Context-Free Languages

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Context-Free Languages• Not all languages are regular.

• L1 = {anbn | n 0} is not regular.

L2 = {(), (()), ((())), ...} is not regular.

some properties of programming languages

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Context-Free GrammarsG = (V, T, S, P)Productions are of the form:

A xA V and x (V T)*

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Context-Free Grammars• A regular language is also a context-free

language.

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Context-Free Grammars• A regular language is also a context-free

language.

• Why the name?

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Example• G = ({S}, {a, b}, S, P)

P = { S aSa S bSb S }

S aSa aaSaa aabSbaa aabbaa

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Example• G = ({S}, {a, b}, S, P)

P = { S aSa S bSb S }

S aSa aaSaa aabSbaa aabbaa

L(G) = {wwR | w {a, b}*}

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Example• G = ({S, A, B}, {a, b}, S, P)

P = { S abB A aaBb B bbAa A }

S abB abbbAa abbbaaBba abbbaabbAaba abbbaabbaba

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Example• L = {anbm | n m} is context-free.

G = (?, {a, b}, S, ?)

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Example• L = {anbm | n m} is context-free.

G = (?, {a, b}, S, ?)

P = { S AS1 | S1B S1 aS1b |

A aA | a B bB | b }

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Example• G = ({S}, {a, b}, S, P)

P = { S aSb | SS | }

S aSb aaSbb aaSSbb aaabSbb aaababbb

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Example• G = ({S}, {a, b}, S, P)

P = { S aSb | SS | }

S aSb aaSbb aaSSbb aaabSbb aaababbb

L(G) = ?

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Derivations• G = ({S, A, B}, {a, b}, S, {S AB, A aaA, A , B Bb, B

}) 1 2 3 4 5

S AB aaAB aaB aaBb aab

S AB ABb aaABb aaAb aab

1 2 3 4 5

1 4 2 5 3

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Derivations• Leftmost: in each step the leftmost variable in

the sentential form is replaced.

• Rightmost: in each step the rightmost variable in the sentential form is replaced.

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Example• G = ({S, A, B}, {a, b}, S, {S AB, A aaA, A , B Bb, B

}) 1 2 3 4 5

S AB aaAB aaB aaBb aab leftmost

S AB ABb aaABb aaAb aab

1 2 3 4 5

1 4 2 5 3

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Derivation TreesA abABc

A

ba cBA

ordered tree

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Derivation Trees• Let G = (V, T, S, P) be a context-free grammar.

An ordered tree is a derivation tree iff:

1. The root is labeled S.2. Every leaf has a label from T {}.3. Every interior vertex has a label from V.4. A vertex has label AV and its children are labeled a1, a2, ..., an iff

P contains the production A a1 a2 ... an.5. A leaf labeled has no siblings.

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Derivation Trees• Let G = (V, T, S, P) be a context-free grammar.

An ordered tree is a partial derivation tree iff:

1. The root is labeled S.2. Every leaf has a label from T {}.

Every leaf has a label from V T {}.1. Every interior vertex has a label from V.2. A vertex has label AV and its children are labeled a1, a2, ..., an iff

P contains the production A a1 a2 ... an.5. A leaf labeled has no siblings.

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Derivation TreesThe string of symbols obtained by reading the leaves of a tree from left to right (omitting any 's encountered) is called the yield of the tree.

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Derivation Trees

S aABA bBbB A |

yield: abbbb

S

b

a

Bb

A

B

b b

A

B

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Derivation Trees

S aABA bBbB A |

S

b

a

Bb

A

B

b b

A

B

partial derivation tree

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Sentential Forms & Derivation Trees

• Let G = (V, T, S, P) be a context-free grammar. wL(G) iff there exists a derivation tree of G whose yield

is w.

If tG is a partial derivation tree of G whose root label is S, then the yield of tG is a sentential form of G.

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Membership and Parsing• Membership algorithm: tells if wL(G).

• Parsing: finding a sequence of productions by whichwL(G) is derived.

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Membership and ParsingS SS | aSb | bSa | w = aabb

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Exhaustive Search Parsing• Top-down parsing.

• S SS | aSb | bSa | w = aabb

1. S SS S SS SSS S aSb aSSb2. S aSb S SS aSbS S aSb aaSbb3. S bSa S SS bSaS S aSb abSab4. S S SS S S aSb ab

S aSb aaSbb aabb

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Exhaustive Search Parsing• It is not efficient.

• If w L(G) then it may never terminate, due to:

A B A

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Exhaustive Search ParsingSuppose G = (V, T, S, P) is a context-free grammar which does not have any rule of the form A B or A .

Then the exhaustive search parsing method can decide if wL(G) or not.

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Exhaustive Search ParsingSuppose G = (V, T, S, P) is a context-free grammar which does not have any rule of the form A B or A .

Then the exhaustive search parsing method can decide if wL(G) or not.

Proof: no more than |w| rounds are involved.

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Exhaustive Search ParsingSuppose G = (V, T, S, P) is a context-free grammar which does not have any rule of the form A B or A .

Then the exhaustive search parsing method can decide if wL(G) or not.

Proof: no more than |w| rounds are involved.

Complexity: |P||w|.

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Theorem• For every context-free grammar G, there exists an

algorithm that parses any wL(G) in a number of steps proportional to |w|3.

• Not satisfactory as linear time parsing algorithm (proportional to the length of a string)

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Simple Grammars (S-Grammars)

G = (V, T, S, P)Productions are of the form:

A axA V, a T, and x V*, and any pair (A,a) can occur in at most one rule.

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Simple Grammars (S-Grammars)

Any wL(G) can be parsed in at most |w| steps.

w = a1a2 ... an S a1A1A2 ... Am a1a2B1B2 ... BkA1A2 ... Am

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Simple Grammars (S-Grammars)

Many features of Pascal-like programming languages can be expressed with s-grammars.

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AmbiguityA context-free grammar G is said to be ambiguous if there exits some wL(G) that has at least two distinct derivation trees.

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AmbiguityS aSb | SS |

w = aabb

S

a

a b

b

S

S

S

a

a b

b

S

S

S

S

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Ambiguity• G = ({E, I}, {a, b, c, +, *, (, )}, E, P) w = a + b * c

E IE E + EE E * EE (E)

I a | b | c

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Ambiguity• G = ({E, T, F, I}, {a, b, c, +, *, (, )}, E, P) w = a + b * c

E T | E + TT F | T * FE E * EF I | (E)

I a | b | c

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AmbiguityIf L is a context-free language for which there exists an unambiguous grammar, then L is said to be unambiguous.

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AmbiguityIf L is a context-free language for which there exists an unambiguous grammar, then L is said to be unambiguous.

If every grammar that generates L is ambiguous, then L is called inherently ambiguous.

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Ambiguity• L = {anbncm} {anbmcm} is inherently ambiguous.

L = L1 L2

S S1 | S2

S1 S1c | A S2 aS2 | BA aAb | B bBc |

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CFGs and Programming Languages

• Important uses of formal languages: to define precisely a programming language. to construct an efficient translator for it.

• RLs are used for simple patterns, while CFLs for more complicated aspects.

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CFGs and Programming Languages

• S-grammars:<if-statement> ::= if <expression> <then_clause> <else_clause><then_clause> ::= then <statement><else_clause> ::= else <statement>

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Homework• Exercises: 2, 3, 4, 6, 7, 9, 15, 17, 22 of Section

5.1 - Linz’s book.

• Exercises: 1, 2, 5, 6, 8, 10, 11, 12, 13, 15 of Section 5.2 - Linz’s book.

• Exercises: 1, 2, 3 of Section 5.3 - Linz’s book.