Bottom up parser
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Bottom-up parsing builds a derivation by working from the input sentence back toward the start symbol S. It is preferred in practice and also called LR parsing, where L means tokens are read left to right and R means it constructs a rightmost derivation. The two main types are operator-precedence parsing and LR parsing, which covers a wide range of grammars through techniques like SLR, LALR, and LR parsing. LR parsing reduces a string to the start symbol by inverting productions through identifying handles and replacing them.
![The Function GOTO
GOTO (I, X) is defined to be the closure of the set of all items [A -> αX.β] such that
[A -> α.Xβ] is in I.
If I is the set of two items { [E' -> E·] , [E -> E· + T] } , then
GOTO(I, +) contains the items
E -> E + ·T
T -> ·T * F
T -> ·F
F -> · (E)
F -> ·id](https://image.slidesharecdn.com/bottomupparser-160604143840/85/Bottom-up-parser-41-320.jpg)
![Structure of the LR Parsing
Table
1. The ACTION function takes as arguments a state i and a terminal
a (or $, the input endmarker). The value of ACTION[i, a] can have
one of four forms:
a) Shift j , where j is a state. The action taken by the parser shifts
input a to the stack, but uses state j to represent a .
b) Reduce A ->β. The action of the parser reduces β on the top of
the stack to head A.
c) Accept. The parser accepts the input and finishes parsing;
d) Error. The parser discovers an error in its input and takes some
corrective action.
2. Extend the GOTO function, defined on sets of items, to states: if
GOTo [Ii , A] = Ij , then GOTO also maps a state i and a
nonterminal A to state j .](https://image.slidesharecdn.com/bottomupparser-160604143840/85/Bottom-up-parser-49-320.jpg)
![Construction of SLR Parse
Table
1. Construct C={I0,I1…….In} the collection sets of LR(0)
items for G’.
2. Initial state of the parser is constructed from the set of items
for [S’->S]
3. State I is constructed from Ii. The parsing actions are
determined as follows.
1. If [A->α.aβ] is in Ii and GOTO(Ii,a]=Ij, then ACTION[i,a]
is set to ‘shift j’ here ‘a’ must be a terminal.
2. If[A->α.] is in Ii, then ACTION[i,a] is set to reduce A->a
for all ‘a’ Follow(A).
3. If S’->S is in Ii, than action[i, $] is set to ‘accept’.](https://image.slidesharecdn.com/bottomupparser-160604143840/85/Bottom-up-parser-50-320.jpg)
Bottom up parser
- 2. Bottom-Up Parsing Bottom-up parsing is more general than top- down. A bottom-up parser builds a derivation by working from the input sentence back toward the start symbol S Preferred method in practice Also called LR parsing L means that tokens are read left to right R means that it constructs a rightmost derivation
- 3. Bottom-up Parsing • Two types of bottom-up parsing: • Operator-Precedence parsing • LR parsing • covers wide range of grammars. • SLR – simple LR parser • LR – most general LR parser • LALR – intermediate LR parser (Lookahead LR parser)
- 4. Bottom-Up Parsing LR parsing reduces a string to the start symbol by inverting productions: Str input string of terminals repeat Identify β in str such that A →β is a production Replace β by A in str until str = S (the start symbol) OR all possibilities are exhausted
- 5. Bottom-Up Parsing int + (int) + (int) E + (int) + (int) E + (E) + (int) E + (int) E + (E) E E → E + ( E ) | int A rightmost derivation in reverse
- 7. Reductions Bottom-up parsing as the process of "reducing" a string w to the start symbol of the grammar. At each reduction step, a substring that matches the body of a production is replaced by the non-terminal at the head of that production.
- 8. Handle Handle of a string: Substring that matches the RHS of some production AND whose reduction to the non-terminal on the LHS is a step along the reverse of some rightmost derivation.
