Computation, Computers, and Programs CFGPDA

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CS20a summary (Oct 24, 2002)

• Context-free languages
• Grammars G (V, T, P, S)
• Pushdown automata
• N-PDA CFG
• D-PDA lt CFG
• Today
• What languages are context-free?
• Pumping lemma (similar to pumping lemma for
  regular languages)
• Ogdens lemma

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Regular languages

• Intuition if a FA accepts a string that is long
  enough, it must repeat a state
• But it cant remember that the state was repeated
• So it can be forced to repeat the state over and
  over

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Pumping lemma for regular languages
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Pumping lemma for CFL
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Pumping lemma

• Suppose L is in Chomsky Normal Form
• A ? a
• A ? BC

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Derivation trees
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Size of derivation trees
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Pumping
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Pumping example 1
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Pumping lemma is too weak
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Ogdens lemma
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Building a path
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Building the path
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Branch points
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An example (part 1)
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An example (part 2)
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Closure properties
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Intersection
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Complement
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Intersection with regular languages
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Intersection with regular languages
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Determining if a string is in a language
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CYK algorithm
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LR(k) Parsing

• Left-to-right parsing, rightmost derivation, k
  tokens lookahead
• Stack machine
• Example
• S -gt E
• E -gt number
• E -gt E E
• E -gt E E

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Parsing 1 2 3
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LR(0)

• How do we build the parser table?
• New grammar
• S -gt E E -gt numberE -gt ( L )L -gt EL -gt L E

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LR(0) parsing

• Build a DFA
• States are sets of productions position info
• state 1S -gt . S E -gt . numberE -gt . ( L )
• Actions
• shift t add token t to the stack
• reduce i apply the ith production to the stack
• goto i goto state i

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LR(0) DFA
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Operations

• closure state -gt state
• goto state -gt symbol -gt state

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Goto

• goto I X moves the . past the X symbol

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Building the table
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Building the table

• If E contains I -gtX J
• if X is a terminal, add shift X to state I
• if X is a non-terminal, add goto J to state I
• If I contains A -gt alpha . add reduce A -gt
  alpha to state I\

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LR(1) parsing

• An LR(1) item contains
• a production
• the position (represented by a dot)
• a lookahead token

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FIRST function

• Returns the set of terminals that may begin a
  production

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Closure function

• Modified to get the next lookahead symbol

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Goto function

• Modified for LR(1) items

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An example

• E -gt number
• E -gt E E
• E -gt E E

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LR(1) state diagram
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Disambiguating expressions

• E -gt number E E E E ( E )
• Inherently ambiguous
• Rewrite into a two-level grammar
• E general expression
• E -gt T E T
• T pure multiplication
• T -gt S T S
• S simple expression
• S -gt number ( E )

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How can we disambiguate ifthenelse?

• E -gt number
• E -gt E E E E ( E )
• E -gt if E then E
• E -gt if E then E else E

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Matched expression

• E -gt M U
• M -gt number M M M M ( E )
• M -gt if E then M else M
• U -gt if E then U
• U -gt if E then M else U