CSC 3130: Automata theory and formal languages Tutorial 4

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CSC 3130 Automata theory and formal
languagesTutorial 4

• KN Hung
• Office SHB 1026

Department of Computer Science Engineering
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Agenda

• Context Free Grammar (CFG)
• Design
• Parse Tree
• Cocke-Younger-Kasami (CYK) algorithm
• Parsing CFG in normal form
• Pushdown Automata (PDA)
• Design

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Context-Free Grammar (Recap)

• A context free grammar is consisted of

4) Start Variable
3) Production Rule
S ? AB ba A ? aA a B ? b
Another Production Rule
1) Variable
2) Terminal
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Context-Free Grammar (Recap)

• A string is said to belong to the language (of
  the CFG) if it can be derived from the start
  variable

Apply Production Rule
CFG Example
Derivation
S ? AB ba A ? aA a B ? b

• AB
• aAB
• aaB
• aab

S
Therefore, aab belongs to the language
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Why CFG?

• L w 0n1n n is an positive integer
• L is not a regular language
• Proved by Pumping Lemma
• A Context-Free Grammar can describe it
• Thus, CFG is more general than regular expression
• NFA ? Regular Expression ? DFA

S ? 0S1 S ? 01
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CFG Design

• Given a context-free language, design the CFG
• L ab-string, w Number of as lt Number of
  bs
• Some time for you to get into think 1 min

S ? ?
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CFG Design (Cont)

• Trial Bottom-up
• Shortest string in L b
• Given a string in L, we can expand it, s.t. it is
  still in L
• i.e., Add terminals, while not violating the
  constraints

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CFG Design (Cont)

• One Wrong Trial
• S ? b
• S ? bS Sb
• S ? abS baS bSa aSb

However, cannot parse strings like aabbbbbaa
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CFG Design (Cont)

• Approach 1
• S ? b
• S ? SS
• S ? SaS aSS SSa

But, is it sufficient to say the grammar is
correct?
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CFG Design (Cont)

• Approach 2
• Start with the grammar for ab-strings with same
  number of as and bs
• Call the start symbol of this grammar E
• Now, we generate all strings of type
• EbE EbEbE EbEbEbE
• Thus, we have the grammar

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CFG Design (Cont)

• Approach 2 (Cont)
• S ? EbET
• T ? bET e
• E ?

For the pattern EbE EbEbE
E generates ab-strings with same number of as
and bs (c.f. 09L7.pdf Slide 32)
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CFG Design (Cont)

• After designing the grammar, G, you may have to
  prove (if required) that the language of this
  grammar is equivalent to the given language
• i.e., Prove that L(G) L
• Proof
• Part 1) L(G) ? L
• Part 2) L ? L(G)
• Due to time limit, I will not do this part

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Parse Tree

• How to parse aab in this grammar? (Previous
  example)

CFG Example
S ? AB ba A ? aA a B ? b
S
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Parse Tree (Cont)

• Idea Production Rule Node Children
• Should be very intuitive to understand

Derivation

• AB
• aAB
• aaB
• aab

S
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Parse Tree (Cont)

• Ambiguity

3 - 1 - 2
String
CFG
3 1 2
3 (1 2)
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Parse Tree (Cont)

• Useful in programming language
• CSC3180
• Useful in compiler
• CSC3120

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Cocke-Younger-Kasami Algorithm

• Used to parse context-free grammar in Chomsky
  normal form (or simply normal form)

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CYK Algorithm - Idea

• Algorithm 2 in Lecture Note (09L8.pdf)
• Idea Bottom Up Parsing
• Algorithm
• Given a string s of length N
• For k 1 to N
• For every substring of length k
• Determine what variable(s) can derive it
• sub(x,y) starts at index x, ends at index y

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CYK Algorithm - Init

• Base Case k 1
• The possible choices of variable(s) can be known
  by scanning through each production

B
A,C
A,C
B
A,C
a
b
a
b
a
We want to parse this string
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CYK Algorithm Table

• Each cell Variables deriving the substring

a
b
a
b
a
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CYK Algorithm Loop (kgt1)

• When k 2
• Example
• sub(1,2) ba
• ba b a sub(1,1) sub(2,2)

• Possible
• BA BC
• ? Variable A,S
• Since A?BA, S?BC

S,A
B
A,C
A,C
B
A,C
a
b
a
b
a
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CYK Algorithm Loop (kgt1)

• For each substring
• Decompose into two substrings
• Example
• sub(2,4) aab

• Possible
• AS, AC, CS, CC

, BB
a
b
a
b
a
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CYK Algorithm Loop (kgt1)

• How about sub(3,5) ?
• Give you 1 min

S,A
B
S,C
S,A
B
A,C
A,C
B
A,C
a
b
a
b
a
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CYK Algorithm Parse Tree

• Parse Tree is known from the table
• See 09L8.pdf - Slide 21

Length of Substring
a
b
a
b
a
Start Index of Substring
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CYK Algorithm (Conclusion)

• Start from shortest substring to the longest
• i.e., from single-character-string to the
  whole string
• For Context-free grammar, G
• 1) Convert G into normal form
• Remove e-productions
• Remove unit-productions
• 2) Apply CYK algorithm
• Con Loss in intuition

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End

• Thanks for coming!
• Any questions?