle Reducing form production id 1 +id 2 *id 3 id 1 E id E+id 2 *id 3 id 2 E id E+E*id 3 id 3 E id E+E*E E*E E E*E E+E E+E E E+E E 17

15 LR Parsing as Handle Pruning (2nd reduction sequence) rightmost derivation in reverse can be obtained by handle pruning. rm Let G = E E+E E*E (E) id (ambiguous!) Right-sentential Handle Reducing form production id 1 +id 2 *id 3 id 1 E id E+id 2 *id 3 id 2 E id E+E*id 3 E+E E E+E E*id 3 id 3 E id E*E E*E E E*E E 18

16 Shift-Reduce Parsing rm rm S shift Input Handle reduce) Parsing program Output Stack Parsing table 20

17 Stack Implementation of Shift-Reduce Parsers convenient way to implement a shift-reduce parse is to use a stack to hold grammar symbols and an input buffer to hold the string to be parsed. a push-down machine with a tape The parser operates by shifting zero or more symbols onto the stack until a handle is on top of the stack. The parser then replaces/reduces with/to the left side of the appropriate production. This procedure repeats until the stack contains the start symbol and the input is empty. 21

18 Stack Operations Shift: shift the next input symbol onto the top of the stack Reduce: replace the handle at the top of the stack with the corresponding nonterminal ccept: announce successful completion of the parsing Error: call an error recovery routine 22

19 n Example ction Stack Input S \$ a b b c d e \$ S \$ a b b c d e \$ R \$ a b b c d e \$ S \$ a b c d e \$ S \$ a b c d e \$ R \$ a b c d e \$ S \$ a d e \$ R \$ a d e \$ S \$ a B e \$ R \$ a B e \$ \$ S \$ 24

20 Configurations of shift-reduce parser on inputid 1 +id 2 *id 3 Step Stack Input ction 1 \$ id 1 +id 2 *id 3 \$ shift 2 \$id 1 +id 2 *id 3 \$ reduce by E id 3 \$E +id 2 *id 3 \$ shift 4 \$E+ id 2 *id 3 \$ shift 5 \$E+id 2 *id 3 \$ reduce by E id 6 \$E+E *id 3 \$ shift 7 \$E+E* id 3 \$ shift 8 \$E+E*id 3 \$ reduce by E id 9 \$E+E*E \$ reduce by E E*E 10 \$E+E \$ reduced by E E+E 11 \$E \$ accept *Note: The grammar is ambiguous. Therefore, there is another possible reduction sequence. 25

21 G: S a B e b c b B d Input: abbcde LR Parsing States S 10 How to represent parsing states so we can tell the right parsing actions to take? B a b b c d e \$

22 G: S a B e b c b B d Input: abbcde LR Parsing States S 10 How to represent parsing states so we can tell the right parsing actions to take? B a b b c d e \$

