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CS 1622 Lecture 10 1
CS1622
Lecture 10Parsing (5)
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LL(1) Parsing Table Example Left-factored grammar
E → T X X → + E | εT → ( E ) | int Y Y → * T | ε
The LL(1) parsing table:
εεε* TY( E )int YT
εε+ EXT XT XE
$)(+*int
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LL(1) Parsing Table Example(Cont.) Consider the [E, int] entry
“When current non-terminal is E and next input isint, use production E → T X
This production can generate an int in the firstplace
Consider the [Y,+] entry “When current non-terminal is Y and current token
is +, get rid of Y” Y can be followed by + only in a derivation in
which Y → ε
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LL(1) Parsing Tables. Errors Blank entries indicate error situations
Consider the [E,*] entry “There is no way to derive a string starting
with * from non-terminal E”
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Using Parsing Tables Method similar to recursive descent, except
For each non-terminal S We look at the next token a And chose the production shown at [S,a]
We use a stack to keep track of pending non-terminals - instead of procedure stack!
We reject when we encounter an error state We accept when we encounter end-of-input
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LL(1) Parsing Algorithminitialize stack = <S $> and nextrepeat case stack of <X, rest> : if T[X,*next] = Y1…Yn then stack ← <Y1… Yn
rest>; else error (); <t, rest> : if t == *next ++ then stack ← <rest>; else error ();until stack == < >
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LL(1) Parsing ExampleStack Input ActionE $ int * int $ T XT X $ int * int $ int Yint Y X $ int * int $ terminalY X $ * int $ * T* T X $ * int $ terminalT X $ int $ int Yint Y X $ int $ terminalY X $ $ εX $ $ ε$ $ ACCEPT
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Constructing Parsing Tables LL(1) languages are those defined by a
parsing table for the LL(1) algorithm No table entry can be multiply defined We want to generate parsing tables
from CFG
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Constructing Parsing Tables(Cont.) If A → α, where in the line of A we place α ? In the column of t where t can start a string
derived from α α →* t β We say that t ∈ First(α)
In the column of t if α is ε and t can follow anA S →* β A t δ We say t ∈ Follow(A)
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Computing First SetsDefinition First(X) = { t | X →* tα} ∪ {ε | X →* ε}Algorithm: First(t) = { t } ε ∈ First(X) if X → ε is a production ε ∈ First(X) if X → A1 … An
and ε ∈ First(Ai) for 1 ≤ i ≤ n First(α) ⊆ First(X) if X → A1 … An α
and ε ∈ First(Ai) for 1 ≤ i ≤ n
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First Sets. Example Recall the grammar
E → T X X → + E | ε T → ( E ) | int Y Y → * T | ε
First sets First( ( ) = { ( } First( T ) = {int, ( } First( ) ) = { ) } First( E ) = {int, ( } First( int) = { int } First( X ) = {+, ε } First( + ) = { + } First( Y ) = {*, ε } First( * ) = { * }
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Computing Follow Sets forNonterminals Follow(S) needed for productions that
generate the empty string ε Definition: Follow(X) = { t | S →* β X t δ }
Note that ε CAN NEVER BE IN FOLLOW(X)!! Intuition
If X → A B then First(B) ⊆ Follow(A) and Follow(X) ⊆ Follow(B) Also if B →* ε then Follow(X) ⊆ Follow(A) If S is the start symbol then $ ∈ Follow(S)
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Computing Follow Sets (Cont.)Algorithm sketch: $ ∈ Follow(S) For each production A → α X β
First(β) - {ε} ⊆ Follow(X)
For each production A → α X β where ε ∈First(β)Follow(A) ⊆ Follow(X)
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Follow Sets. Example Recall the grammar
E → T X X → + E | ε T → ( E ) | int Y Y → * T | ε
Follow sets
Follow( E ) = {), $}Follow( X ) = {$, ) }Follow( T ) = {+, ) , $}Follow( Y ) = {+, ) , $}
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Constructing LL(1) ParsingTables Construct a parsing table T for CFG G
For each production A → α in G do: For each terminal t ∈ First(α) do
T[A, t] = α If ε ∈ First(α), for each t ∈ Follow(A) do
T[A, t] = α If ε ∈ First(α) and $ ∈ Follow(A) do
T[A, $] = α
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Notes on LL(1) Parsing Tables If any entry is multiply defined then G is not
LL(1). Possible reasons: If G is ambiguous If G is left recursive If G is not left-factored Grammar is not LL(1)
