minimization of dfa

Pumping Lemma?Regular or not?

Minimization of FA

Examples on Pumping Lemma andMinimization of DFA

Zhao [email protected]

Department of Computer Science & EngineeringThe Chinese University of Hong Kong

September 28, 2008

Zhao Qiao [email protected] Examples on Pumping Lemma and Minimization of DFA


Minimization of FA

Outline

1 Pumping Lemma?Adversary ArgumentExplanationExamples

2 Regular or not?General MethodExamples

3 Minimization of FAExample



Minimization of FA

Adversary ArgumentExplanationExamples

Adversary Argument

Pumping Lemma

L is regular ⇒ (∃n)(∀z)(

z ∈ L, |z| ≥ n ⇒

(∃u, v , w)((z = uvw , |uv | ≤ n, |v | ≥ 1) and (∀i)uv iw ∈ L

))m

Adversary Argument

L is not regular ⇐ (∀n)(∃z)(

z ∈ L, |z| ≥ n,

(∀u, v , w)((z = uvw , |uv | ≤ n, |v | ≥ 1) ⇒ (∃i)uv iw ∈/L

))



Minimization of FA


Explanation

Using the adversary argument,we can verify a non-regularlanguage L by the following game:

Game Proofthe adversary pick an arbitrary n to challenge us for astring z.

? we construct a special string z in L with length greater thanor equal to n.the adversary arbitrarily break z into u, v and w ,where v isnot empty and uv ’s length less or equal to n.

? if we can always choose a i to show him that uv iw is not inL,then we win.



Minimization of FA


Palindromes over {a, b}

{wwR|w ∈ {a, b}∗

}J the adversary pick an arbitrary n to challenge us for a

string z.

I u,v only contain a; w contains a trailing substring bban,andmaybe some leading a’s.If we set i = 0(pump v out),thenuv iw = uw will have less leading a’s than its trailing a’s,souw is not a palindrome.,

In fact,we can choose any i other than 1.



Minimization of FA



{wwR|w ∈ {a, b}∗


string z.I? How to choose z in L?The following moves will mess with

the first n symbols of our z,and we have to make sure theoutcome is not in L.





Minimization of FA



{wwR|w ∈ {a, b}∗


string z.I we choose z = anbban





Minimization of FA



{wwR|w ∈ {a, b}∗


string z.I we choose z = anbban

J the adversary arbitrarily break z into u, v and w ,where v isnot empty and uv ’s length less than or equal to n.





Minimization of FA


Twin strings over over {a, b}

{ww |w ∈ {a, b}∗


string z.

I we choose z = anbanbJ the adversary arbitrarily break z into u, v and w ,where v is

not empty and uv ’s length less than or equal to n.I u,v only contain a; w contains a trailing substring banb,and

maybe some leading a’s.If we set i = 0(pump v out),thenuv iw = uw will have less leading a’s before the first b thanits a’s between 2 b’s,so uw is not a twin string.,

can we choose other i ’s to win?



Minimization of FA



{ww |w ∈ {a, b}∗


string z.I we choose z = anbanb

J the adversary arbitrarily break z into u, v and w ,where v isnot empty and uv ’s length less than or equal to n.

I u,v only contain a; w contains a trailing substring banb,andmaybe some leading a’s.If we set i = 0(pump v out),thenuv iw = uw will have less leading a’s before the first b thanits a’s between 2 b’s,so uw is not a twin string.,




Minimization of FA



{ww |w ∈ {a, b}∗


string z.I we choose z = anbanbJ the adversary arbitrarily break z into u, v and w ,where v is

not empty and uv ’s length less than or equal to n.

I u,v only contain a; w contains a trailing substring banb,andmaybe some leading a’s.If we set i = 0(pump v out),thenuv iw = uw will have less leading a’s before the first b thanits a’s between 2 b’s,so uw is not a twin string.,




Minimization of FA



{ww |w ∈ {a, b}∗


string z.I we choose z = anbanbJ the adversary arbitrarily break z into u, v and w ,where v is

not empty and uv ’s length less than or equal to n.I u,v only contain a; w contains a trailing substring banb,and

maybe some leading a’s.If we set i = 0(pump v out),thenuv iw = uw will have less leading a’s before the first b thanits a’s between 2 b’s,so uw is not a twin string.,




Minimization of FA

General MethodExamples

General Method

To prove a language to be regular,we can use regularexpression,DFA,NFA or ε- NFA to construct it directly.

We can also use the closure properties of regularlanguages: union,concatenation,Kleene closure,complement,intersection,substitution(quotient).To prove a language to be non-regular,we can usepumping lemma and the closure properties of regularlanguages.



Minimization of FA


General Method

To prove a language to be regular,we can use regularexpression,DFA,NFA or ε- NFA to construct it directly.We can also use the closure properties of regularlanguages: union,concatenation,Kleene closure,complement,intersection,substitution(quotient).

To prove a language to be non-regular,we can usepumping lemma and the closure properties of regularlanguages.



