formal language and automata theory (cs21004)cse.iitkgp.ac.in/~soumya/flat/min.pdf · formal...

Formal Languageand Automata

Theory (CS21004)

Soumyajit DeyCSE, IITKharagpur

Pumping Lemma

Minimization

Myhill-NerodeTheorem

Pumping Lemma Minimization Myhill-Nerode Theorem

Formal Language and Automata Theory(CS21004)

Soumyajit DeyCSE, IIT Kharagpur

Soumyajit Dey CSE, IIT Kharagpur Formal Language and Automata Theory (CS21004)


Theory (CS21004)


Pumping Lemma

Minimization



Announcements

The slide is just a short summary

Follow the discussion and the boardwork

Solve problems (apart from those we dish out in class)



Theory (CS21004)


Pumping Lemma

Minimization



Table of Contents

1 Pumping Lemma

2 Minimization

3 Myhill-Nerode Theorem



Theory (CS21004)


Pumping Lemma

Minimization



Languages that are not regular

L = {anbn | n ≥ 0} = {ǫ, ab, aabb, aaabbb, · · · }

needs to remember number of a-sand match with b-s.Infinite number of possibilities

cannot remember with finite number of states

We further provide a formal arguement



Theory (CS21004)


Pumping Lemma

Minimization



Languages that are not regular

Let a DFA with k states accept L. consider some n >> k

starting from initial state i , anbn leads to the acceptstate f

some state must have been visited more than once, letit be p

Let anbn = uvw , j = |v | > 0 where

δ̂(i , u) = p, δ̂(p, v) = p, δ̂(p,w) = f

Hence δ̂(i , uw) = f

uw = an−jbn /∈ L

Similarly, δ̂(i , uv3w) = f but uv3w = an+2jbn /∈ L

♠ Such a DFA does not exist



Theory (CS21004)


Pumping Lemma

Minimization



Pumping Lemma for Regular Languages

Let L be a regular language. Then there exists an integerp ≥ 1 such that every string w in L of length at least p (p iscalled the ”pumping length”) can be written as w = xyz

(i.e., w can be divided into three substrings), satisfying thefollowing conditions:

|y | ≥ 1

|xy | ≤ p

∀i ≥ 0, xy iz ∈ L



Theory (CS21004)


Pumping Lemma

Minimization



Pumping Lemma for Regular Languages : general

version

Let L be a regular language. Then there exists an integerp ≥ 1 such that every string uwv in L with |w | ≥ p can bewritten as uwv = uxyzv such that

|y | ≥ 1

|xy | ≤ p

∀i ≥ 0, uxy izv ∈ L

standard version is a special case with u, v being empty.Since the general version imposes stricter requirements onthe language, it can be used to prove the non-regularity ofmany more languages, such as {ambncn : m ≥ 1, n ≥ 1}



Theory (CS21004)


Pumping Lemma

Minimization



Pumping Lemma for Regular Languages

Necessary but not sufficient condition

Cannot be used to prove language as regular

There are non-regular languages which satisfy thelemma

Violation can be used to prove language as non-regular



Theory (CS21004)


Pumping Lemma

Minimization



Pumping Lemma Ex

{a2n| n ≥ 0}.

Let this be accepted by a k state DFA. Choose n suchthat n >> k

Thus 2n > k . Hence we may decompose the string a2n

to parts of length i , j , l such that 2n = i + j + l and theintermediate j symbols form a cycle in the DFA

The DFA will accept a2n+j

Note, i + j ≤ k < n ⇒ j < n

2n + j < 2n + n < 2n + 2n = 2n+1

♠ such a DFA cannot exist



Theory (CS21004)


Pumping Lemma

Minimization



Pumping Lemma Ex

We can show using Pumping Lemma

{anbm | n ≥ m} is not regular

{an! | n ≥ 0} is not regular



Theory (CS21004)


Pumping Lemma

Minimization



More examples of languages not regular

Alternate lines of argument :

A = {w | na(w) = nb(w)} ⇒ if A is regular thanA ∩ L(a∗b∗) = {anbn | n ≥ 0} is regular

{anbm | n ≥ m} is regular⇒ AR = {bman | n ≥ m} is regular⇒ C = AR [a 7→ b, b 7→ a] = {ambn | n ≥ m} is regular⇒ A ∩ C = {anbn | n ≥ 0} is regular



Theory (CS21004)


Pumping Lemma

Minimization



More examples of languages not regular

U ⊆ N is an ultimately periodic set, i.e. ∃n ≥ 0,∃p > 0, ∀m ≥ n, m ∈ U iff m+ p ∈ U. We call p is theperiod of U. Every such U is regular

Ex. {0, 3, 7, 9, 19, 20, 23, 26, 29, 32, 35, · · · } :(n = 20, p = 3), (n = 21, p = 6) : n, p need not beunique

Let A ⊆ {a}∗. A is regular iff {m | am ∈ A} isultimately periodic



Theory (CS21004)


Pumping Lemma

Minimization



Table of Contents

1 Pumping Lemma

2 Minimization




Theory (CS21004)


Pumping Lemma

Minimization



Equivalence of FAs

When we convert NFA to DFA,

We ignore unreachable states, keep them you simplyhave a larger DFA for the same language !!

