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Lecture 16 Oct 18. Context-Free Languages (CFL) - basic definitions Examples. Context-Free Languages (Ch. 2). Context-free languages allow us to describe non-regular languages like { 0 n 1 n | n  0}.

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lecture 16 oct 18
Lecture 16 Oct 18
  • Context-Free Languages (CFL) - basic definitions
  • Examples
context free languages ch 2
Context-Free Languages (Ch. 2)

Context-free languages allow us to describe non-regular languages like { 0n1n | n0}

General idea: CFL’s are languages that can be recognized by automata that have one stack:

{ 0n1n | n0} is a CFL

{ 0n1n0n | n0} is not a CFL

context free grammars
Context-Free Grammars

Start symbol S with rewrite rules:

1) S  0S1

2) S  e

S yields 0n1n :

S  0S1  00S11  …  0nS1n  0n1n

context free grammars def
Context-Free Grammars (Def.)
  • A context free grammar G=(V,,R,S) is defined by
  • V: a finite set variables (non-terminals)
  • : finite set terminals (with V=)
  • R: finite set of substitution rules V  (V)*
  • S: start symbol V
  • The language of grammar G is denoted by L(G):
  • L(G) = { w* | S * w }
derivation
Derivation *

A single step derivation “” consist of the substitution of a variable by a string accordingto a substitution rule.

Example: with the rule “ABB”, we can havethe derivation “01AB0  01BBB0”.

A sequence of several derivations (or none)is indicated by “ * ”Same example: “0AA * 0BBBB”

some remarks
Some Remarks

The language L(G) = { w* | S * w } contains only strings of terminals, not variables.

Notation: we summarize several rules, like A  B

A  01 by A  B | 01 | AA

A  AA

Unless stated otherwise: topmost rule has the start variable on the left side.

context free grammars ex
Context-Free Grammars (Ex.)

Consider the CFG G=(V,,R,S) with V = {S}  = {0,1} R: S  0S1 | 0Z1 Z 0Z | 

Then L(G) = {0i1j | ij and j > 0}

S yields0j+k1j according to:S  0S1  …  0jS1j  0jZ1j  0j0Z1j …  0j+kZ1j  0j+k1j = 0j+k1j

importance of cfl
Importance of CFL
  • Model for natural languages (Chomsky)
  • Specification of programming languages: “parsing of a computer program”
  • parser for HTML (and some special cases of SGML)
  • Describes mathematical structures.
  • Intermediate between regular languages andcomputable languages (Chapters 3,4,5 and 6)
set of boolean expressions is a cfl
Set of boolean expressions is a CFL

Consider the CFG G=(V,,R,S) with V = {S}  = {0,1,(,),,,} R: S  0 | 1 | (S) | (S)(S) | (S)(S)

Some elements of L(G): 0 (((0))(1)) (1)((0)(0))

Note: Parentheses prevent “100” confusion. This language requires full-parenthesizing.

a very small subset of english
A very small subset of English

Rules:<SENTENCE>  <NOUN-PHRASE><VERB-PHRASE> <NOUN-PHRASE>  <CMPLX-NOUN> | <CMPLX-NOUN><PREP-PHRASE><VERB-PHRASE>  <CMPLX-VERB> | <CMPLX-VERB><PREP-PHRASE> <CMPLX-NOUN>  <ARTICLE><NOUN> <CMPLX-VERB>  <VERB> | <VERB><NOUN-PHRASE> …

<ARTICLE>  a | the<NOUN>  boy | girl | house<VERB>  sees | ignores

A string that can be generated by this grammar:

the boy sees the girl

parse trees

(

)

S

(

)

S

0

S

(

)

(

)

S

0

1

Parse Trees

The parse tree of (0)((0)(1)) via rule

S  0 | 1 | (S) | (S)(S) | (S)(S):

S

ambiguity
Ambiguity

A grammar is ambiguous if some strings are derived ambiguously.

A string is derived ambiguously if it has more than one leftmost derivations or more than one parse tree.