Most programming language grammars arenot LL(1) Some can be made LL(1) though but others can’t
There are tools that build LL(1) tables E.g. LLGen
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Bottom-Up Parsing Bottom-up parsing is more general than top-
down parsing And just as efficient Builds on ideas in top-down parsing
Bottom-up is the preferred method in practice
Concepts first, algorithms next time
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An Introductory Example Bottom-up parsers don’t need left-factored
grammars
Hence we can revert to the “natural”grammar for our example:
E → T + E | TT → int * T | int | (E)
Consider the string: int * int + int
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Bottom-Up Parsing
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The IdeaBottom-up parsing reduces a string to the
start symbol by inverting productions:
EE → T + ET + EE → TT + TT → intT + intT → int * Tint * T + intT → intint * int + int
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Observation Read the productions in parse in
reverse (i.e., from bottom to top) This is a rightmost derivation!
EE → T + ET + EE → TT + TT → intT + intT → int * Tint * T + intT → intint * int + int
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Important Fact #1
Important Fact #1 about bottom-upparsing:
A bottom-up parser traces a rightmostderivation in reverse
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A Bottom-up Parse
ET + ET + TT + intint * T + intint * int + int E
T E
+ int*int
T
int
T
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A Bottom-up Parse in Detail(1)
+ int*int int
int * int + int
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A Bottom-up Parse in Detail(2)
int * T + intint * int + int
+ int*int int
T
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A Bottom-up Parse in Detail(3)
T + intint * T + intint * int + int
T
+ int*int int
T
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A Bottom-up Parse in Detail(4)
T + TT + intint * T + intint * int + int
T
+ int*int
T
int
T
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A Bottom-up Parse in Detail(5)
T + ET + TT + intint * T + intint * int + int
T E
+ int*int
T
int
T
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A Bottom-up Parse in Detail(6)
ET + ET + TT + intint * T + intint * int + int E
T E
+ int*int
T
int
T
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A Trivial Bottom-Up ParsingAlgorithmLet I = input string
repeatpick a non-empty substring β of I
where X→ β is a productionif no such β, backtrackreplace one β by X in I
until I = “S” (the start symbol) or allpossibilities are exhausted
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Questions Does this algorithm terminate?
How fast is the algorithm?
Does the algorithm handle all cases?
How do we choose the substring to reduce ateach step?
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Where Do Reductions HappenImportant Fact #1 has an interesting
consequence: Let αβω be a step of a bottom-up parse Assume the next reduction is by X→ β Then ω is a string of terminals
Why? Because αXω → αβω is a step in aright-most derivation
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Notation Idea: Split string into two substrings
Right substring is as yet unexamined by parsing(a string of terminals)
Left substring has terminals and non-terminals
The dividing point is marked by a | The | is not part of the string
Initially, all input is unexamined |x1x2 . . . xn
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Shift-Reduce ParsingBottom-up parsing uses only two kinds of
actions:
Shift
Reduce
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Shift Shift: Move | one place to the right
Shifts a terminal to the left string
ABC|xyz ⇒ ABCx|yz
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Reduce Apply an inverse production at the right
end of the left string If A → xy is a production, then
Cbxy|ijk ⇒ CbA|ijk
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The Example with ReductionsOnly
reduce T → intT + int |
E |reduce E → T + ET + E |reduce E → TT + T |
reduce T → int * Tint * T | + intreduce T → intint * int | + int
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The Example with Shift-Reduce Parsing
reduce T → intT + int |shiftT + | int
shiftint | * int + intshiftint * | int + int
shift|int * int + int
E |reduce E → T + ET + E |reduce E → TT + T |
shiftT | + intreduce T → int * Tint * T | + intreduce T → intint * int | + int
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A Shift-Reduce Parse inDetail (1)
+ int*int int↑
