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CS453 Lecture Context Free Grammar Intro 1
Plan for This Today
Context Free Grammars – model for specifying programming languages – why not just use regular expressions? – example grammar – derivations
Parse trees
Syntax-directed translation – using syntax-directed translation to interpret MiniSVG
Top-down Predictive Parsing
Our Next Class of Languages
Regular Languages
}{ nnba }{ Rww
Context-Free Languages
**ba *)( ba +
Context-Free Languages
Pushdown Automata
Context-Free Grammars
stack
automaton
We will start here
Example
A context-free grammar :
!
S"aSbS"#
aabbaaSbbaSbS !!!
G
A derivation:
aaabbbaaaSbbbaaSbbaSbS !!!!Another derivation:
!"
"
SaSbS
=)(GL
(((( ))))
}0:{ !nba nn
Describes parentheses:
An Application of This Language Deriving another grammar
Regular Languages
}{ nnba }{ Rww
Context-Free Languages Can we derive a Grammar for:
Gave a grammar for:
!
S"aSaS"bSbS"#
abbaabSbaaSaS !!!
A context-free grammar : G
A derivation:
Example
abaabaabaSabaabSbaaSaS !!!!Another derivation:
!"
"
SaSbS
=)(GL
(((( ))))
}0:{ !nba nn
Describes parentheses:
Representing All Properly Nested Parentheses
Can we build a grammar to include any valid combination of ()? For example ( () (()) )
!
S"(S)S"SSS"#
!
S" SS" (S)S" ()S" ()
A context-free grammar : G
A derivation:
The Possible Grammar
!
S" SS" (S)S" ()S" ()(S)" ()()Another derivation:
Context-Free Grammars
Grammar
Productions of the form: xA!
String of symbols, Nonterminals and terminals
),,,( PSTVG =
Nonterminals Terminals Start symbol
Nonterminal
Derivation Order
ABS!.1
!
2. A"aaA3. A"#
!
4. B"Bb5. B"#
aabaaBbaaBaaABABS54321!!!!!
Leftmost derivation:
Given a grammar with rules:
Always expand the leftmost non-terminal
Derivation Order
aabaaAbAbABbABS32541!!!!!
Rightmost derivation:
Always expand the rightmost non-terminal
Given a grammar with rules:
ABS!.1
!
2. A"aaA3. A"#
!
4. B"Bb5. B"#
Leftmost derivation:
Rightmost derivation:
Stm --> id := Exp Exp --> num Exp --> ( Stm, Exp )
Grammar
a := ( b := ( c := 3, 2 ), 1 )
String
Stm ==> a := Exp ==> a := ( Stm, Exp ) ==> a := ( b := Exp, Exp ) ==> a := ( b := ( Stm, Exp ), Exp ) ==> a := ( b := ( c := Exp, Exp ), Exp ) ==> a := ( b := ( c := 3, Exp ), Exp ) ==> a := ( b := ( c := 3, 2), Exp ) ==> a := ( b := ( c := 3, 2), 1)
Stm ==> a := Exp ==> a := ( Stm, Exp ) ==> a := ( Stm, 1) ==>
ABS!
ABS!
!
A"aaA |#
!
B"Bb |#
S
BA
Parse Trees
aaABABS !!
a a A
S
BA
ABS!
!
A"aaA |#
!
B"Bb |#Parse Trees
aaABbaaABABS !!!S
BA
a a A B b
Parse Trees
ABS!
!
A"aaA |#
!
B"Bb |#
aaBbaaABbaaABABS !!!!S
BA
a a A B b
!
"
Parse Trees
ABS!
!
A"aaA |#
!
B"Bb |#aabaaBbaaABbaaABABS !!!!!
S
BA
a a A B b
!
"
!
"
ABS!
!
A"aaA |#
!
B"Bb |#
yield
!
aa""b= aab
Parse Trees
Sentential forms
ABS!
S
BA
Partial parse tree
ABS!
!
A"aaA |#
!
B"Bb |# aaABABS !!
S
BA
a a A
Partial parse tree
sentential form
aabaaBbaaBaaABABS !!!!!
aabaaAbAbABbABS !!!!!S
BA
a a A B b
Same parse tree
Sometimes, derivation order doesn’t matter
Leftmost:
Rightmost:
!
"
!
" CS453 Lecture Context Free Grammar Intro 22
Does it matter here?
a := ( b := ( c := 3, 2 ), 1 )
Stm --> id := Exp Exp --> num Exp --> ( Stm, Exp )
Grammar
String
Parse Tree
CS453 Lecture Context Free Grammar Intro 23
How about here?
