# Bottom-Up Parsing II (Different types of Shift-Reduce Conflicts) Lecture 10. Prof. Aiken (Modified by Professor Vijay Ganesh.

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1 Bottom-Up Parsing II Different types of Shift-Reduce Conflicts) Lecture 10 Ganesh. Lecture 10) 1

2 Review: Bottom-Up Parsing Bottom-up parsing is more general than topdown parsing And just as efficient Doesn t have issues with left-recursion Doesn t require left-factoring Many well-known parser generators Yacc, Bison, ) Can handle many more grammars without backtracking than otherwise LRk) parsers are bottom-up 2

3 Review: An Introductory xample Bottom-up parsers don t need left-factored grammars Revert to the natural grammar for our example: + * ) Consider the string: * + 3

4 Review: he Idea Bottom-up parsing reduces a string to the start symbol by inverting productions: * + * * + 4

5 Review: Bottom-up Parsers and Rightmost derivations in reverse Read the productions in reverse from bottom to top) his is a rightmost derivation! * + * * + 5

6 Review: Notation useful for Shift-Reduce Parsers Idea: Split string o two substrings Right substring is as yet unexamined by parsing a string of terminals) Left substring has terminals and non-terminals he dividing po is marked by a he is not part of the string Initially, all input is unexamined x 1 x 2... x n 6

7 Review: Shift-Reduce Parsing Bottom-up parsing uses only two kinds of actions: Shift Reduce 7

8 Review: Shift Shift: Move one place to the right Shifts a terminal to the left string ABC xyz ABCx yz 8

9 Review: Reduce Apply an inverse production at the right end of the left string If D Cx is a production, then ABCx yz ABD yz 9

10 he Stack Left string can be implemented by a stack op of the stack is the Shift pushes a terminal on the stack Reduce pops 0 or more symbols off of the stack production rhs) and pushes a nonterminal on the stack production lhs) Prof. Aiken CS 143 Lecture 7 10

11 Review: he xample with Reductions Only * + * + reduce reduce * reduce reduce reduce + 11

12 Review: he xample with Shift-Reduce Parsing * + * + * + * + * shift shift shift reduce reduce * shift shift reduce reduce reduce + 12

13 Conflicts In a given state, more than one action shift or reduce) may lead to a valid parse If it is legal to shift or reduce, there is a shiftreduce conflict If it is legal to reduce by two different productions, there is a reduce-reduce conflict Prof. Alex Aiken Lecture 9 Modified by Professor Vijay Ganesh) 13

14 Key Issue: o Shift or Reduce? How do we decide when to shift or reduce? xample grammar: + * ) Consider step * + We could reduce by giving * + A fatal mistake! No way to reduce to the start symbol 14

15 Handles: Symbols replaced by Reduction Intuition: Want to reduce only if the result can still be reduced to the start symbol Handle: Informally, represents the RHS of a production. Let X β be a production in G. hen β in the position after α is a handle of αβω 15

16 Handles Cont.) Handles formalize the uition A handle is a string that can be reduced and also allows further reductions back to the start symbol using a particular production at a specific spot) We only want to reduce at handles Note: We have said what a handle is, not how to find handles 16

17 Important Fact #2 Important Fact #2 about bottom-up parsing: In shift-reduce parsing, handles appear only at the top of the stack, never inside Using the symbol already shifted o the stack left context) and the next k lookahead symbols right context), decide to shift or reduce 17

18 A Shift-Reduce Parse in Detail 1) * + * + Prof. Alex Aiken Lecture 9 Modified by Professor Vijay Ganesh) 18

19 A Shift-Reduce Parse in Detail 2) * + * + * + Prof. Alex Aiken Lecture 9 Modified by Professor Vijay Ganesh) 19

20 A Shift-Reduce Parse in Detail 3) * + * + * + * + Prof. Alex Aiken Lecture 9 Modified by Professor Vijay Ganesh) 20

21 A Shift-Reduce Parse in Detail 4) * + * + * + * + * + Prof. Alex Aiken Lecture 9 Modified by Professor Vijay Ganesh) 21

22 A Shift-Reduce Parse in Detail 5) * + * + * + * + * + * + Prof. Alex Aiken Lecture 9 Modified by Professor Vijay Ganesh) 22

23 A Shift-Reduce Parse in Detail 6) * + * + * + * + * + + * + Prof. Alex Aiken Lecture 9 Modified by Professor Vijay Ganesh) 23

