Context-Free and Beyond

8/3/2019 Context-Free and Beyond

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Finishing up context-free

languagesReview

properties of cflsmaterial covered in Chapters 5-8


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Context-free Languages

Review

A language iscontext-free if it can be generatedby acontext-freegrammar (cfg) or recognized by

apushdownautomaton (pda) A context-free grammar is one in which every

production is of the form:X w

that is, of the form in which the left-hand side is asingle variable and the right-hand side is any stringof variables and terminals.


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Review

a cfg productionXxXygenerates from the

center and produces an equal number ofxs and

ys

the cfg productionsX xY, Y xYy will

generate morexs thanys.

the cfg productionsX aXa | bXb generatespalindromic strings.

x y is the same asx > y or x < y


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Review

Derivation trees

Derivations

Left-most and right-most derivations

Ambiguity (grammars)

Inherent Ambiguity (languages)

{an

bm

cm

} {an

bn

cm

} is INHERENTLY AMBIGUOUS! There has to be at least two ways to generateabc since it

could be in the language because theb's equal thec's, orbecause thea's equal theb's


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Review

rules (X )

unit rules

useless productions

in general, hard to remove because of cycles

(XY, YX). But in simple cases can be removed

by merely crossing them out and adding productions

of the formX w where there is a production of

the form Yw

we can get rid of a rule by crossing it out and adding

new rules with the erased variable (X) removed foreach case where it appears on a right-hand side

a variable is said to be useless if the variable can not be

generated from the start symbol orif no string containing only

terminal symbols can be generated from it. Any production

mentioning a useless variable anywhere in it is said to be

useless and can be removed.


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Review

Chomsky Normal Form

all rules are of the formX YZ whereX, YandZ are

all variables. (They dont have to be different variables)orXa. We also allow the rule S

I expect you to understand this and to be able to

transform simple cfgs into Chomsky Normal Form.

The technique is to create new variables wherever aterminal appears on the right-hand side and then to

break upX WYZ intoX VZ, V WY


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Review

Greibach Normal Form

all rules are of the formX aY...Zwherea is a

terminal and Y...Z is any string of zero or morevariables. We also allow the rule S

I expect you to understand this and to be able totransform simple cfgs into Greibach Normal Form.

This is generally hard to do (when the right-hand side

begins with a variable), but in simple cases, technique isto remove unit rules and add variables corresponding toany terminal on the right-hand side that is not the firstsymbol.


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Review

Pushdown automata

A pushdown automaton is merely a finite automaton

with a stack added to it. The stack allows for unbounded memorization.

The transition function maps (q,a,s) to a finite numberof possible pairs (q,s*). We also allowtransitions.The transition (q,a,s) (r, uts) means that if we are in

state q and the next input symbol isa and the symbolsis on top of the stack, then go into stater and replace thesymbols on the stack with the symbols u,t ands (one ata time); that is, push ut. The stack string read from leftto right is the same from top downward.


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Review

Pushdown automata

A string is accepted by a pda if after reading that string,

the pda could wind up in an accepting state. Thecontents of the stack is irrelevant. We always start the

stack with a special stack symbol z0 to avoid the

situation of having an empty stack and therefore cant

write a transition rule for the situation.


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Review

Pushdown automata

The language {anbmcn+m}

why does this work?

a, z0, xz0a, x, xx

, z0, z0

,x, x

b, z0, xz0

b, x, xx

, z0 , z0,x, x

c, x,

, z0 , z0


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Review Pushdown automata

The language {anbm | n m}

why does this work?

a, z0, xz0a, x, xx

, z0, z0

,x, x

b, x,

, x, xb, z0, z0

too manyas

too manybs


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Review

Pushdown automata

The language of balanced parentheses

why does this work?

(, z0, (z0

(, (, ((

), (,

, z0 , z0


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Review

Pushdown automata

Adeterministicpushdownautomaton (dpda) only

allows one transition for each triple (q,a,s) and if thereare transitions (q,,s)then there are no corresponding

non- transitions (q,c,s)for anyc

Not every context-free language can be recognized by a

deterministic push-down automaton. How would adpda generate the language {wwR}, the

language of unmarked palindromes? It cant. Why?


