COSC 3340: Introduction to Theory of Computation

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COSC 3340 Introduction to Theory of Computation

• University of Houston
• Dr. Verma
• Lecture 9

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Context Free Languages

• Strictly bigger class than regular languages

anbn n gt 0
context-free languages
regular languages
?
?
?
wwR w in a,b
arithmetic expressions
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A simple grammar for some sentences

• ltSentencegt ? ltnoungt ltverbgt ltobjectgt
• ltnoungt ? Alice John
• ltverbgt ? eats eat ate
• ltobjectgt ? apple orange mango
• The goal is to generate sentences in English over
  the English alphabet. Example
• ltSentencegt ? ltnoungtltverbgtltobjectgt ? Alice
  ltverbgtltobjectgt ?
• Alice eatsltobjectgt ? Alice eats apple

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Context-free grammars (CFGs)

• Informally, a CFG is a finite set of rules.
• Each rule is of the form
• ltnonterminal symbolgt ? string over terminals and
  nonterminals.
• Terminals- symbols that the desired strings
  should contain.
• Example a...z, ' ',...
• Nonterminals- symbols to which rules can be
  applied.
• Example ltnoungt, ltverbgt, ...
• A special nonterminal is called start symbol.
• Example ltSentencegt

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A grammar for arithmetic expressions

• E ? E E E E (E) x y
• Start symbol - E.
• Terminals?
• Ans x, y, , , (, ), ' '
• Nonterminals?
• Ans E
• A derivation
• E ? E E ? E E E ? x E E ? x x E ? x
  x y

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Another grammar for arithmetic exprs

• E ? E T T
• T ? T F F
• F ? (E) x y
• A derivation for x x y ?
• E ?E T ?T T ?F T ?x T ?
• x T F ?x F F ?x x F ?x x y
• Why two different grammars for arithmetic
  expressions?

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Context Free Grammar Definition

• A CFG G (V, T, P, S) where V ? T ?,
• V -- A finite set of symbols called nonterminals
• T -- A finite set of symbols called terminals
• P is a finite subset of V X (V ? T) called
  productions or rules
• We write A ? w whenever (A, w) ? P.
• S ? V -- start symbol

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Derivations and L(G)

• One step derivation
• u ? v if u xAy, v xwy and A ? w in P
• 0 or more steps derivation
• u ? v if u u0 ? u1 ? .... ? un v (n ? 0)
• L(G) w in T S ? w .
• A language L is context-free if there is a CFG G
  with L(G) L

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Example

• S ? aSb ?
• Derivation
• S ? aSb ? aaSbb ? aabb.
• L(G) ?
• Ans anbn n ? 0

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Parse trees

• All derivations can be shown in the form of
  trees.
• Order of rule application is lost.

S
a
b
S
a
b
S
?
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Parse trees contd.

• In general, if we apply rule
• A ? w0w1...wn, then we add nodes for wi as
  children of node labeled A

A
w0
w1
w2
wn
? ? ?
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E ? E E ? E E E ? x E E ? x x E ? x
x y
E
E
E

x
E

E
x
y
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E ?E T ?T T ?F T ?x T ?x T F ?x F
F ?x x F ?x x y
E
E
T

F
T

T
F
y
F
x
x
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Leftmost and Rightmost Derivations

• Derivation is leftmost if the nonterminal
  replaced in every step is the leftmost
  nonterminal.
• Consider E ? E E ? E x.
• Is it leftmost derivation?
• Derivation is rightmost if the nonterminal
  replaced in every step is the rightmost
  nonterminal.
• Consider E ? E E ? x E.
• Is it rightmost derivation?

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Ambiguity

• A CFG is ambiguous if there is a string with at
  least two leftmost derivations
• Example
• E ? E E E E (E) x y is
  ambiguous
• A CFL is inherently ambiguous if every CFG that
  generates it is ambiguous.
• Example
• anbncm n, m ? 0 ? ambncn n, m ?
  0