Formal Languages

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Formal Languages Simplifications of Context-Free
Languages Slides based on RPI CSCI 2400 Thanks to
Petros Drineas
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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
(other 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
(other 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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Conversion 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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exercise
Find CNF for this grammar S -gt 0A0 1B1 BB A
-gt C B -gt S A C -gt S epsilon (exercise
7.1.3, Hopcroft, Motwani, Ullman)
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Greibach Normal Form
All productions have form
symbol
variables
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examples
Greibach Normal Form
Not Greibach Normal Form
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Conversion to Greibach Normal Form
Greibach Normal Form
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Theorem
For any context-free grammar (which doesnt
produce ) there is an equivalent grammar in
Greibach Normal Form
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Observations

• Greibach normal forms are very good
• for parsing

• For many context-free grammars,
• it's hard to find the Greibach normal form.

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• Why?
• Try to find Greibach normal form
• for grammar in CNF exercise ...

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Compilers

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Machine Code
Program
Add v,v,1 cmp v,5 jmplt eLSe THeN add x,
12,v eLSe WHILe cmp x,3 ...
v 1 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)
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The following slides are courtesy of Professor
Papp, University of Debrecen.
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1,7


1,1
7,7
a a b b a b a
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S
S B,C
S A,D S D
S,B C B S
S A,S D S
D A A B,C B,C A B,C
A
a a b b a b a
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S
S B,C
S A,D S D
S,B C B S
S A,S D S
D A A B,C B,C A B,C
A
a a b b a b a
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S
S B,C
S A,D S D
S,B C B S
S A,S D S
D A A B,C B,C A B,C
A
a a b b a b a
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S
S B,C
S A,D S D
S,B C B S
S A,S D S
D A A B,C B,C A B,C
A
a a b b a b a
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S
S B,C
S A,D S D
S,B C B S
S A,S D S
D A A B,C B,C A B,C
A
a a b b a b a
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S
S B,C
S A,D S D
S,B C B S
S A,S D S
D A A B,C B,C A B,C
A
a a b b a b a
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S
S B,C
S A,D S D
S,B C B S
S A,S D S
D A A B,C B,C A B,C
A
a a b b a b a
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S
S B,C
S A,D S D
S,B C B S
S A,S D S
D A A B,C B,C A B,C
A
a a b b a b a
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S
S B,C
S A,D S D
S,B C B S
S A,S D S
D A A B,C B,C A B,C
A
a a b b a b a
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S
S B,C
S A,D S D
S,B C B S
S A,S D S
D A A B,C B,C A B,C
A
a a b b a b a
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S
S B,C
S A,D S D
S,B C B S
S A,S D S
D A A B,C B,C A B,C
A
a a b b a b a
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Word is in language if S appears here
S
S B,C
S A,D S D
S,B C B S
S A,S D S
D A A B,C B,C A B,C
A
a a b b a b a
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exercise
Parse baaba for this grammar S -gt AB BC A -gt
BA a B -gt CC b C -gt AB a (Hopcroft,
Motwani, Ullman, p301)