24 A Shift-Reduce Parse in Detail 7) * + * + * + * + * * + Prof. Alex Aiken Lecture 9 Modified by Professor Vijay Ganesh) 24

25 A Shift-Reduce Parse in Detail 8) * + * + * + * + * * + Prof. Alex Aiken Lecture 9 Modified by Professor Vijay Ganesh) 25

26 A Shift-Reduce Parse in Detail 9) * + * + * + * + * * + Prof. Alex Aiken Lecture 9 Modified by Professor Vijay Ganesh) 26

27 A Shift-Reduce Parse in Detail 10) * + * + * + * + * * + Prof. Alex Aiken Lecture 9 Modified by Professor Vijay Ganesh) 27

28 A Shift-Reduce Parse in Detail 11) * + * + * + * + * * Prof. Alex Aiken Lecture 9 Modified by Professor Vijay Ganesh) + 28

29 heorem: Handle always on top of Stack Proof Method: Informal induction on # of reduce moves Proof: rue initially, stack is empty Immediately after reducing a handle right-most non-terminal on top of the stack next handle must be to right of right-most non-terminal, because this is a right-most derivation Sequence of shift moves reaches next handle 29

30 Summary of Handles In shift-reduce parsing, handles always appear at the top of the stack Handles are never to the left of the rightmost non-terminal herefore, shift-reduce moves are sufficient; the need never move left Bottom-up parsing algorithms are based on recognizing handles 30

31 Recognizing Handles here are no known efficient algorithms to recognize handles Solution: use heuristics to guess which stacks are handles On some CFGs, the heuristics always guess correctly For the heuristics we use here, these are the SLR grammars Other heuristics work for other grammars 31

32 Grammars All CFGs Unambiguous CFGs will generate conflicts SLR CFGs 32

33 Viable Prefixes It is not obvious how to detect handles At each step the parser sees only the stack, not the entire input; start with that... α is a viable prefix if there is an ω such that α ω is a state of a shift-reduce parser 33

34 Huh? What does this mean? A few things: A viable prefix does not extend past the right end of the handle It s a viable prefix because it is a prefix of the handle As long as a parser has viable prefixes on the stack no parsing error has been detected 34

35 Important Fact #3 Important Fact #3 about bottom-up parsing: For any grammar, the set of viable prefixes is a regular language 35

36 Important Fact #3 Cont.) Important Fact #3 is non-obvious We show how to compute automata that accept viable prefixes 36

37 Items An item is a production with a. somewhere on the rhs he items for ) are.).).) ). 37

38 Items Cont.) he only item for X is X. Items are often called LR0) items 38

39 Intuition he problem in recognizing viable prefixes is that the stack has only bits and pieces of the rhs of productions If it had a complete rhs, we could reduce hese bits and pieces are always prefixes of rhs of productions 39

40 xample Consider the input ) hen ) is a state of a shift-reduce parse is a prefix of the rhs of ) Will be reduced after the next shift Item.) says that so far we have seen of this production and hope to see ) 40

41 Generalization he stack may have many prefixes of rhs s Prefix 1 Prefix 2... Prefix n-1 Prefix n Let Prefix i be a prefix of rhs of X i α i Prefix i will eventually reduce to X i he missing part of α i-1 starts with X i i.e. there is a X i-1 Prefix i-1 X i β for some β Recursively, Prefix k+1 Prefix n eventually reduces to the missing part of α k 41

42 An xample Consider the string * ): * ) is a state of a shift-reduce parse is a prefix of the rhs of ) is a prefix of the rhs of * is a prefix of the rhs of * 42

43 An xample Cont.) he stack of items.). *. Says We ve seen of ) We ve seen of We ve seen * of * 43

44 Recognizing Viable Prefixes Idea: o recognize viable prefixes, we must Recognize a sequence of partial rhs s of productions, where ach sequence can eventually reduce to part of the missing suffix of its predecessor 44

45 An NFA Recognizing Viable Prefixes 1. Add a dummy production S S to G 2. he NFA states are the items of G Including the extra production 3. For item α.xβ add transition α.xβ X αx.β 4. For item α.xβ and production X γ add α.xβ X.γ 45