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Context-free grammars and Pushdown

Automata

Context-free grammars recognize exactly the

same class of languages as non-deterministic

pushdown automata.

Proof:

Consider any context-free grammar. There is an

equivalent Greibach Normal Form grammar. Construct

annpda that takes the form below:

,z0, Sz0


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Automata

Proof(continued):

Consider any context-free grammar. There is an

equivalent Greibach Normal Form grammar. For each

productionY

aX

1X

2...X

n, wheren 1

, add the labela, Y,X1X2...Xnto the loop on the middle state. For each

production Ya , adda, Y, to the transition going to

the accepting state. This machine accepts the same

language as the original context-free grammar!!!

,z0, Sz0


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Automata

Example:

Consider the grammar

SaXY

XaXB | aB YbYA | bA

Aa

Bb

L(G) = {aanbnbmam | m, n > 0}

a, S, XY

a, X, XB

a, X,B

b, Y, YA

b,Y, A

a, A,

b,B,

Consider aaabbba

S

,z0, Sz0

X

YA

B

z0

X

B


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Automata

Context-free grammars recognize exactly the

same class of languages as non-deterministic

pushdown automata.

Proof:

We leave the proof that any push-down automaton can

be converted into a context-free grammar to the

"reader."


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The Pumping Lemma for Context-Free

Languages

LetL be context-free. There exists some

integerm such that for all w inLwith |w| m,

w = uvxyz with |vxy|m and |vy|> 0 such that

uvixyizL for all i = 0, 1, 2, 3 ....


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Languages

This is similar to our previous Pumping

Lemma, except we are allowed to pump

equally in two different spots in the middle of

the string. These two spots must be within a

distance ofm from one another.

We use this Pumping Lemma for a very

similar reason: to show that a language is NOTcontext-free


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Languages

Show thatL = {anbncn: n 0} is not context-

free.

Proof: Assume thatL is context-free.

Then the Pumping Lemma applies.

Letmbe the magic number.

Consider the stringam

bm

cm

which is clearly inL and is longer thanm.

By the Pumping Lemma,


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Languages

aaaaa....aaaaabbb...bbbbbccc...ccccc

m m m

u v x y z


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Languages

ambmcmcan be written as uvxyz with |vxy| m and

|vy| > 0 such that uvixyiz is inL for all i = 0, 1, 2, 3 ....

There are only a limited number of possibilities for v

andy. v crosses the border between theas andbs (ory crosses

the border betweenbs andcs)

v is allas andy is allbs

v is allbs andyis allcs

etc. etc.


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Languages

Take each case separately. Clearly ifv ory

crosses a boundary uv2xy2z will not be of the

form as followed by bs followed by cs

But ifv is all a's theny can not contain c's. So

pumping up will leave too few c's.

Ifv is all b's then pumping up will leave too

few a's.

Therefore the language is not context-free.


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What are the properties of Context-Free

Languages?

There is an algorithm to determine if a given

Context-Free Language is empty.

Proof: Construct any grammar for the

language. Remove all useless productions. IfS

is found to be useless, then the language isempty; otherwise it is not.


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What are the properties of Context-FreeLanguages?

There is an algorithm to determine if a given

Context-Free Language is infiniteProof: LetGbe a grammar forL that has no

unit rules, productions or useless productions.Let Ghaven variables. For each variable,

consider all the derivations of lengthn startingwith that variable. If any sentential form in anyof these derivations contains its starting symbol,the language is infinite.


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Languages?

There exists an algorithm to determine if a

string belongs to a given context-free

language.

Proof: Construct a Greibach Normal Form

grammar. Construct the corresponding npda. Ifthe npda accepts the string the string is in the

language. If it doesn't, the string it isn't in the

language.


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Languages?

Context-free Languages are closed under

union.

Proof: LetG1 be a grammar forL1and G2 a

grammar forL2. We can modify one of these

to ensure that they have no variables in commonwithout affecting the language. Change the start

symbol ofG1 to S1, and the start symbol ofG2

to S2. Add S S1 | S2. Voila!