- 9. Handles Formally: Handle of a right sentential form is <A , location of in > i.e. A is a handle of at the location immediately after the end of , if:S => A => A certain sentential form may have many different handles. Right sentential forms of a non-ambiguous grammar have one unique handle
- 10. Example S aABe aAde aAbcde abbcde S aABe A Abc | b B d S aABe is a handle of aABe in location 1. B d is a handle of aAde in location 3. A Abc is a handle of aAbcde in location 2. A b is a handle of abbcde in location 2.
- 11. Handle Pruning A rightmost derivation in reverse can be obtained by “handle-pruning.” abbcde Find the handle = b at loc. 2 aAbcde b at loc. 3 is not a handle: aAAcde ... blocked.
- 12. Handle-pruning The process of discovering a handle & reducing it to the appropriate left-hand side is called handle pruning. Handle pruning forms the basis for a bottom-up parsing method. To construct a rightmost derivation S = 0 1 2 … n-1 n = w Apply the following simple algorithm for i n to 1 by -1 Find the handle Ai i in i Replace i with Ai to generate i-1
- 13. 1. S -> 0 S1|01 indicate the handle in each of the following right-sentential forms: 1. 000111 2. 00S11 2. For the grammar S -> S S + | S S * | a indicate the handle in each of the following right-sentential forms: 1. SSS + a * + 2. SS + a * a+ 3. aaa * a + +. 3. Give bottom-up parses for the following input strings 1. The input 000111 according to the grammar of Exercise 1 2. The input aaa * a + + according to the grammar of 2
- 14. Shift Reduce Parsing with a Stack Two problems: locate a handle decide which production to use (if there are more than two candidate productions).
- 15. Shift-reduce Parsing A shift-reduce parser is a stack automaton with four actions Shift — next word is shifted onto the stack Reduce — right end of handle is at top of stack Locate left end of handle within the stack Pop handle off stack & push appropriate lhs Accept — stop parsing & report success Error — call an error reporting/recovery routine Accept & Error are simple Shift is just a push and a call to the scanner Reduce takes |rhs| pops & 1 push
- 16. x - 2 * y Stack Input Handle Action $ id - num * id None shift $ id - num * id 1. Shift until the top of the stack is the right end of a handle 2. Find the left end of the handle and reduce 0 Goal Expr 1 Expr Expr + Term 2 | Expr - Term 3 | Term 4 Term Term * Factor 5 | Term / Factor 6 | Factor 7 Factor number 8 | id 9 | ( Expr )
- 17. Back to x - 2 * y Stack Input Handle Action $ id - num * id none shift $ id - num * id 8,1 reduce 8 $ Factor - num * id 6,1 reduce 6 $ Term - num * id 3,1 reduce 4 $ Expr - num * id 1. Shift until the top of the stack is the right end of a handle 2. Find the left end of the handle and reduce 0 Goal Expr 1 Expr Expr + Term 2 | Expr - Term 3 | Term 4 Term Term * Factor 5 | Term / Factor 6 | Factor 7 Factor number 8 | id 9 | ( Expr )
- 18. Back to x - 2 * y Stack Input Handle Action $ id - num * id none shift $ id - num * id 8,1 reduce 8 $ Factor - num * id 6,1 reduce 6 $ Term - num * id 3,1 reduce 4 $ Expr - num * id 1. Shift until the top of the stack is the right end of a handle 2. Find the left end of the handle and reduce 0 Goal Expr 1 Expr Expr + Term 2 | Expr - Term 3 | Term 4 Term Term * Factor 5 | Term / Factor 6 | Factor 7 Factor number 8 | id 9 | ( Expr ) Expr is not a handle at this point because it does not occur at this point in the derivation.
