MELJUN CORTES Automata Theory 8

8/13/2019 MELJUN CORTES Automata Theory 8

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CSC 3130: Automata theory and formal languages

Context-free languages

MELJUN P. CORTES MBA MPA BSCS ACS

MELJUN CORTES


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Context-free grammar

This is an a different model for describinglanguages

The language is specified by productions

(substitution rules) that tell how strings can be

obtained, e.g.

Using these rules, we can derivestrings like this:

A 0A1

A B

B #

A, Bare variables

0, 1, #are terminals

Ais the start variable

A 0A1 00A11 000A111 000B111 000#111


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Some natural examples

Context-free grammars were first used for naturallanguages

a girl with a flower likes the boy

SENTENCE

NOUN-PHRASE VERB-PHRASE

ART NOUN PREP ART NOUN VERB ART NOUN

CMPLX-NOUNCMPLX-NOUN CMPLX-NOUN

PREP-PHRASE

CMPLX-VERB

NOUN-PHRASE


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Natural languages

We can describe (some fragments) of the Englishlanguage by a context-free grammar:

SENTENCE NOUN-PHRASE VERB-PHRASE

NOUN-PHRASE CMPLX-NOUN

NOUN-PHRASE CMPLX-NOUN PREP-PHRASE

VERB-PHRASE CMPLX-VERBVERB-PHRASE CMPLX-VERB PREP-PHRASE

PREP-PHRASE PREP CMPLX-NOUN

CMPLX-NOUN ARTICLE NOUN

CMPLX-VERB VERB NOUN-PHRASE

CMPLX-VERB VERB

ARTICLE a

ARTICLE the

NOUN boy

NOUN girlNOUN flower

VERB likes

VERB touches

VERB sees

PREP with

variables: SENTENCE, NOUN-PHRASE,

terminals: a, the, boy, girl, flower, likes, touches, sees, with

start variable:SENTENCE


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Programming languages

Context-free grammars are also used to describe(parts of) programming languages

For instance, expressions like (2 + 3) * 5or

3 + 8 + 2 * 7can be described by the CFG

+

*

()

0 1

9

Variables:

Terminals:+, *, (, ), 0, 1, , 9


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Motivation for studying CFGs

Context-free grammars are essential forunderstanding the meaning of computer

programs

They are used in compilers

code:(2 + 3) * 5

meaning: add 2and 3, and then multiply by 5


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Definition of context-free grammar

A context-free grammar(CFG) is a 4-tuple(V, T, P, S)where

Vis a finite set of variablesor non-terminals

Tis a finite set of terminals(VT= )

Pis a set of productionsor substitution rulesof theform

whereAis a symbol in Vand a is a stringover VT Sis a variable inVcalled the start variable

A a


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Shorthand notation for productions

When we have multiple productions with thesame variable on the left like

we can write this in shorthandas

E E + E

E E * E

E (E)

E N

E E + E | E * E | (E) | 0 | 1

N 0N | 1N | 0 | 1

Variables:E, N

Terminals:+, *, (, ), 0, 1

Start variable: E

N 0N

N 1N

N 0

N 1


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Derivation

A derivationis a sequential application ofproductions:

E

derivation

E * E

(E) * E

(E) * N

(E + E ) * N

(E + E ) * 1

(E + N) * 1

(N + N) * 1

(N + 1N) * 1(N + 10) * 1

(1 + 10) * 1

a b

means bcan be obtained

from awith one production

a b*

means bcan be obtained

from aafter zero or more

productions


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Language of a CFG

The language of a CFG(V, T, P, S)is the set of allstrings containing only terminals that can bederived from the start variable S

This is a language over the alphabet T

A language Lis context-freeif it is the language

of some CFG

L = {| T* and S }*


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Example 1

Is the string 00#11in L?

How about 00#111, 00#0#1#11?

What is the language of this CFG?

A 0A1 | B

B #

variables:A, B

terminals: 0, 1, #

start variable:A

L= {0n#1n: n 0}


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Example 2

Give derivations of (), (()())

How about ())?

