1 context-free languages not all languages are regular. l 1 = {a n b n | n 0} is not regular. l 2 =...

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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 Grammars

G = (V, T, S, P)

Productions are of the form:

A x

A V and x (V T)*

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• A regular language is also a context-free language.

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• 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 Trees

A 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


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 Trees

The 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 aAB

A bBb

B A |

yield: abbbb

S

b

a

Bb

A

B

b b

A

B

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Derivation Trees

S aAB

A bBb

B A |

S

b

a

Bb

A

B

b b

A

B

partial derivation tree

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Sentential Forms & Derivation Trees


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 which

wL(G) is derived.

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Membership and Parsing

S 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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• It is not efficient.

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

A B

A

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Suppose 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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wL(G) or not.

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

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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 ax

A V, a T, and x V*, and any pair (A,a) can occur

in at most one rule.

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

w = a1a2 ... an

S a1A1A2 ... Am a1a2B1B2 ... BkA1A2 ... Am

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Many features of Pascal-like programming languages can

be expressed with s-grammars.

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Ambiguity

A 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 I

E E + E

E E * E

E (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 + T

T F | T * F

E E * E

F I | (E)

I a | b | c

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Ambiguity

If L is a context-free language for which there exists an

unambiguous grammar, then L is said to be unambiguous.

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Ambiguity

If 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 | B

A 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.

1 context-free languages not all languages are regular. l 1 = {a n b n | n 0} is not regular. l 2 =...

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s asa s bsb s

s asb ss

s asb s ss asbs s asb

s ss s ss sss s asb

s bsa s ss bsas s asb

b bb b

s asa aasaa aabsbaa

s as1 s1b s1 as1b