- 19. Back to x - 2 * y Stack Input Handle Action $ id - num * id none shift $ id - num * id 8,1 reduce 8 $ Factor - num * id 6,1 reduce 6 $ Term - num * id 3,1 reduce 3 $ Expr - num * id none shift $ Expr - num * id none shift $ Expr - num * id 1. Shift until the top of the stack is the right end of a handle 2. Find the left end of the handle and reduce 0 Goal Expr 1 Expr Expr + Term 2 | Expr - Term 3 | Term 4 Term Term * Factor 5 | Term / Factor 6 | Factor 7 Factor number 8 | id 9 | ( Expr )
- 20. Back to x - 2 * y Stack Input Handle Action $ id - num * id none shift $ id - num * id 8,1 reduce 8 $ Factor - num * id 6,1 reduce 6 $ Term - num * id 3,1 reduce 3 $ Expr - num * id none shift $ Expr - num * id none shift $ Expr - num * id 7,3 reduce 7 $ Expr - Factor * id 6,3 reduce 6 $ Expr - Term * id 1. Shift until the top of the stack is the right end of a handle 2. Find the left end of the handle and reduce 0 Goal Expr 1 Expr Expr + Term 2 | Expr - Term 3 | Term 4 Term Term * Factor 5 | Term / Factor 6 | Factor 7 Factor number 8 | id 9 | ( Expr )
- 21. Back to x - 2 * y Stack Input Handle Action $ id - num * id none shift $ id - num * id 8,1 reduce 8 $ Factor - num * id 6,1 reduce 6 $ Term - num * id 3,1 reduce 3 $ Expr - num * id none shift $ Expr - num * id none shift $ Expr - num * id 7,3 reduce 7 $ Expr - Factor * id 6,3 reduce 6 $ Expr - Term * id none shift $ Expr - Term * id none shift $ Expr - Term * id 1. Shift until the top of the stack is the right end of a handle 2. Find the left end of the handle and reduce 0 Goal Expr 1 Expr Expr + Term 2 | Expr - Term 3 | Term 4 Term Term * Factor 5 | Term / Factor 6 | Factor 7 Factor number 8 | id 9 | ( Expr )
- 22. Back to x - 2 * y 5 shifts + 9 reduces + 1 accept Stack Input Handle Action $ id - num * id none shift $ id - num * id 8,1 reduce 8 $ Factor - num * id 6,1 reduce 6 $ Term - num * id 3,1 reduce 3 $ Expr - num * id none shift $ Expr - num * id none shift $ Expr - num * id 7,3 reduce 7 $ Expr - Factor * id 6,3 reduce 6 $ Expr - Term * id none shift $ Expr - Term * id none shift $ Expr - Term * id 8,5 reduce 8 $ Expr - Term * Factor 4,5 reduce 4 $ Expr - Term 2,3 reduce 2 $ Expr 0,1 reduce 0 $ Goal none accept 1. Shift until the top of the stack is the right end of a handle 2. Find the left end of the handle and reduce 0 Goal Expr 1 Expr Expr + Term 2 | Expr - Term 3 | Term 4 Term Term * Factor 5 | Term / Factor 6 | Factor 7 Factor number 8 | id 9 | ( Expr )
- 23. Goal <id,x> Term Fact. Expr – Expr <id,y> <num,2> Fact. Fact.Term Term * Stack Input Action $ id - num * id shift $ id - num * id reduce 8 $ Factor - num * id reduce 6 $ Term - num * id reduce 3 $ Expr - num * id shift $ Expr - num * id shift $ Expr - num * id reduce 7 $ Expr - Factor * id reduce 6 $ Expr - Term * id shift $ Expr - Term * id shift $ Expr - Term * id reduce 8 $ Expr - Term * Factor reduce 4 $ Expr - Term reduce 2 $ Expr reduce 0 $ Goal accept Back to x - 2 * y Corresponding Parse Tree
- 24. Conflicts During Shift-Reduce Parsing Conflicts “shift/reduce” or “reduce/reduce” Example: stmt if expr then stmt | if expr then stmt else stmt | other (any other statement) Stack Input if … then stmt else … Shift/ Reduce Conflict We can’t tell whether it is a handle
- 25. LR Parsing Bottom-up parser based on a concept called LR(k) parsing "L" is for left-to-right scanning of the input. "R" for constructing a rightmost derivation in reverse, “k” for the number of input symbols of lookahead that are used in making parsing decisions.