Typical example: rule S  0 | 1 | S+S | SS S  S+S  SS+S  0S+S  01+S  01+1

versus

S  SS  0S  0S+S  01+S  01+1

ambiguity and parse trees

S

S

S

S

S

+

S

0

1

S

S

+

S

S

0

1

1

1

Ambiguity and Parse Trees

The ambiguity of 01+1 is shown by the two different parse trees:

more on ambiguity
More on Ambiguity

The two different derivations: S  S+S  0+S  0+1and

S  S+S  S+1  0+1 do not constitute an ambiguous string 0+1 (they will have the same parse tree)

Languages that can only be generated by ambiguous grammars are “inherently ambiguous”

context free languages
Context-Free Languages

Any language that can be generated by a contextfree grammar is a context-free language (CFL).

The CFL { 0n1n | n0 } shows us that certain CFLs are nonregular languages.

Q1: Are all regular languages context free?Q2: Which languages are outside the class CFL?

slide16

Example. A context-free grammar for the set of strings over {0,1} with an equal number of 0’s and 1’s.

We need a grammar for the language

L = { w | w has an equal number of 0’s and 1’s}

Consider the grammar:

S  0 S 1 | 1 S 0 | e

This does not cover all the strings.

Exhibit a string in L that is not generated by G.

slide17

We need to add the rule S  SS

The complete grammar is: S 0S1 | 1S0 | e | SS

How do we get a string like 011010?

S  SS  S S S  0S1SS  01SS 011S0S 

0110S  01101S0  011010

How do we show that L = L(G)?

We need the following

Lemma: Let x be a string in L. Then, (exactly) one of the following holds: (a) x = 0 y 1 for some y in L, (b) x = 1 y 0 for some y in L, or (c) x = yz for some y and z (both of them non-null) such that both y and z are in L.

chomsky normal form
Chomsky Normal Form

A context-free grammar G = (V,,R,S) is in Chomsky normal form if every rule is of the form

A  BC

or A  x

with variables AV and B,CV \{S}, and x 

For the start variable S we also allow the rule

S  

Advantage: Grammars in this form are far easier to analyze.

theorem 2 6
Theorem 2.6

Every context-free language can be describedby a grammar in Chomsky normal form.

Outline of Proof:We rewrite every CFG in Chomsky normal form.We do this by replacing, one-by-one, every rulethat is not in Chomsky Normal Form (CNF).

We have to take care of: Starting Symbol,  symbol, all other violating rules.

proof theorem 2 6
Proof Theorem 2.6

Given a context-free grammar G = (V,,R,S),rewrite it to Chomsky Normal Form by

1) New start symbol S0 (and add rule S0S)

2) Remove A rules (from the tail): before: BxAy and A, after: B xAy | xy

3) Remove unit rules AB (by the head): “AB” and “BxCy”, becomes “AxCy” and “BxCy”

4) Shorten all rules to two: before: “AB1B2…Bk”, after: AB1A1, A1B2A2,…, Ak-2Bk-1Bk

5) Replace ill-placed terminals “a” by Ta with Taa

careful removing of rules
Careful Removing of Rules

Do not introduce new rules that you removedearlier.

Example: AA simply disappears

When removing A rules, insert all newreplacements:

BAaA becomes B AaA | aA | Aa | a

example of cnf
Example of CNF

Initial grammar: S aSb | 

In Chomsky normal form:

S0   | TaTb | TaX X  STbS  TaTb | TaX Ta  a Tb  b

rl cfl
RL  CFL

Every regular language can be expressed by a context-free grammar.

Proof Idea:Given a DFA M = (Q,,,q0,F), we construct a corresponding CF grammar GM = (V,,R,S)with V = Q and S = q0

Rules of GM: qi x(qi,x) for all qiV and all xqi  for all qiF

example rl cfl

0

1

1

0

q1

q2

q3

0,1

Example RL  CFL

The DFA

leads to thecontext-free grammarGM = (Q,,R,q1) with the rules q1 0q1 | 1q2 q2 0q3 | 1q2 |  q3 0q2 | 1q2

picture thus far
Picture Thus Far

??

context-free languages

Regularlanguages

{ 0n1n }