23 G: S a B e b c b B d Input: abbcde LR Parsing States S0:. S. a B e S 10 B a b b c d e \$

24 G: S a B e b c b B d Input: abbcde LR Parsing States S 10 S1: S a. B e (shift a).bc.b B a b b c d e \$

25 G: S a B e b c b B d Input: abbcde LR Parsing States S 10 S2: b. (shift b, to reduce b) B a b b c d e \$

26 G: S a B e b c b B d Input: abbcde LR Parsing States S 10 S3: S a. B e B. d.bc B a b b c d e \$

27 G: S a B e b c b B d Input: abbcde LR Parsing States S 10 S4: b.c (shift b) B a b b c d e \$

28 G: S a B e b c b B d Input: abbcde LR Parsing States S5: bc. (shift c, reduce bc) S 10 B a b b c d e \$

29 G: S a B e b c b B d Input: abbcde LR Parsing States S 10 S6: S a. B e B. d B a b b c d e \$

30 G: S a B e b c b B d Input: abbcde LR Parsing States S 10 S7: B d. (shift d, reduce B d) B a b b c d e \$

31 G: S a B e b c b B d Input: abbcde LR Parsing States S8: S a B. e S 10 B a b b c d e \$

32 G: S a B e b c b B d Input: abbcde LR Parsing States S9: S a B e. (shift e, reduce S a B e ) S 10 B a b b c d e \$

33 G: S a B e b c b B d Input: abbcde LR Parsing States S10: S S. (S reduced) S 10 B a b b c d e \$

34 G: S a B e b c b B d Input: abbcde LR Parsing States S0:. S. a B e S1: S a. B e (shift a) a.bc.b.bc.b (closed, no further expansion) S2: b. (shift b, reduce b) a S3: S a. B e B. d.bc S4: b.c (shift b) a b S5: bc. (shift c, reduce bc) S6: S a. B e B. d S7: B d. (shift d, reduce B d) a b S8: S a B. e S9: S a B e. (shift e, reduce S a B e ) S10: S S. (S reduced) b b b b c b c d e B b c c d e d e d e a S B b b c d e 39

35 mbiguity: Sources of Conflicts When trying to reduce a sub-string of the current sentential form: Not all reducible substrings are handles mbiguous: More than one substring as a handle Sources of Conflicts non-lr Grammar Shift-reduce conflicts Reduce-reduce conflicts 40

36 Shift/Reduce Conflict stmt if expr then stmt if expr then stmt else stmt other Stack Input \$ if expr then stmt * else stmt \$ Shift if expr then stmt else stmt Reduce if expr then stmt 41

37 Reduce/Reduce Conflict (1) stmt id ( para_list ) // func(a,b) (2) stmt expr := expr (3) para_list para_list, para (4) para_list para (5) para id (6) expr id ( expr_list ) // array(a,b) (7) expr id (8) expr_list expr_list, expr (9) expr_list expr -Need a complex lexical analyzer to identify id vs. procid - Reduction depends on stack[sp-2] Stack Input (a) \$ id ( id, id ) \$ [Q: r5? r7?] [Sol: use stmt procid ( para_list ) => (a) r7 (b) r5] (b) \$- - - procid ( id, id ) \$ [r5] 42

38 LR(k) Grammars Only some classes of grammars, known as the LR(k) Grammars, can be parsed deterministically by a shift-reduce parser CFG s that are non-lr may need some adaptation to make them deterministically parsed with a shift-reduce parser Parsing Table Construction Predict handles at each positions (after shifts) 43

39 LR(k) Parsing The L stands for scanning the input from left to right The R stands for constructing a rightmost derivation in reverse The k stands for the number of lookahead input symbols used to make parsing decisions 44

40 LR Parsing The LR parsing algorithm Constructing SLR(1) parsing tables Constructing LR(1) parsing tables Constructing LLR(1) parsing tables 45

41 Model of an LR Parser State after action handle Stack S m X m S m-1 Shift/Reduce LR Parsing Program Input Output State after Reduction State before action Initial State X m-1 S 0 ction Goto Parsing table 46

42 Parsing Table for Expression Grammar (0) E E (1) E E + T (2) E T (3) T T * F (4) T F (5) F ( E ) (6) F id Follow(E)={+,),\$} Follow(T)={+,),\$,*} Follow(F)={+,),\$,*} State ction Goto id + * ( ) \$ E T F 0 s5 s s6 acc 2 r2 s7 r2 r2 3 r4 r4 r4 r4 4 s5 s r6 r6 r6 r6 6 s5 s s5 s s6 s11 9 r1 s7 r1 r1 10 r3 r3 r3 r3 11 r5 r5 r5 r5 47

43 GOTO ctions I0: E. E E.E+T E.T T.T*F T.F F.(E) F.id E T F I1: E E. E E.+T I2: E T. T T.*F I3: T F. 0 E 1 0 T 2 0 F 3 0 id 5 ( id fter reduction Before reduction I4: F (.E) E.E+T E.T T.T*F T.F F.(E) F.id I5: F id. 48

44 LR Parsing lgorithm Input: n input string and an LR parsing table with functions action and goto for a grammar G. Output: If is in L(G), a bottom-up parse for ; otherwise, an error indication. Method: Initially, the parser has s 0 on its stack, where s 0 is the initial state, and \$ in the input buffer. Shift/reduce according to the parsing table (See next Page) 49

