Lesson 2

CDT301 – Compiler Theory, Spring 2011Teacher: Linus Källberg

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Outline

• Context-free languages– Context-free grammars– Parse trees– Push down automata

Grammars

S → a BB → εB → b B

• Regex: ab*• Derivation of “abb”:

S a B a b B a b b B a b b⇒ ⇒ ⇒ ⇒

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Parse trees

S a B⇒ a b B⇒ a b b B⇒ a b b⇒

S

B

B

B

ε

a

b

b

Parse trees

S → a BB → εB → b B

String: “abb”

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S

Parse trees

S → a BB → εB → b B

String: “abb”

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S

Ba

Parse trees

S → a BB → εB → b B

String: “abb”

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S

B

B

a

b

Parse trees

S → a BB → εB → b B

String: “abb”

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S

B

B

B

a

b

b

Parse trees

S → a BB → εB → b B

String: “abb”

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S

B

B

B

ε

a

b

b

Parse trees

S → a BB → εB → b B

String: “abb”

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S

B

B

B

εa b b

Exercise (1)

• Write grammars that producea) strings over { a, b } that start and end with the

same letter. Draw the parse tree for “abaaa”.b) palindromes over { a, b }. Draw the parse tree

for “ababa”.c) valid e-mail addresses. Draw the parse tree

for “ab@c.d”. Try to derive the string “a@b@c” (this should not be possible).

• Recall: S+(.S+)*@S+(.S+)* 11

PDA

• Push down automata• FA + stack

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PDA example

• Language: { (), (()), ((())), … }

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Exercise (2)

• Write grammars fora) general parenthesis expressions:

{ ε, (), ()(), (()), ()(()), ((()))()(), … }.b) simple infix addition between single-digit

numbers, with support for prioritizing with parentheses:{ 7, 1 + 2, 1 + (9 + 1), ((6) + 0), … }.

c) Create PDAs for the above languages.d) Create a PDA that accepts palindromes.

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Conclusion

• Context-free languages– Context-free grammars– Parse trees– Push down automata

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Next time

• Specifying language syntax using CFGs• Introduction to parsing• Ambiguous grammars

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