Minimization of FA


General Method

To prove a language to be regular,we can use regularexpression,DFA,NFA or ε- NFA to construct it directly.We can also use the closure properties of regularlanguages: union,concatenation,Kleene closure,complement,intersection,substitution(quotient).To prove a language to be non-regular,we can usepumping lemma and the closure properties of regularlanguages.



Minimization of FA


Q1

L is a regular language over {a,b,c},Deicide whether thefollowing languages are regular.

Problemsa {w |w ∈ L, a ∈/w}

b {waw |w ∈ L}c {uv |u ∈ L, v ∈/L}

Hints

a concatenation & complementb like palindromes→pumping

lemmac concatenation & complement



Minimization of FA


Q1


Problemsa {w |w ∈ L, a ∈/w}

b {waw |w ∈ L}c {uv |u ∈ L, v ∈/L}

Hintsa concatenation & complement

b like palindromes→pumpinglemma

c concatenation & complement



Minimization of FA


Q1


Problemsa {w |w ∈ L, a ∈/w}b {waw |w ∈ L}

c {uv |u ∈ L, v ∈/L}

Hintsa concatenation & complement

b like palindromes→pumpinglemma




Minimization of FA


Q1


Problemsa {w |w ∈ L, a ∈/w}b {waw |w ∈ L}

c {uv |u ∈ L, v ∈/L}

Hintsa concatenation & complementb like palindromes→pumping

lemma




Minimization of FA


Q1


Problemsa {w |w ∈ L, a ∈/w}b {waw |w ∈ L}c {uv |u ∈ L, v ∈/L}


lemma




Minimization of FA


Q1


Problemsa {w |w ∈ L, a ∈/w}b {waw |w ∈ L}c {uv |u ∈ L, v ∈/L}


lemmac concatenation & complement



Minimization of FA


Q2

Prove that the following languages are non-regular.

Problemsa all strings over {a, b} with the

same number of a’s and b’s.

b all strings over (, ) in which theparentheses are paired.

c all strings over {a, b} in whichthe number of a’s is a perfectcube.

d all non-palindromes over {a,b}.

Hints

a anbn

b (n)n

c n < (n + 1)3 − n3

d closure property ofcomplement



Minimization of FA


Q2



same number of a’s and b’s.

b all strings over (, ) in which theparentheses are paired.



Hintsa anbn

b (n)n

c n < (n + 1)3 − n3




Minimization of FA


Q2



same number of a’s and b’s.b all strings over (, ) in which the

parentheses are paired.



Hintsa anbn

b (n)n

c n < (n + 1)3 − n3




Minimization of FA


Q2



same number of a’s and b’s.b all strings over (, ) in which the

parentheses are paired.c all strings over {a, b} in which

the number of a’s is a perfectcube.


Hintsa anbn

b (n)n

c n < (n + 1)3 − n3




Minimization of FAExample

Minimization of FA

initial mark

bcdef X X X X X

g X X X X Xa b c d e f

mark final and non-final pair




Minimization of FA

mark ab

b Xcdef X X X X X


δ(a, 0) = c, δ(b, 0) = f




Minimization of FA

mark ac

b Xc cd,bedef X X X X X


δ(a, 0) = c, δ(c, 0) = d

δ(a, 1) = b, δ(c, 1) = e




Minimization of FA

mark ad

b Xc cd,bed cd,beef X X X X X


δ(a, 0) = c, δ(d , 0) = d

δ(a, 1) = b, δ(d , 1) = e




Minimization of FA

mark ae

b Xc cd,bed cd,bee Xf X X X X X


δ(a, 0) = c, δ(e, 0) = f




Minimization of FA

mark bc

b Xc cd,be Xd cd,bee Xf X X X X X


δ(b, 0) = f , δ(c, 0) = d




Minimization of FA

mark bd

b Xc cd,be Xd cd,be Xe Xf X X X X X


δ(b, 0) = f , δ(d , 0) = d




Minimization of FA

mark be

b Xc cd,be Xd cd,be Xe X -f X X X X X


δ(b, 0) = δ(e, 0) = f

δ(b, 1) = δ(e, 1) = g




Minimization of FA

mark dc

b Xc cd,be Xd cd,be X -e X -f X X X X X


δ(d , 0) = δ(c, 0) = d

δ(d , 1) = δ(c, 1) = e




Minimization of FA

mark ec

b Xc cd,be Xd cd,be X -e X - Xf X X X X X


δ(e, 0) = f , δ(c, 0) = d




Minimization of FA

mark de

b Xc cd,be Xd cd,be X -e X - X Xf X X X X X


δ(d , 0) = d , δ(e, 0) = f




Minimization of FA

mark fg


g X X X X X Xa b c d e f

δ(f , 0) = d , δ(g, 0) = f




Minimization of FA

merge non-distingushable states


g X X X X X Xa b c d e f

merge a,c,d and b,e


minimization of dfa

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