45 LR Parsing Program while (1) do { s := the state of top of the stack; a := get input token; if (action[s,a] == shift s ) { push a then s on top of the stack; a = get input token; } else if (action[s,a] == reduce -> ) { pop 2* symbols off the stack; s = the state now on top of the stack; push then goto[s,] on top of the stack; output the production -> ; } else if (action[s,a] == accept) return; else error(); } 50

46 Stack Input LR Parsing onid 1 *id 2 +id 3 shift/reduce+goto ction (1) 0 id * id + id \$ (0,id):s5 Shift (2) 0 id 5 * id + id \$ (5,*):r6; (0,F):3 Reduce by F id (3) 0 F 3 * id + id \$ (3,*):r4; (0,T):2 Reduce by T F (4) 0 T 2 * id + id \$ (2,*):s7 Shift (5) 0 T 2 * 7 id + id \$ (7,id):s5 Shift (6) 0 T 2 * 7 id 5 + id \$ (5,+):r6; (7,F):10 Reduce by F id (7) 0 T 2 * 7 F 10 + id \$ (10,+):r3; (0,T):2 Reduce by T T*F (8) 0 T 2 + id \$ (2,+):r2; (0,E):1 Reduce by E T (9) 0 E 1 + id \$ (1,+):s6 Shift (10) 0 E id \$ (6,id):s5 Shift (11) 0 E id 5 \$ (5,\$):r6; (6,F):3 Reduce by F id (12) 0 E F 3 \$ (3,\$):r4; (6,T):9 Reduce by T F (13) (14) 0 E T 9 0 E 1 \$ \$ (9,\$):r1; (0,E):1 (1,\$):acc Reduce by E E+T ccept 51

47 LR Parsing dvantages Efficient: non-backtracking Efficient Parsing Efficient Error detection (& correction) Coverage: Detect syntax error as soon as one appear during L-o-R scan virtually all programming languages G(LR) > G(TD predictive parsing) Disadvantages: Too much work to construct by hands ( YCC) 52

48 How To: LR Parsing (repeated) LR Parsing =/= Leftmost Reduction The 1 st reducible substring does not always result in successful parse Handle(s): those successfully lead to S Top-Down: Expansion Matching Bottom-Up: Shift/Reduce Locating next handle to reduce [How To??] Handle pruning: hide details below reduced ( N)* 53

49 LR Parsing Table Construction Techniques Parsing Table Construction: SLR(1) Parser - LR(0) Items & States LR(1) Parser - shift/reduce conflict resolution - LR(1) Items & States LLR(1) Parser - LR(1) state merge - reduce-reduce conflict 54

50 SLR Parser Coverage: weakest in terms of #grammars it succeeds Easiest to construct Parser: a DF for recognizing viable prefixes States: Sets of LR(0) Items The items in a set can be viewed as the states of an NF recognizing viable prefixes Grouping items into sets is equivalent to subset construction 56

51 G: S a B e b c b B d Input: abbcde LR Parsing States S0:. S. a B e S1: S a. B e (shift a) a.bc.b.bc.b (closed, no further expansion) S2: b. (shift b, reduce b) a S3: S a. B e B. d.bc S4: b.c (shift b) a b S5: bc. (shift c, reduce bc) S6: S a. B e B. d S7: B d. (shift d, reduce B d) a b S8: S a B. e S9: S a B e. (shift e, reduce S a B e ) S10: S S. (S reduced) b b b b c b c d e B b c c d e d e d e a S B b b c d e 57

52 Viable Prefix The set of prefixes of c.s.f. s (canonical/right sentential forms) that can appear on the stack of a shift-reduce parser are called viable prefixes. Equivalently, it is a prefix of a right-sentential form that does not continue past the right end of the rightmost handle of that sentential form If is a viable prefix, then w * w is a c.s.f. 59