42 + 7 * 6
(1) exp --> exp * exp (2) exp --> exp + exp (3) exp --> NUM
Grammar
String
CS453 Lecture Context Free Grammar Intro 24
Syntax Directed Translation or Interpretation
Use parse tree to … – Translate one language to another – Create a data structure of the program – Interpret, or evaluate, the program
Works conceptually by… – Parser discovers the parse tree – Parser executes certain actions while “traversing” the parse tree using a
depth-first, post-order traversal
CS453 Lecture Context Free Grammar Intro 25
MiniSVG Grammar
svg -> SVG_START elem_list SVG_END
elem_list -> elem elem_list | epsilon
elem -> RECT_START KW_X EQ NUM KW_Y EQ NUM KW_WIDTH EQ NUM KW_HEIGHT EQ NUM KW_FILL EQ COLOR ELEM_END
| CIRCLE_START KW_CX EQ NUM KW_CY EQ NUM KW_R EQ NUM KW_FILL EQ COLOR ELEM_END
| LINE_START KW_X1 EQ NUM KW_Y1 EQ NUM KW_X2 EQ NUM KW_Y2 EQ NUM KW_STROKE EQ COLOR ELEM_END
CS453 Lecture Context Free Grammar Intro 26
Example Parse Tree for MiniSVG
CS453 Lecture Context Free Grammar Intro 27
Interpret this program
a := ( b := ( c := 3, 2 ), 1 )
Stm --> id := Exp Exp --> num Exp --> ( Stm, Exp )
Grammar
String
Parse Tree
CS453 Lecture Context Free Grammar Intro 28
How about here?
42 + 7 * 6
(1) exp --> exp * exp (2) exp --> exp + exp (3) exp --> NUM
Grammar
String
grammar
Parser input string
derivation
Example:
Parser derivation
!
S"SSS"aSbS"bSaS"#
input
? aabb
Exhaustive Search
!
S"SS | aSb |bSa |#
Phase 1:
!
S" SSS" aSbS" bSaS"#
aabb
All possible derivations of length 1
Find derivation of aabb
!
S" SSS" aSbS" bSaS"#
Phase 2
aSbSSSS
!
!
aabb
SSSSbSaSSSSaSbSSSSSSSSSS
!!
!!
!!
!!
Phase 1
abaSbSabSabaSbSaaSbbaSbSaSSbaSbS
!!
!!
!!
!!
!
S"SS | aSb |bSa |# Phase 2
SSSSaSbSSSSSSSSSS
!!
!!
!!
aaSbbaSbSaSSbaSbS
!!
!!
Phase 3 aabbaaSbbaSbS !!!
aabb
!
S"SS | aSb |bSa |#
Final result of exhaustive search
Parser
derivation
input
aabb
aabbaaSbbaSbS !!!
(top-down parsing)
!
S" SSS" aSbS" bSaS"#
Time complexity of exhaustive search
Suppose there are no productions of the form
!
A"#
BA!
Number of phases for string : w ||2 w
Time for phase 1: k
k possible derivations
For grammar with rules k
Time for phase 2: 2k
possible derivations 2k
Time for phase : ||2wk
possible derivations ||2wk
||2 w
Total time needed for string : w
||22 wkkk +++ !
Extremely bad!!!
phase 1 phase 2 phase 2|w|
For general context-free grammars:
There exists a parsing algorithm that parses a string in time
||w3||w
For LL(1) grammars we can use Predictive parsing and parse in ||w
CS453 Lecture Context Free Grammar Intro 42
Example Predictive Parser
void start() { switch(m_lookahead) { case NUM: mesh(); match(Token.Tag.EOF); break; default: throw new ParseException(…); }} void mesh() { switch(this.m_lookahead) { case NUM: num_nodes = ((Num)m_lookahead).value; match(NUM); nodelist(); num_elem = ((Num)m_lookahead).value; match(NUM); elemlist(); break; default: throw new ParseException(…); }} void nodelist() { switch(m_lookahead) { case NUM: break; // nodelist -> epsilon case NODE: node(); nodelist(); break; // nodelist -> node nodelist default: throw new ParseException(…); }}
(1) start -> mesh EOF(2) mesh -> NUM nodelist NUM elemlist(3a & b) nodelist -> ϵ | node nodelist(4) node -> NODE NUM REAL REAL // node_id, x, y(5a & b) elemlist -> ϵ | elem elemlist(6a) elem -> TRI NUM NUM NUM NUM // elem_id, 3 node ids(6b) elem -> SQR NUM NUM NUM NUM NUM //elem_id,4 node ids
CS453 Lecture Context Free Grammar Intro 43