- 26. Why LR Parsers? For a grammar to be LR it is sufficient that a left-to-right shift-reduce parser be able to recognize handles of right-sentential forms when they appear on top of the stack.
- 27. Why LR Parsers? LR parsers can be constructed to recognize all programming language constructs for which context-free grammars can be written. The LR-parsing method is the most general non- back-tracking shift-reduce parsing method. An LR parser can detect a syntactic errors. The class of grammars that can be parsed using LR methods is a proper superset of the class of grammars that can be parsed with predictive or LL methods.
- 28. Components of LR Parser X Y Z $ A + B $ Parsing Program Parse Table Action Goto output stack Input buffer
- 29. Techniques for Creating Parse Table SLR: Construct parsing table for small set of grammars called SLR(1). Easy to develop. CLR(CANONICAL LR) : Most powerful. Generates a large parse table. More difficult develop. Works for all types of CFG May detect position of error. LALR(LOOK AHEAD LR) :Widely used method. Optimizes the size of parse table, by combining some states. Information may get lost.
- 30. SLR Parsers 1. Formation of augmented grammar G’ for the given grammar G 2. Construction of LR(0) collection of items. To find LR(0) collection of items Closure(I) and Goto(I,X) have to be computed. 3. Finding first and follow of non-terminals 4. Construction of parse table.
- 31. Formation of Augmented Grammar The augmented grammar G’, is G with a new start symbol S’ and an additional production S’ -> S E->E+T|T T->T*F|F The augmented grammar G’ is given by E’->E E->E+T|T T->T*F|F
- 32. Items and the LR(O) Automaton How does a shift-reduce parser know when to shift and when to reduce? How does the parser know that symbol on the top of the stack is not a handle? An LR parser makes shift-reduce decisions by maintaining states to keep track of all the operations. States represent sets of "items."
- 33. Items and the LR(O) Automaton An LR(O) item of a grammar G is a production of G, with a dot at some position in the body. Production A -> XYZ yields the four items A -> ·XYZ A -> X·YZ A -> XY·Z A -> XYZ· A ->ε generates only one item, A ->. An item indicates how much of a production we have seen at a given point in the parsing process.
- 34. Items and the LR(O) Automaton The item A -> ·XYZ indicates that we hope to see a string derivable from XYZ next on the input. Item A -> X·YZ indicates that we have just seen on the input a string derivable from X and that we hope next to see a string derivable from YZ. Item A -> XY·Z indicates that we have just seen on the input a string derivable from XY and that we hope next to see a string derivable from Z. Item A -> XYZ· indicates that we have seen the body XY Z and that it may be time to reduce XYZ to A
- 35. Items and the LR(O) Automaton Collection of sets of LR(0) items, called the canonical LR(0) collection. Provides the basis for constructing a deterministic finite automaton that is used to make parsing decisions. Automaton is called an LR(0) automaton. Each state of the LR(0) automaton represents a set of items in the canonical LR(0) collection.
- 36. Construction of LR(0) Items Items are viewed as states in NFA. Grouped to form same states. Process of grouping together is called subset Construction Algorithm. Closure and goto operations have to be computed.
- 37. Closure of Item Sets If I is a set of items for a grammar G, then CLOSURE(I) is the set of items constructed from I by the two rules: Initially, add every item in I to CLOSURE(I). If A -> α·Bγ is in CLOSURE(I) and B -> γ is a production, then add the item B -> γ to CLOSURE(I), if it is not already there. Apply this rule until no more new items can be added to CLOSURE (I).