53 Item and Valid Item n LR(0) item (item for short) is a marked production [ 1 2 ] (dotted rule: production with a dot at RHS) n item [ 1 2 ] is said to be valid for some viable prefix 1 iff w * S * w 1 2 w The represents where we are now during parsing Left of dot: those scanned Right of dot: those to be visited later S w

54 Example of Valid Item Consider the grammar: S 1C D C 3 4 D 1B B 2 S S or S D 1 C Valid items for the viable prefix : [S 1C], [S D], and [D 1B] 1 B 61

55 Example of Valid Item (cont.) ssume 1, i.e., S ' could be S 1 C or S 1 C D S 1 B Valid items for the viable prefix 1 : [S 1 C], [C 3], [C 4], [D 1 B], and [B 2] 62

56 Example of Valid Item (cont.) ssume S 1 C 3 Valid item for viable prefix 13 : [C 3 ] Valid item for viable prefix 1C : [S 1C ] 63

57 Closure: ll Valid Items Enumerable from G Given a grammar E E E E+T T T T*F F F (E) id What are valid items for the viable prefix E+? [E E+ T], but also [T... F] since 1 2 E * E+T T F E+ F 1 2 Likewise, [T T*F], [T F], [F (E)], [F id] called Closure of [E E+ T] (inclusive)

58 Computation of Closure Given a set, I, of items Initially Closure(I) = I Loop: for all items [ B ] If [ B ] is in Closure(I) and B is in P, then include [B ] into Closure(I). Repeat the Loop until no new dotted rules can be added Initial set of items for a grammar: I 0 = Closure({[S S] }) (S: start symbol, S : augmented start symbol) 65

59 GOTO Computation Let I be a set of items which are valid for some viable prefix. Then goto(i,x), where X (N or Σ), is the set of items which are valid for the viable prefix X. So [ X ] in I implies Closure({[ X ]}) in goto(i,x) S * ]w X w X w ([]: set of items I, including [ X ] others) = 66

60 Sets of LR(0) Items Construction ugment the grammar with: S S Let I 0 = Closure({[S S] }), C = {I 0 } while (not all elements of C are marked) { } -select an unmarked item set of C (say I ) and mark it; - X (V or Σ), if goto(i,x) is not already in C, then add goto(i,x) to C (unmarked); also called Characteristic Finite State Machine (CFSM) Construction lgorithm. 67

61 SLR(1) Parsing ctions Compute the CFSM states C={I 0,I 1,,I n }. 1. If [ a ] I i and goto(i i,a) = I j then set action(i i,a) = shift,i j (where a is a terminal) 2. If [ ] I i then set action(i i,a) = reduce for all a in Follow() 1. terminal a in Follow() does not guarantee that will result in a successful parse. (not necessarily a handle ) 2. But, a terminal NOT in Follow() will definitely indicate an impossible parse. 3. So reduction on symbols in Follow() is only a loose criterion for possible success parse. 3. If [S S ] I i then set action(i i,\$) = accept 4. Other action(*,*) = error 68

62 Conflicts Shift-reduce conflicts: both a shift action and a reduce action are possible in the same Closure. E.g., state 2 in Figure 4.37 (p.229) [ho 86] Reduce-reduce conflicts: two or more distinct reduce actions are possible in the same Closure. 69

63 Example: Grammar G for Math Expressions (0) E E (1) E E+T (2) E T (3) T T*F (4) T F (5) F (E) (6) F id Follow(E)={+,),\$}, Follow(T)={+,),\$,*}, Follow(F)={+,),\$,*} 70

64 Computing SLR(1) States for G an SLR(1) State = a set of LR(0) items (See the next slide, Fig. 4.35, page 225, [ho 86]) 71

65 Canonical LR(0) Collection for G I0: E. E E.E+T E.T T.T*F T.F F.(E) F.id ( ( E T F T I1: E E. E E.+T I2: E T. T T.*F I3: T F. I4: F (.E) E.E+T E.T T.T*F T.F F.(E) F.id F + * E I6: E E+.T T.T*F T.F F.(E) F.id F I7: T T*.F F.(E) F.id + I8: F (E.) E E.+T T ( id ( id I9: E E+T. T T.*F * I10: T T*F. F I11: F (E). ) id id I5: F id. 72