Even some reachable states can be merged preservinglanguage equivalence !!



Theory (CS21004)


Pumping Lemma

Minimization



Examplea, b

a b a

a, b

Apply standard NFA to DFA conversion, remove unreachable states

b

a b

b

b

b

b

a

a

a

a

a

a

a

a

a

b

b

b

b

Accept states can be merged !

Not all such cases are as obviousSoumyajit Dey CSE, IIT Kharagpur Formal Language and Automata Theory (CS21004)


Theory (CS21004)


Pumping Lemma

Minimization



Example

a

bb b

a

a,b

a,b

a,b

a

0

2

1 3

4

5

a,b a,ba,b

6 7 8



Theory (CS21004)


Pumping Lemma

Minimization



Example

a

b

a,b

a,b

a,b a,ba,b

a,b



Theory (CS21004)


Pumping Lemma

Minimization



How to decide which states to collapse

Intuitively two states are mergeable if they behavesimilarly (in terms of language acceptance) for the sameinput string

Starting from respective states, with the same inputstring, either both lead to respective final states or nonelead to respective final states

Turns out to be a necessary and sufficient condition

Such relations among state pairs are equivalence

relations



Theory (CS21004)


Pumping Lemma

Minimization



Equivalence relation on states

∀p, q ∈ Q, p ≈ q iff

∀x ∈ Σ∗ [δ̂(p, x) ∈ F ⇔ δ̂(q, x) ∈ F ]

reflexive, symmetric, transitive

for any state p, [p] = {q | p ≈ q}

by definition, equivalence classes are mutually exclusiveand exhaustive : every state is in exactly oneclass/partition,

p ≈ q ⇔ [p] = [q]



Theory (CS21004)


Pumping Lemma

Minimization



Quotient Automaton M/ ≈ for DFA M

Given M = (Q,Σ, δ, s,F ), M/ ≈def= (Q ′,Σ, δ′, s ′,F ′)

Q ′ = {[p] | p ∈ Q}

δ′([p], a) = [δ(p, a)]

s ′ = [s]

F ′ = {[p] | p ∈ F}



Theory (CS21004)


Pumping Lemma

Minimization



Is δ′ well defined ??

If p, q,∈ [p], is δ′([p], a) = [δ(p, a)] = δ′([q], a) = [δ(q, a)] ?For any a ∈ Σ, y ∈ Σ∗

δ̂(δ(p, a), y) ∈ F ⇔ δ̂(p, ay) ∈ F by definition of δ̂

⇔ δ̂(q, ay) ∈ F since p ≈ q

⇔ δ̂(δ(q, a), y) ∈ F by definition of δ̂

Hence, δ(p, a) ≈ δ(q, a) by definition of ≈. So,[δ(p, a)] = [δ(q, a)]



Theory (CS21004)


Pumping Lemma

Minimization



p ∈ F ⇔ [p] ∈ F ′

p ∈ F ⇒ [p] ∈ F ′ by definition of F ′. What about the otherdirection, i.e. [p] ∈ F ′ ⇒ p ∈ F ??

What is there to prove ??

Note that you have [p], that does not specify any p butthe overall equivalence class.

Need to show that all elements of the class are in F

rather than one specific member.

Prove that any such equivalence class is either subset ofF or disjoint.



Theory (CS21004)


Pumping Lemma

Minimization



[p] ∈ F ′ ⇒ p ∈ F

[p] ∈ F ′ ⇒ ∃x ∈ [p], x ∈ F

⇒ δ̂(x , ǫ) = x ∈ F by defn. of δ̂

⇒ ∀q ≈ x , δ̂(q, ǫ) ∈ F by defn. of ≈

⇒ ∀q ≈ x , δ̂(q, ǫ) = q ∈ F by defn. of δ̂

⇒ ∀q ∈ [p], q ∈ F ∀q ≈ x ∈ [p], q ∈ [p]



Theory (CS21004)


Pumping Lemma

Minimization



Prove the following

∀x ∈ Σ∗, δ̂′([p], x) = [δ̂(p, x)]

L(M/ ≈) = L(M)

M/ ≈ cannot be collapsed any further



Theory (CS21004)


Pumping Lemma

Minimization



Here is an algorithm for computing the collapsing relation ≈for a given DFA M with no inaccessible states. Thealgorithm will mark (unordered) pair of states {p, q}. A pair{p, q} will be marked as soon as a reason is discovered whyp and q are not equivalent.