|int * int + int
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A Shift-Reduce Parse inDetail (2)
+ int*int int↑
int | * int + int|int * int + int
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A Shift-Reduce Parse inDetail (3)
+ int*int int↑
int | * int + intint * | int + int
|int * int + int
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A Shift-Reduce Parse inDetail (4)
+ int*int int↑
int | * int + intint * | int + int
|int * int + int
int * int | + int
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A Shift-Reduce Parse inDetail (5)
+ int*int int
T
int | * int + intint * | int + int
|int * int + int
int * T | + intint * int | + int
↑
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A Shift-Reduce Parse inDetail (6)
T
+ int*int int
T
int | * int + intint * | int + int
|int * int + int
T | + intint * T | + intint * int | + int
↑
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A Shift-Reduce Parse inDetail (7)
T
+ int*int int
T
T + | int
int | * int + intint * | int + int
|int * int + int
T | + intint * T | + intint * int | + int
↑
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CS 1622 Lecture 10 46
A Shift-Reduce Parse inDetail (8)
T
+ int*int int
T
T + int |T + | int
int | * int + intint * | int + int
|int * int + int
T | + intint * T | + intint * int | + int
↑
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A Shift-Reduce Parse inDetail (9)
T
+ int*int
T
int
T
T + int |T + | int
int | * int + intint * | int + int
|int * int + int
T + T |
T | + intint * T | + intint * int | + int
↑
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A Shift-Reduce Parse inDetail (10)
T E
+ int*int
T
int
T
T + int |T + | int
int | * int + intint * | int + int
|int * int + int
T + E |T + T |
T | + intint * T | + intint * int | + int
↑
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CS 1622 Lecture 10 49
A Shift-Reduce Parse inDetail (11)
E
T E
+ int*int
T
int
T
T + int |T + | int
int | * int + intint * | int + int
|int * int + int
E |T + E |T + T |
T | + intint * T | + intint * int | + int
↑
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The Stack Left string can be implemented by a stack
Top of the stack is the |
Shift pushes a terminal on the stack
Reduce pops 0 or more symbols off of thestack (production rhs) and pushes a non-terminal on the stack (production lhs)
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Key Issue (will be resolved byalgorithms) How do we decide when to shift or
reduce? Consider step int | * int + int We could reduce by T → int giving T | * int
+ int A fatal mistake: No way to reduce to the
start symbol E
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Conflicts Generic shift-reduce strategy:
If there is a handle on top of the stack, reduce Otherwise, shift
But what if there is a choice? If it is legal to shift or reduce, there is a
shift-reduce conflict If it is legal to reduce by two different productions,
there is a reduce-reduce conflict
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Source of Conflicts Ambiguous grammars always cause
conflicts
But beware, so do many non-ambiguous grammars
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Conflict ExampleConsider our favorite ambiguous
grammar:
int|(E)|E * E|
E + E→E
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One Shift-Reduce Parse
E |reduce E → E + EE + E |
. . .. . .reduce E → E * EE * E | + int
shift|int * int + int
reduce E → intE + int|shiftE + | intshiftE | + int
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Another Shift-Reduce Parse
E |reduce E → E * EE * E |
. . .. . .shiftE * E | + int
shift|int * int + int
reduce E → E + EE * E + E|reduce E → intE * E + int |shiftE * E + | int
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Example Notes In the second step E * E | + int we can either
shift or reduce by E → E * E
Choice determines associativity of + and * As noted previously, grammar can be
rewritten to enforce precedence Precedence declarations are an alternative
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Precedence DeclarationsRevisited Precedence declarations cause shift-reduce
parsers to resolve conflicts in certain ways Declaring “* has greater precedence than +”
causes parser to reduce at E * E | + int More precisely, precedence declaration is
used to resolve conflict between reducing a *and shifting a +
CS 1622 Lecture 10 59
Precedence DeclarationsRevisited (Cont.) The term “precedence declaration” is
misleading These declarations do not define
precedence; they define conflictresolutions Not quite the same thing!