66 GOTO ctions I0: E. E E.E+T E.T T.T*F T.F F.(E) F.id E T F I1: E E. E E.+T I2: E T. T T.*F I3: T F. 0 E 1 0 T 2 0 F 3 0 id 5 ( id fter reduction Before reduction I4: F (.E) E.E+T E.T T.T*F T.F F.(E) F.id I5: F id. 74

67 Parsing Table for Expression Grammar (0) E E (1) E E + T (2) E T (3) T T * F (4) T F (5) F ( E ) (6) F id Follow(E)={+,),\$} Follow(T)={+,),\$,*} Follow(F)={+,),\$,*} State ction Goto id + * ( ) \$ E T F 0 s5 s s6 acc 2 r2 s7 r2 r2 3 r4 r4 r4 r4 4 s5 s r6 r6 r6 r6 6 s5 s s5 s s6 s11 9 r1 s7 r1 r1 10 r3 r3 r3 r3 11 r5 r5 r5 r5 76

68 Transition Diagram of DF D for Viable Prefixes State transition in terms of sets of LR(0) items (Fig. 4.36) SLR(1) Parsing Table: (Fig. 4.31) I i = a => I j : action(i,a) = shift-j I i = => I j : goto(i,) = j I i : [. ] action(i,follow()) = reduce [ If = S (augmented start symbol ) action(i,\$)=accept 77

69 Visualizing Transitions in the Transition Diagram Shift: moving forward one step along arc Equivalent to pushing input symbols Reduce LHS RHS : moving backward to a previous state s along arcs labeled with the RHS symbols Then GOTO(s, LHS) equivalent to popping RHS symbols from stack then pushing LHS, then redefining current state 78

70 Parsing Table for Expression Grammar State action goto id + * ( ) \$ E T F 0 s5 s s6 acc 2 r2 s7 r2 r2 3 r4 r4 r4 r4 4 s5 s r6 r6 r6 r6 6 s5 s s5 s s6 s11 9 r1 s7 r1 r1 10 r3 r3 r3 r3 11 r5 r5 r5 r5 79

71 LR Parsing Table Construction Techniques Canonical LR Parsing Table LLR Parsing Table (See Textbook ) 80

72 Canonical LR Parser SLR(1) parser does NOT always work SLR(1) Grammar => Unambiguous Unambiguous CFG =/=> SLR(1) Grammar E.g., Shift-reduce conflicts in the SLR(1) parsing table may NOT be a real shift-reduce conflict (e.g., impossible reduce ) Need more specific & additional information to define states [to avoid false reductions] use LR(1) items, instead of LR(0) items Much more states than SLR(1) Need (canonical) LR(1) or LLR(1) Parsers (Parsing Table construction methods) 81

73 (0) S S (1) S L = R (2) S R Example: non-slr(1) Grammar for ssignment (3) L * R (content of R) (4) L id (5) R L L I2: Follow(R) = { =, } (1) S L. = R (5) R L. R ction(2, = ) = reduce 5 I3: ction(2, = ) =shift 6 (2) S R. = Follow(S) S => L = R => *R = R IF: Reduce on = Goto I3 Error ( Follow(S)) 82 NOT Really Reducible

74 Example: non-slr(1) Grammar for ssignment Problem: G is unambiguous SLR Shift/Reduce conflict is false, but SLR parsing table is unable to remember enough left context to decide proper action on = when seeing a string reducible to L 83

75 Why Unambiguous Yet Non-SLR(1) Some reduce actions are not really reducible by checking input against Follow(LHS) Not all symbols in FOLLOW(LHS) result in successful reduction to S. May fail after a few steps of reductions. SLR(1) states does not resolve such conflicts by using LR(0)-item defined states Need more specific constraints to rule out a subset of Follow(LHS) from indicating a reduction action 84