46 An NFA Recognizing Viable Prefixes Cont.) 5. very state is an accepting state 6. Start state is S.S 46

47 NFA for Viable Prefixes of the xample ). ).).) ). S.. + S *..* * *. 47

48 NFA for Viable Prefixes in Detail 1) S. 48

49 NFA for Viable Prefixes in Detail 2) S. S

50 NFA for Viable Prefixes in Detail 3). ) S.. + S

51 NFA for Viable Prefixes in Detail 4). ) S.. + S

52 NFA for Viable Prefixes in Detail 5). ).) S.. + S

53 NFA for Viable Prefixes in Detail 6). ).) S.. + S...)

54 NFA for Viable Prefixes in Detail 7). ).).) ) ). S.. + S

55 NFA for Viable Prefixes in Detail 8). ).).) ) ). S.. + S

56 NFA for Viable Prefixes in Detail 9). ).) S.. + S...).+ + ) )

57 NFA for Viable Prefixes in Detail 10). ).) S.. + S...).+. ) + )

58 NFA for Viable Prefixes in Detail 11). ).) S.. + S...).+. ) + ) *. 58

59 NFA for Viable Prefixes in Detail 12). ).) S.. + S...).+. ) + ) *..* * 59

60 NFA for Viable Prefixes in Detail 13). ).) S.. + S...).+. ) + ) *..* * *. 60

61 +. ranslation to the DFA S * S *.) *..).. +.). *..) +.. ).. +.)... ). ) 61

62 Lingo he states of the DFA are canonical collections of items or canonical collections of LR0) items he Dragon book gives another way of constructing LR0) items 62

63 Valid Items Item X β.γ is valid for a viable prefix αβ if S * αxω αβγω by a right-most derivation After parsing αβ, the valid items are the possible tops of the stack of items 63

64 Items Valid for a Prefix An item I is valid for a viable prefix α if the DFA recognizing viable prefixes terminates on input α in a state s containing I he items in s describe what the top of the item stack might be after reading input α 64

65 Valid Items xample An item is often valid for many prefixes xample: he item.) is valid for prefixes... 65

66 +. Valid Items for S * S *.) *..) ). *..) +.. ).. +.)... ). ) 66

67 LR0) Parsing Idea: Assume stack contains α next input is t DFA on input α terminates in state s Reduce by X β if s contains item X β. Shift if s contains item X β.tω equivalent to saying s has a transition labeled t 67

68 LR0) Conflicts LR0) has a reduce/reduce conflict if: Any state has two reduce items: X β. and Y ω. LR0) has a shift/reduce conflict if: Any state has a reduce item and a shift item: X β. and Y ω.tδ 68

69 +. LR0) Conflicts S * S *.) *..) ).. ).. + *..).) wo shift/reduce conflicts with LR0) rules... ). ) 69

70 SLR LR = Left-to-right scan SLR = Simple LR SLR improves on LR0) shift/reduce heuristics Fewer states have conflicts 70

71 SLR Parsing Idea: Assume stack contains α next input is t DFA on input α terminates in state s Reduce by X β if s contains item X β. t FollowX) Shift if s contains item X β.tω 71

72 SLR Parsing Cont.) If there are conflicts under these rules, the grammar is not SLR he rules amount to a heuristic for detecting handles he SLR grammars are those where the heuristics detect exactly the handles 72

73 +. SLR Conflicts S... + S... *.. + *.). *..) ). *..) ) ). +.. ).. + Follow) = { ), \$ }.) Follow) = { +, ), \$ }. No conflicts with SLR rules! 73

74 Precedence Declarations Digression Lots of grammars aren t SLR including all ambiguous grammars We can parse more grammars by using precedence declarations Instructions for resolving conflicts 74

75 Precedence Declarations Cont.) Consider our favorite ambiguous grammar: + * ) he DFA for this grammar contains a state with the following items: *.. + shift/reduce conflict! Declaring * has higher precedence than + resolves this conflict in favor of reducing 75

76 Precedence Declarations Cont.) he term precedence declaration is misleading hese declarations do not define precedence; they define conflict resolutions Not quite the same thing! 76

77 Naïve SLR Parsing Algorithm 1. Let M be DFA for viable prefixes of G 2. Let x 1 x n \$ be initial configuration 3. Repeat until configuration is S \$ Let α ω be current configuration Run M on current stack α If M rejects α, report parsing error Stack α is not a viable prefix If M accepts α with items I, let a be next input Shift if X β. a γ I Reduce if X β. I and a FollowX) Report parsing error if neither applies 77

78 Notes If there is a conflict in the last step, grammar is not SLRk) k is the amount of lookahead In practice k = 1 78

79 SLR xample Configuration DFA Halt State Action * \$ 1 shift 79

80 2 Configuration * \$ S.. S... +.) *. * *..) ) ). *..) ) ).. +.)