- 38. Computation of CLOSURE E’ E E E + T | T T T * F | F F ( E ) | id I0 E’ .E E .E + T E .T T .T * F T .F F .( E ) F .id
- 39. Computation of CLOSURE SetOfltems CLOSURE (I) { J = I; repeat for ( each item A -> a·Bγ in J ) for ( each production B ->γ of G ) if (B ->.γ is not in J ) add B ->.γ to J; until no more items are added to J on one round; return J; }
- 40. Kernel items : The initial item, S' ->·S, and all items whose dots are not at the left end. Non-kernel items : All items with their dots at the left end, except for S' -> ·S.
- 41. The Function GOTO GOTO (I, X) is defined to be the closure of the set of all items [A -> αX.β] such that [A -> α.Xβ] is in I. If I is the set of two items { [E' -> E·] , [E -> E· + T] } , then GOTO(I, +) contains the items E -> E + ·T T -> ·T * F T -> ·F F -> · (E) F -> ·id
- 43. Grammar S E + S | E E num E num . S’ S . $ num ES’ . S $ S .E + S S . E E .num 1 S E . +S S E . 2 S E + S . 5 S E + . S S . E + S S . E E . num 3 S’ S $ . 7 4 S S $ + E num
- 44. num + $ E S 1 s4 g2 g6 2 s3 SE 3 s4 g2 g5 4 Enum Enum 5 SE+S 6 s7 7 accept
- 45. S' S S L=R S R L *R L id R L id S' S S L=R S R L *R L id R L L id S L =R R L S' S I0 I1 I2 I3 S R I4 L * R R L L id L * R I5 S L * id R S L= R R L L *R L id I6 = R S L=R R L L L I7 id I3 * * L *R R I8 I9
- 46. state action goto id = * $ S L R 0 s3 s5 1 2 4 1 accept 2 s6/r(RL) 3 r(Lid) r(Lid) 4 r(SR) 5 s3 s5 7 8 6 s3 s5 7 9 7 r(RL) r(RL) 8 r(L*R) r(L*R) 9 r(SL=R)
- 49. Structure of the LR Parsing Table 1. The ACTION function takes as arguments a state i and a terminal a (or $, the input endmarker). The value of ACTION[i, a] can have one of four forms: a) Shift j , where j is a state. The action taken by the parser shifts input a to the stack, but uses state j to represent a . b) Reduce A ->β. The action of the parser reduces β on the top of the stack to head A. c) Accept. The parser accepts the input and finishes parsing; d) Error. The parser discovers an error in its input and takes some corrective action. 2. Extend the GOTO function, defined on sets of items, to states: if GOTo [Ii , A] = Ij , then GOTO also maps a state i and a nonterminal A to state j .
- 50. Construction of SLR Parse Table 1. Construct C={I0,I1…….In} the collection sets of LR(0) items for G’. 2. Initial state of the parser is constructed from the set of items for [S’->S] 3. State I is constructed from Ii. The parsing actions are determined as follows. 1. If [A->α.aβ] is in Ii and GOTO(Ii,a]=Ij, then ACTION[i,a] is set to ‘shift j’ here ‘a’ must be a terminal. 2. If[A->α.] is in Ii, then ACTION[i,a] is set to reduce A->a for all ‘a’ Follow(A). 3. If S’->S is in Ii, than action[i, $] is set to ‘accept’.
- 51. Construction of SLR parsing table 4. GOTO transitions are constructed for all non-terminals. If GOTO(Ii,A) =Ij, then goto(i,A)of the parse table is set to j. 5. All other entries are error entries.
- 52. LR-Parser Configurations Helps to have complete state o its stack and the remaining input. A configuration of an LR parser is a pair:(s0s1 ………… sm, aiai+1…………an$). where the first component is the stack contents and the second component is the remaining input.
- 53. Behavior of the LR Parser
- 54. LR-parsing algorithm. INPUT: An input string w and an LR-parsing table with functions ACTION and GOTO for a grammar G. OUTPUT: If w is in L ( G), the reduction steps of a bottom-up parse for W ; otherwise, an error indication. METHOD: Initially, the parser has So on its stack, where So is the initial state, and w$ in the input buffer.