MELJUN CORTES Automata Theory 4

8/13/2019 MELJUN CORTES Automata Theory 4

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CSC 3130: Automata theory and formal languages

Properties of regular languages

MELJUN P. CORTES MBA MPA BSCS ACS

MELJUN CORTES


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Operations that preserve regularity

In the last lecture we saw three operations thatpreserve regularity:

Union: If L, Lare regular languages, so is L L

Concatenation: If L, Lare regular languages, so is LL

Star: If Lis a regular language, so is L*

Exercise: If Lis regular, is L4also regular?

Answer: Yes, because

L4= ((LL)L)L


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Example

The language Lof strings that end in 101isregular

How about the language Lof strings that do not

end in 101?

(0+1)*101


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Complement

The complementLof a language Lis the set ofall strings (over S) that are not in L

Examples (S= {0, 1}) L1= all strings that end in 101

L1=all strings that do not endin 101

= all strings end in 000, , 111or have length 0, 1, or

2

L2= 1* = {e, 1, 11, 111, }

L2= all strings that contain at least one 0

= (0 + 1)*0(0 + 1)*


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Closure under complement

If Lis a regular language, is Lalso regular? Previous examples indicate answer should be

yes

TheoremIf Lis a regular language, so is L.


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Proof of closure under complement

To prove this in general, we can use any of theequivalent definitionsfor regular languages:

In this proof DFA definitionwill be mostconvenient

We will assume L is accepted by a DFA, and show the

same forL

regularexpression

DFANFA


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Suppose Lis regular, then it is accepted by aDFA M

Now consider the DFA Mwith the accepting and

rejecting states ofMreversed


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Now for every input xS*:Maccepts x

After processing x,

Mends inan

accepting state

After processing x, Mends in

anrejecting state

Mrejects x

Language of Mis L

Lis regular


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A warning

NFA for language of strings ending in 101

Give NFA that accepts strings that do notend in101

1 0 1

0, 1

q0 q1 q2 q3

1 0 1

0, 1

q0

q1

q2

q3wrong!


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Intersection

The intersection LLis the set of strings thatare in both Land L

Examples:

If L, Lare regular, is LLalso regular?

L= (0 + 1)*111 L = 1* LL = ?

L= (0 + 1)*101 L = 1* LL = ?


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Closure under intersection

Theorem

Proof:

If L andLare regular languages, so is L

L.

Lregular Lregular

Lregular Lregular

L Lregular

But L L = LL

LL regular

LL regular.


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Reversal

The reversalwR

of a string wis wwrittenbackwards

The reversalLRof a languageLis the language

obtained by reversing all its strings

w= cave wR= evac

L= {push, pop} LR= {hsup, pop}


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Reversal of regular languages

L= all strings that end in 101is regular

How about LR?

This is the language of all strings beginningin 101

Yes, because it is represented by

(0+1)*101

101(0+1)*


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Closure under reversal

Theorem

Proof

We will use the representation of regular languages by

regular expressions

If Lis a regular language, so is LR.

regular

expressionDFANFA


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Proof of closure under reversal

If L is regular, then there is a regular expressionEthat describes it

We will give a systematic way of reversingE

Recall that a regular expression can be of thefollowing types:

Special expressions and e

Alphabet symbols a, b,

The union, concatenation, or star of simplerexpressions

In each of these cases we show how to do a

reversal


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Proof of closure under reversal

regular expressionE

e

a (alphabet symbol)

E1+ E2

reversalER

E1E2

E1*

e

a

E1R+ E2

R

E2RE1

R

(E1R)*


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Applications: text search

Text search: grep Looks for occurrences of a regular expressionin a file

Syntax of regular expressions in grep

[atr]means the set {a, t, r}

[b-e]means {b, c, d, e} (in ASCII/Unicode ordering)

|means +(union), *means star

? means zero or one: R?is e+ R

+meansone or more

: R+is RR*

{n}means ncopies of: R{5}is RRRRR


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Regular expressions in grep

Say we have a file w.txt

Want to know all sunny days> grep sunny w.txt

Any cloudy days in April 08?> grep 4/[0-9]*/08 ([0-9]|C| )* cloudy w.txt

Any consecutive string of 7 sunny days?> grep (([0-9]|/|C| )* sunny\n){7} w.txt

1/1/08 14C rain\n

1/2/08 17C sunny\n

1/3/08 18C sunny\n


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Implementation of grep

One way to implement grepis to convert theregular expression into a DFA and then run the

DFA on this text file

Two issues that arise:

grep looks for patterns inside text, while the DFA

processes the input as a whole

DFA only produces yes/noanswer, while grepoutputs linesthat contain given pattern

How would you resolve these issues?

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MELJUN CORTES Automata Theory 4