81 SLR xample Configuration DFA Halt State Action * \$ 1 shift * \$ 3 * not in Follow) shift 81

82 2 Configuration * \$ S.. S... +.) *. * *..) ) ). *..) ) ).. +.)

83 2 Configuration * \$ S.. S... +.) *. * *..) ) ). *..) ) ).. +.)

84 SLR xample Configuration DFA Halt State Action * \$ 1 shift * \$ 3 * not in Follow) shift * \$ 11 shift 84

85 2 Configuration * \$ S.. S... +.) *. * *..) ) ). *..) ) ).. +.)

86 2 Configuration * \$ S.. S... +.) *. * *..) ) ). *..) ) ).. +.)

87 2 Configuration * \$ S.. S... +.) *. * *..) ) ). *..) ) ).. +.)

88 SLR xample Configuration DFA Halt State Action * \$ 1 shift * \$ 3 * not in Follow) shift * \$ 11 shift * \$ 3 \$ Follow) red. 88

89 2 Configuration * \$ S.. S... +.) *. * *..) ) ). *..) ) ).. +.)

90 2 Configuration * \$ S.. S... +.) *. * *..) ) ). *..) ) ).. +.)

91 2 Configuration * \$ S.. S... +.) *. * *..) ) ). *..) ) ).. +.)

92 2 Configuration * \$ S.. S... +.) *. * *..) ) ). *..) ) ).. +.)

93 SLR xample Configuration DFA Halt State Action * \$ 1 shift * \$ 3 * not in Follow) shift * \$ 11 shift * \$ 3 \$ Follow) red. * \$ 4 \$ Follow) red. * 93

94 2 Configuration * \$ S.. S... +.) *. * *..) ) ). *..) ) ).. +.)

95 2 Configuration * \$ S.. S... +.) *. * *..) ) ). *..) ) ).. +.)

96 2 Configuration * \$ S.. S... +.) *. * *..) ) ). *..) ) ).. +.)

97 2 Configuration * \$ S.. S... +.) *. * *..) ) ). *..) ) ).. +.)

98 SLR xample Configuration DFA Halt State Action * \$ 1 shift * \$ 3 * not in Follow) shift * \$ 11 shift * \$ 3 \$ Follow) red. * \$ 4 \$ Follow) red. * \$ 5 \$ Follow) red. 98

99 Configuration \$ S.. S... +.) *. * *..) ). *..) ) ) ).. +.) 8. 99

100 Configuration \$ S.. S... +.) *. * *..) ). *..) ) ) ).. +.)

101 SLR xample Configuration DFA Halt State Action * \$ 1 shift * \$ 3 * not in Follow) shift * \$ 11 shift * \$ 3 \$ Follow) red. * \$ 4 \$ Follow) red. * \$ 5 \$ Follow) red. \$ accept 101

102 Notes Skipped using extra start state S in this example to save space on slides Rerunning the automaton at each step is wasteful Most of the work is repeated 102

103 An Improvement Remember the state of the automaton on each prefix of the stack Change stack to contain pairs Symbol, DFA State 103

104 An Improvement Cont.) For a stack sym 1, state 1... sym n, state n state n is the final state of the DFA on sym 1 sym n Detail: he bottom of the stack is any,start where any is any dummy symbol start is the start state of the DFA 104

105 Goto able Define goto[i,a] = j if state i A state j goto is just the transition function of the DFA One of two parsing tables 105

106 Refined Parser Moves Shift x Push a, x on the stack a is current input x is a DFA state Reduce X α As before Accept rror 106

107 Action able For each state s i and terminal a If s i has item X α.aβ and goto[i,a] = j then action[i,a] = shift j If s i has item X α. and a FollowX) and X S then action[i,a] = reduce X α If s i has item S S. then action[i,\$] = accept Otherwise, action[i,a] = error 107

108 SLR Parsing Algorithm Let I = w\$ be initial input Let j = 0 Let DFA state 1 have item S.S Let stack = dummy, 1 repeat case action[top_statestack),i[j]] of shift k: push I[j++], k reduce X A: pop A pairs, push X, goto[top_statestack),x] accept: halt normally error: halt and report error 108

109 Notes on SLR Parsing Algorithm Note that the algorithm uses only the DFA states and the input he stack symbols are never used! However, we still need the symbols for semantic actions 109

110 More Notes Some common constructs are not SLR1) LR1) is more powerful Build lookahead o the items An LR1) item is a pair: LR0) item x lookahead [. *, \$] means After seeing * reduce if lookahead is \$ More accurate than just using follow sets ake a look at the LR1) automaton for your parser! 110

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