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Languages?

Context-free languages are not closed under

intersection.

Proof: Consider the following two grammars:

L1 = {anbncm, n > 0, m > 0}

L2 = {anbmcm, n > 0, m > 0}

L1 L2 = {anbncn, n > 0}. We will shortly show

that this language is not context-free.


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Review

Context-free languages

What kinds of languages are beyond the scopeof context-free languages?

anbncn

wwanbmcnm


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Homework

8.1/3, 7c and d, 8

8.2/12, 18

You are NOT responsible for any material

regarding linear languages.


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Where do we go from here?

From the earlier example, we see that not all

languages are context-free.

Is there a mathematical model for even more

complex computations?

There are many.

We will study one in particular: an automaton

called a Turing machine.


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Turing machines

A Turing machine (tm) is a finite automaton

with a twist:

The tape head can move in either direction (left

OR right)

A square on the input tape can be overwritten with

another symbol

By the way, there is no stack!


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Turing machines

Remember our finite automaton?

inputtape

on/offswitch accept

or

rejecttape read

head

"guts"of the

machine


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Turing machines

A Turing machine

inputtape

on/offswitch accept

or

rejecttape read

head

"guts"of the

machine

tape head

can move

left or right.

tape head

can read or

write!


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Turing machines

A Turing machine (tm) is a finite automaton

with a twist:

The tape head can move in either direction (left

OR right)

A square on the input tape can be overwritten with

another symbol

By the way, there is no stack!


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Turing machines

What can a Turing machine do? Clearly it can

accept any regular language. Just limit the

instructions to read from left to right and

never change the contents of the input tape.

Can it accept the language {anbn}? How?

How about {wcwR}?

How about {wwR}?

How about {ww}?


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Turing machines

Can it accept the language {anbn}? How?

Lay out a strategy

Let's mark off eacha and matchingb with anX

Let's work with thea's andb's from left to right

Whenever we mark off ana, move right until we find a

b. Mark off theb and head back left looking for the

left-most remaininga. Repeat as long as you can.

If there are no morea's, go to the right and check thatthere are no moreb's.

Let the machine die if there is no matchingb for ana.


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Turing machines

a, X, R

a, a, R

X, X, R

X, X, R

b, b, L

X, X, L

a, a, L

a, a, L

X, X, R

, , R

X, X, R

, , R

, , R

L = {anbn, n 0}


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Turing machines

Can it accept the language {wcwR}? How?

Lay out a strategy

Let's find the centerc.

Find the next unmarked symbol on the left. Head right,remembering whether it was an a or a b.

If the next unmarked symbol on the right matches, head

back to the middle and start again.

If from the centerc, there are no more unmarkedsymbols on the left, make sure there are no more on the

right as well.


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Turing machines

How about {wcwR}?

a,a,R

b,b,R

a,X,R c,c,R

X,X,R

a,X,LX,X,L

c,c,L

b,X,R

X,X,R

X,X,R

X,X,R

c,c,Rb,X,L

c,c,L

c,c,L

X,X,L

X,X,L

,,R

X,X,R

c,c,R

X,X,R

,,R


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Turing machines

How about {wwR}?

Describe in words how a Turing machine can

recognize this language.

Instead of going to the middle and working

outwards like the last example, start with the

outermost symbols and keep matching until

you hit the middle.


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Turing machines

How about {anbncn}?

YES!!!!!!!!!!!

consider our first machine and how it can be

extended to handle matchingcs as well.

Turing machines are more powerful thanpushdown automata!


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Turing machines

The Church-Turing Thesis:

If something can be computed it can be

computed by a Turing machine.

Note that this is called a Thesis, not a theorem.

It cant be proved, because the term can be

computed is too vague.

But it is universally accepted as a true statement.


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Turing machines

Given the Church-Turing Thesis:

What does this say about "computability"?

Are there things even a Turing machine can't do?

If there are, then there are things that simply can't

be "computed."

Not with a Turing machine

Not with your laptop Not with a supercomputer

There ARE things that a Turing machine can't do!!!

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Context-Free and Beyond