76 LR(1) Parsing Table Construction SLR: reduce on input a if Ii contains [.] & a FOLLOW() Not really reducible for all a FOLLOW() Only a subset (maybe proper subset) But on some cases: S a =/=> a Reduce does not produce a right sentential form E.g., S L = R =/=> S R = R although S *R = R = in follow(r) 85

77 LR(1) Parsing Table Construction Solution: Define each state by including more specific information to rule out invalid reductions Sometimes results in splitting states of the same core LR(0) items: [. ] Only dotted production (the core ) LR(1) items: [., L s] Dotted production(the core ), plus lookaheads that allow reduction upon [ ] 1 : length of L symbols 86

78 LR(1) Parsing Table Construction [., a] (& ) : L ( a ) has no effect on items of this form [., a] (i.e., = ): L has effect on items of this form Reduction is called for only when next input is a (not all terminal symbols in Follow()) Only a subset in Follow() will be the right L s Initially, only one restriction is known: [S. S, \$] Infer other restrictions by closure computation 87

79 LR(1) Item and Valid Item n LR(1) item is a dotted production plus lookahead symbols: [,, a] n LR(1) item [,, a] is said to be valid for a viable prefix if r.m. derivation S * w w, where 2. a First(w) (or w= && a = \$ ) The represents where we are now during parsing Left of dot: those scanned Right of dot: those to be visited later 88

80 LR(1) Parsing Table Construction Change the closure() and goto() functions of SLR parsing table construction, with initial collection: C = {closure({s. S, \$})} [ B a] valid implies [B, b] valid if b is in FIRST( a) Construction method for set of LR(1) items See next few pages 89

81 LR(1): Closure(I) Given a set, I, of items Initially Closure(I) = I Repeat: for each items [ B a] in I, each production B is in G, and each terminal b in FIRST( a), include [B, b] to Closure(I). Until no more items can be added to I 90

82 LR(1): GOTO(I,X) Let J = {[ X, a] such that [ X a] is in I}. goto(i,x) = closure(j) That is: J = {} For all [ X a] in I, J += {[ X, a]} Return(closure(J)) I: [ X, a] [ X, a ] X J: [ X, a] [ X, a ] Goto(I,X) = Closure ({[ X, a], [ X, a ]}) 91

83 Sets of LR(1) Items Construction ugment the grammar with: S S, call it G Let I 0 = Closure({[S S, \$] }), C = {I 0 } Repeat { } - I C, - X (N or Σ), if goto(i,x) is not already in C, then add goto(i,x) to C Until no more sets of items can be added to C 92

84 Example: resolving shift/reduce conflicts with LR(1) items G : {S S, S CC, C cc d} L(G)={ c m d c n d } => I0 ~ I9 (Fig. 4.39, p. 235 [ho 86]) I3 vs. I6: same set of LR(0) items with different lookaheads Conditions for reduction are different I3: reduce on c/d (when constructing 1 st C ) I6: reduce on \$ (when constructing 2 nd C ) 93

85 SLR(1) Goto Graph S I1: S S. [\$] I0: S. S S.CC C.cC C.d C I2: S C.C C.cC C.d C I5: S CC.[\$] c d Follow Sets: S: {\$} C: {c,d,\$} c c I3: C c.c C.cC C.d C I8: C cc.,c/d/\$ G: S S S CC C cc C d d d I4: C d.,c/d/\$ 94

86 LR(1) Goto Graph S I1: S S., \$ I0: S. S, \$ S.CC, \$ C.cC, c/d C.d,c/d C I2: S C.C, \$ C.cC, \$ C.d,\$ C c I5: S CC., \$ c I6: C c.c, \$ C.cC, \$ C.d,\$ C I9: C cc.,\$ c c I3: C c.c, c/d C.cC, c/d C.d,c/d C d d I7: C d.,\$ I8: C cc.,c/d G: S S S CC C cc C d d d I4: C d.,c/d 95

87 Construction of Canonical LR(1) Parsing Table lgorithm 4.10 Shift: (same as SLR, ignoring L in item) Reduce on a : [,, a] ccept on \$ : [S S,, \$] Goto: (same as SLR) LR(1) Grammar: a grammar without conflicts (multiply defined actions) in LR(1) Parsing Table 96