1 Write down a table of all pairs {p, q}, initiallyunmarked.

2 Mark {p, q} if p ∈ F and q 6∈ F of vice versa.

3 Repeat the following until no more changes occur:If there exists an unmarked pair {p, q} such that{δ(p, a), δ(q, a)} is marked for some a ∈ Σ, then mark{p, q}.

4 When done, p ≈ q iff {p, q} is not marked.



Theory (CS21004)


Pumping Lemma

Minimization



Table of Contents

1 Pumping Lemma

2 Minimization




Theory (CS21004)


Pumping Lemma

Minimization



DFA : isomorphism

Two deterministic finite automataM = (QM ,Σ, δM , sM ,FM), N = (QN ,Σ, δN , sN ,FN), areisomorphic iff ∃f , f : QM → QN such that

f (sM) = sN

∀p ∈ QM , a ∈ Σ, f (δM(p, a)) = δN(f (p), a)

p ∈ FM iff f (p) ∈ FN

One is just the renamed version of another. Note,M/ ≡, N/ ≡ are also isomorphic. ⇒ We should be able todefine a minimal automata directly from the language itself.All other possible minimal automata will be isomorphic withthis.



Theory (CS21004)


Pumping Lemma

Minimization



Myhill-Nerode Relations

Let R ⊆ Σ∗ be regular with DFA M = (Q,Σ, δ, s,F ) for R .M does not have any unreachable states. A relation ≡M onΣ∗ defined as

x ≡M y ⇔ δ̂(s, x) = δ̂(s, y)

≡M is an equivalence relation. Other properties of ≡M

1 ∀x , y ∈ Σ∗, a ∈ Σ, x ≡ y ⇒ xa ≡ ya : right congruence(show this)

2 ≡M refines R : x ≡M y ⇒ (x ∈ R ⇔ y ∈ R) – every≡M -class has either all its elements in R or none of itselements in R , i.e. R is a union of ≡M -classes

3 The no. of ≡M classes is finite ( = no. of states in M ?)



Theory (CS21004)


Pumping Lemma

Minimization



Myhill-Nerode Relations

Any equivalence relation on Σ∗ which is a right congruenceof finite index refining a regular set R is called aMyhill-Nerode Relation

Just like M →≡M we can ≡→ M≡

Let ≡ be an arbitrary Myhill-Nerode Relation on Σ∗ for someR ⊆ Σ∗, i.e. ≡ is some equivalence Relation on Σ∗ which isalso right congruence of finite index refining a regular set R



Theory (CS21004)


Pumping Lemma

Minimization



≡→ M≡

DFA M≡ = (Q,Σ, δ, s,F )

Q = {[x ] | x ∈ Σ∗} (is finite, why ?)

s = [ǫ]

F = {[x ] | x ∈ R}

δ([x ], a) = [xa] (y ∈ [x ] ⇒ [xa] = [ya] by rightcongruence)

Can show

x ∈ R ⇔ [x ] ∈ F : The ‘⇒’ is by defn of F , for ⇐,y ∈ [x ] ∈ F ⇒ x ∈ F ⇒ y ∈ R by refinement

δ̂([x ], y) = [xy ] : by induction

L(M≡) = R

≡M≡is identical to ≡

M≡Mis isomorphic to M



Theory (CS21004)


Pumping Lemma

Minimization



Myhill-Nerode Theorem

Let R ⊆ Σ∗. The following statements are equivalent:

R is regular

there exists a Myhill-Nerode relation for R

the relation ≡R creates a finite partitioning of Σ∗



Theory (CS21004)


Pumping Lemma

Minimization



Example application

Consider R = {anbn|n ≥ 0} ⊆ Σ∗. Let R be regular with aMyhill-Nerode relation ≡ on Σ∗ for R

Let am ≡ ak for any m 6= k

By right congruence ambk ≡ akbk

Note x ≡ y ⇒ [x ∈ R ⇔ y ∈ R ] , i.e. an equivalencepartition is either inside R or outside R , but cannotspan across

But now we have one equivalence partition containingambk , akbk where ambk /∈ R , akbk ∈ R .

Hence, it is not the case that ak ≡ am

The relation ≡ creates an infinite partitioning of Σ∗

♠ R is not regular


formal language and automata theory (cs21004)cse.iitkgp.ac.in/~soumya/flat/min.pdf · formal...

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