S SS | (S) | convention:variables in uppercase,

terminals in lowercase, start variable first

S (S) (rule 2)

() (rule 3)

S (S) (rule 2)

(SS) (rule 1)((S)S) (rule 2)

((S)(S)) (rule 2)(()(S)) (rule 3)(()()) (rule 3)


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Examples: Designing CFGs

Write a CFG for the following languages Linear equations over x, y, z, like:

x + 5yz = 911xy = 2

Numbers withoutleading zeros, e.g., 109, 0but not 019

The language L= {anbncmdm| n0, m0}

The language L= {anbmcmdn| n0, m0}


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Context-free versus regular

Write a CFG for the language (0 + 1)*111

Can you do so for everyregular language?

Proof:

S A111

A | 0A | 1A

Every regular language is context-free

regular

expressionDFANFA


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From regular to context-free

regular expression

a (alphabet symbol)

E1+ E2

CFG

E1E2

E1*

grammar with no rules

S

S a

S S1| S2

S S1S2

S SS1 |

In all cases, Sbecomes the newstart symbol


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Context-free versus regular

Is every context-free language regular? No! We already saw some examples:

This language is context-free but not regular

A 0A1 | BB # L= {0n#1n: n 0}


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Parse tree

Derivations can also be represented using parsetrees

E E + E

V + E

x + E

x + (E)

x + (E -E)

x + (V -E)x + (y -E)

x + (y -V)

x + (y -z)

E

E E+

V ( E )

E - E

V V

x

y z

E E + E | E - E | (E) | V

V x | y | z


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Definition of parse tree

A parse treefor a CFG Gis an ordered tree withlabels on the nodes such that

Every internal node is labeled by a variable

Every leaf is labeled by a terminal or

Leaves labeled by have no siblings

If a node is labeledAand has childrenA1, , Akfrom

left to right, then the rule

is a production in G.

A A1Ak


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Left derivation

Always derive the leftmost variablefirst:

Corresponds to a left-to-right traversalof parse

tree

E E + E

V + E

x + Ex + (E)

x + (E -E)

x + (V -E)

x + (y -E)

x + (y -V)

x + (y -z)

E

E E+

V ( E )

E - E

V V

x

y z


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Ambiguity

A grammar is ambiguousif some strings havemore than one parse tree

Example: E E + E | E -E | (E) | VV x | y | z

x + y + z

E

E E+

E E+VV Vx

y z

E

E E+

E E+ V

V V

x y

z


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Why ambiguity matters

The parse tree represents the intended meaning:

x + y + z

E

E E+

E E+VV Vx

y z

E

E E+

E E+ V

V V

x y

z

first add yand z,and then add this to x

first add xand y,and then add zto this


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Why ambiguity matters

Suppose we also had multiplication:E E + E | E -E | E E| (E) | VV x | y | z

x y + z

E

E E*

E E+V

V Vx

y z

E

E E+

E E V

V V

x y

z

first x y, then + zfirst y + z, then x


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Disambiguation

Sometimes we can rewrite the grammartoremove the ambiguity

Rewrite grammar so cannot be broken by +:

E E + E | E -E | E E | (E) | VV x | y | z

E T | E + T | E -T

T F | T FF (E) | V

V x | y | z

Tstands for term:x * (y + z)

Fstands for factor: x, (y + z)

A term always splits into factors

A factor is either a variable or a

parenthesized expression


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Disambiguation

Example

x y + z

E T | E + T | E -T

T F | T FF (E) | V

V x | y | z

V V V

F

T

FT

E

E

T

F


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Disambiguation

Can we always disambiguate a grammar?

No, for two reasons

There exists an inherently ambiguouscontext-free L:Every CFG for this language is ambiguous

There is no general procedure that can tell if a

grammar is ambiguous

However, grammars used in programming

languages can typically be disambiguated


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Another Example

Is ab, baba, abbbaain L?

How about a, bba?

What is the language of this CFG?

Is the CFG ambiguous?

S aB | bAA a | aS | bAA

B b | bS | aBB

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MELJUN CORTES Automata Theory 8