88 SLR(1) vs. LR(1) LR(1): more specific states May split into states with the same core but with different lookaheads SLR(1) Grammar LR(1) Grammar Number of states LR(1) >> SLR(1) 97

89 LLR(1) Merge LR(1) states with the same core, while retaining lookahead symbols Considerably smaller than canonical LR tables Most programming language constructs can be expressed by an LLR grammar SLR and LLR have the same number of states Without/with lookahead symbols [full/subset of FOLLOW] Several hundred states for PSCL Several thousands, if using LR(1) G is an LLR(1) Grammar: if no conflicts after state merge 98

90 LLR(1) vs. LR(1) Effect of LR(1) state merge: The merging of states with common cores can never produce a shift-reduce conflict that was not present in one of the original states Because shift actions depend only on the core, not the lookahead However, a merge may produce a reducereduce conflict. Because union of lookaheads may introduce unnecessary reductions 99

91 LLR(1) vs. LR(1) Example: merging that produces reduce-reduce conflicts. LR(1) Grammar: S S S ad bbd abe be c B c Sets of LR(1) items: {[ c.,d], [B c.,e]} (valid for viable prefix ac) {[ c.,e], [B c.,d]} (valid for viable prefix bc) Merging states with common cores {[ c.,d/e], [B c.,d/e]} merging also merges loohaheads Reduce-reduce conflicts: cand B c,on inputs d and e 100

92 LLR(1) vs. LR(1) Effect of LR(1) state merge: Behave like the original, or Declare error later, but before shifting next input symbol For correct input: LR and LLR have the same sequence of shift/reduce For erroneous input: LLR requires extra reduces after LR has detected an error (but before shifting next) 101

93 Example: Merge States with Same Core Fig. 4.39: I4 vs. I7 same reduction with different lookaheads State merge: dotted rules remain, L s merged Examples: I3 + I6 => I36 I4 + I7 => I47 I8 + I9 => I89 Same as SLR(1) table (Fig. 4.41, p239, [ho 86]) 102

94 LLR(1) Parsing Table Construction (I) Method 1: (Naïve Method) [1] Construct LR(1) parsing table Very costly [#states is normally very large] [2] Merge states with the same core 103

95 LLR(1) Parsing Table Construction (II) Method 2: (Efficient Construction Method) [1] Construct kernels set of LR(0) items, from [S S] It is Possible to Compute shift/reduce/goto actions directly from kernel items kernel items: items whose dot is not at the beginning, except [S. S, \$]: those not derived from closure() Can represent a set of items [2] ppend lookaheads Compute initial spontaneous lookaheads, and those item pairs that pass Propagated lookaheads 104

96 LLR(1) Parsing Table Construction (II.1) Compute shift/reduce/goto actions directly from kernel items: (pps ) Reduce: Shift: Goto: Need to pre-compute First (C) = { r.m. C * } for all pairs of nonterminals (C, ) and 105

97 LLR(1) Parsing Table Construction (II.2) Determine spontaneous and propagated lookaheads (Fig. 4.43) Compute closure({core,#}) by assuming a dummy lookahead # 106

98 LLR(1) Parsing Table Construction: Example Example: 4.46/Fig [p. 241, ho 86] Kernels of sets of LR(0) items Fig [with non-kernel items] Example: 4.47 Get Spontaneous & Propagated lookaheads Fig. 4.44: item pairs that propagate lookaheads Fig. 4.45: initial spontaneous lookahead, and multiple passes of lookahead propagation LLR(1) parsing table: Todo by yourself 107

99 LLR(1) Parsing Table Construction LLR(/LR) (Fig 4.45) SLR (Fig. 4.37) SLR: I2: shift/reduce conflict on = I2: (1) S L. = R (5) R L. LLR(/LR): I2: shift on =, reduce on \$, NO conflict I2: (1) S L. = R, \$ (5) R L., \$ 108

100 Using mbiguous Grammar (see Handouts) 109

101 Parser Generators YCC (Slide Part II) 110

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