Limitations - Automata and Complexity Theory - Lecture Slides, Slides of Theory of Automata

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Limitations of finite automata

Many examples of regular languages

q 0 q

0 1 1 0

(^1 0 )

0, 1 q 0 q 1 q 2 q 3

all strings not containing pattern 010 followed by 101

Is every language regular?

Which are regular?

• How about:

• Let’s try to design a DFA for it

… it appears that we need infinitely many states!

L 1 = {0n 1 m: n, m ≥ 0} = 01, so regular

L 2 = {0n 1 n: n ≥ 0} = {ε, 01, 0011, 000111, …}

A non-regular language

• Theorem

• To prove this, we argue by contradiction:

• Suppose we have managed to construct a DFA M

for L 2

• We show something must be wrong with this DFA
• In particular, M must accept some strings not in L 2

The language L 2 = {0n 1 n: n ≥ 0} is not regular.

A non-regular language

• What happens when we run M on input x =

0 n+1 1 n+1?

• M better accept, because x ∈ L 2
• But since M has n states, it must revisit at least one

of its states while reading 0 n+

M

x (^0 0 0 )

0

0 r 1 n+

Pigeonhole principle

• Here, balls are 0 s, bins are states:

Suppose you are tossing n + 1 balls into n

bins. Then two balls end up in the same bin.

If you have a DFA with n states and it reads

n + 1 consecutive 0 s, then it must end up in

the same state twice.

A non-regular language

• The DFA will then also accept strings that go

around the loop multiple times

• But such strings have more 0 s than 1 s, so they

are not in L 2!

M

0 0 0 0

0

0 r 1 n+

General method for showing non- regularity

• Every regular language L has a property:
• For every sufficiently long input z in L, there is

a “middle part” in z that, even if repeated several times, keeps the input inside L

z (^) a (^1) … ak ak+1 … an-

an

an+1…am

Proving non-regularity

• If L is regular, then:

• So to prove L is not regular, it is enough to

show:

There exists n such that for every z in L, we

can write z = u v w where |uv| ≤ n, |v| ≥ 1

and

 For every i ≥ 0, the string u v i^ w is in L.

For every n there exists z in L, such that for

every way of writing z = u v w where

|uv| ≤ n and |v| ≥ 1, the string u v i^ w is not

in L for some i ≥ 0.

Proving non regularity

• This is a game between you and an imagined

adversary

For every n there exists z in L, such that for

every way of writing z = u v w where

|uv| ≤ n and |v| ≥ 1, the string u v i^ w is not

in L for some i ≥ 0.

adversary choose n write z = uvw (|uv| ≤ n,|v| ≥ 1)

you choose z ∈ L choose i you win if uv iw ∉ L

1 2

Example

• L 2 = {0n 1 n: n ≥ 0} Σ = {0, 1}

adversary choose n write z = uvw (|uv| ≤ n,|v| ≥ 1)

you choose z ∈ L choose i you win if uv iw ∉ L

1 2

adversary choose n write z = uvw (|uv| ≤ n,|v| ≥ 1)

you z = 0n+1 1 n+ i = 2 uv iw = 0j+2k+l 1 n+ = 0n+1+k 1 n+ ∉ L

1 2 u = 0j, v = 0k, w = 0l 1 n+ j + k + l = n + 1

More examples

L 3 = {1n: n is divisible by 3} L 4 = {1n: n is prime}

L 5 = {x: x has same number of 0s and 1s}

L 6 = {x: x has same number of patterns 01 and 10} L 7 = {x: x has more 0s than 1s} L 8 = {x: x has different number of 0s and 1s}