Simplifications of Context-Free Grammars

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Simplifications of Context-Free Grammars

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A Substitution Rule
Equivalent grammar
Substitute
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A Substitution Rule
Substitute
Equivalent grammar
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In general
Substitute
equivalent grammar
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Nullable Variables
Nullable Variable
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Removing Nullable Variables
Example Grammar
Nullable variable
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Final Grammar
Substitute
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Unit-Productions
Unit Production
(a single variable in both sides)
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Removing Unit Productions
Observation
Is removed immediately
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Example Grammar
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Substitute
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Remove
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Substitute
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Remove repeated productions
Final grammar
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Useless Productions
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Another grammar
Not reachable from S
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contains only terminals
In general
if
then variable is useful
otherwise, variable is useless
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A production is useless if any of
its variables is useless
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Removing Useless Productions
Example Grammar
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First
find all variables that can produce strings with
only terminals
Round 1
Round 2
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Keep only the variables that produce terminal
symbols
(the rest variables are useless)
Remove useless productions
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Second
Find all variables reachable from
Use a Dependency Graph
not reachable
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Keep only the variables reachable from S
(the rest variables are useless)
Final Grammar
Remove useless productions
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Removing All

• Step 1 Remove Nullable Variables
• Step 2 Remove Unit-Productions
• Step 3 Remove Useless Variables

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Normal FormsforContext-free Grammars

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Chomsky Normal Form
Each productions has form
or
variable
variable
terminal
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Examples
Chomsky Normal Form
Not Chomsky Normal Form
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Convertion to Chomsky Normal Form

• Example

Not Chomsky Normal Form
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Introduce variables for terminals
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Introduce intermediate variable
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Introduce intermediate variable
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Final grammar in Chomsky Normal Form
Initial grammar
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In general
From any context-free grammar (which doesnt
produce ) not in Chomsky Normal Form
we can obtain An equivalent grammar
in Chomsky Normal Form
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The Procedure
First remove Nullable variables Unit
productions
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Then, for every symbol
Add production
In productions replace with
New variable
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Replace any production
with
New intermediate variables
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Theorem
For any context-free grammar (which doesnt
produce ) there is an equivalent grammar in
Chomsky Normal Form
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Observations

• Chomsky normal forms are good
• for parsing and proving theorems

• It is very easy to find the Chomsky normal
• form for any context-free grammar

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Greinbach Normal Form
All productions have form
symbol
variables
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Examples
Greinbach Normal Form
Not Greinbach Normal Form
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Conversion to Greinbach Normal Form
Greinbach Normal Form
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Theorem
For any context-free grammar (which doesnt
produce ) there is an equivalent grammar in
Greinbach Normal Form
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Observations

• Greinbach normal forms are very good
• for parsing

• It is hard to find the Greinbach normal
• form of any context-free grammar

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Compilers

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Machine Code
Program
Add v,v,0 cmp v,5 jmplt ELSE THEN add x,
12,v ELSE WHILE cmp x,3 ...
v 5 if (vgt5) x 12 v while (x !3)
x x - 3 v 10 ......
Compiler
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Compiler
Lexical analyzer
parser
input
output
machine code
program
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A parser knows the grammar of the programming
language
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Parser
PROGRAM STMT_LIST STMT_LIST STMT
STMT_LIST STMT STMT EXPR IF_STMT
WHILE_STMT
STMT_LIST EXPR EXPR EXPR EXPR - EXPR
ID IF_STMT if (EXPR) then STMT
if (EXPR) then STMT else STMT WHILE_STMT
while (EXPR) do STMT
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The parser finds the derivation of a particular
input
derivation
Parser
input
E gt E E gt E E E gt 10 EE gt
10 2 E gt 10 2 5
E -gt E E E E INT
10 2 5
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derivation tree
derivation
E
E gt E E gt E E E gt 10 EE gt
10 2 E gt 10 2 5

E
E
10
E
E

2
5
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derivation tree
E
machine code

E
E
mult a, 2, 5 add b, 10, a
10
E
E

2
5
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Parsing

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Parser
input string
derivation
grammar
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Example
Parser
derivation
input
?
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Exhaustive Search
Phase 1
Find derivation of
All possible derivations of length 1
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(No Transcript)
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Phase 2
Phase 1
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Phase 2
Phase 3
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Final result of exhaustive search
(top-down parsing)
Parser
input
derivation
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Time complexity of exhaustive search
Suppose there are no productions of the form
Number of phases for string
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For grammar with rules
Time for phase 1
possible derivations
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Time for phase 2
possible derivations
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Time for phase
possible derivations
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Total time needed for string
phase 1
phase 2w
phase 2
Extremely bad!!!
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There exist faster algorithms for specialized
grammars
S-grammar
symbol
string of variables
appears once
Pair
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S-grammar example
Each string has a unique derivation
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For S-grammars
In the exhaustive search parsing there is only
one choice in each phase
Time for a phase
Total time for parsing string
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For general context-free grammars
There exists a parsing algorithm that parses a
string in time
(we will show it in the next class)
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The CYK Parser
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The CYK Membership Algorithm
Input

• Grammar in Chomsky Normal Form

• String

Output
find if
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The Algorithm
Input example

• Grammar

• String

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(No Transcript)
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(No Transcript)
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(No Transcript)
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(No Transcript)
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Therefore
Time Complexity
Observation
The CYK algorithm can be easily converted to a
parser (bottom up parser)