LászlóKabódi Languagesandautomata

Languages and automata

László Kabódi

Example

I L₄ ={ww|w ∈ {0,1}^∗} I L₅ ={1ⁿ²}

Solution IV

I L₄ ={ww|w ∈ {0,1}^∗} I Let x=0^p10^p1

I Because |uv| ≤p,u andv can only contain 0.

I If we pump v with k >1, the rst half of x will be dierent from the second, so it won't be ww. Souv^kw ∈/ L₄

I L₄ cannot be a regular language.

Solution V

I L₅ ={1ⁿ²} I Let x=1^p2

I |v| ≥1 and |uv| ≤p, so 1≤ |v| ≤p.

I |uvw|=p². The next square number is (p+1)². But if we usek =2 then

uv²w

can not be(p+1)², but it is more than p², so it is not a square number. uv²w ∈/ L₅

I L₅ cannot be a regular language.

How to use the pumping lemma

To show that a languageLis not regular, you have to I for a chosen word x∈L

I check for all possiblex =uvw partitions, where |uv| ≤p and

|v| ≥1

I that there is at least onek touv^kw ∈/L

But if a language is pumpable, it does not mean that it is regular.

Denition

Grammar

A grammarG is:

I an alphabet (a,b, . . . ,∈Σ, terminals, do not change) I non-terminals or variables (V, they can change during a

deduction)

I starting variable or starting non-terminal (S) I deduction or production or substitution rules

I α→β

I αandβ are strings of terminals and/or non-terminals I αcontain at least one element fromV

I α∈(Σ∪V)^∗V(Σ∪V)^∗ I β∈(Σ∪V)^∗

I derivation: using the substitution rules

I L(G): all the words that can be derived using the substitutions rules from the starting symbol

Examples

Examples I

What languages these grammars dene?

I Grammar I:

S →0S1 S →01 I Grammar II:

S →AB A→0A A→0 B →B1 B →1

Examples

Solutions I

I Grammar I:

S →0S1|01

L₁ ={0ⁿ1ⁿ|n >0} I Grammar II:

S →AB A→0A|0 B →B1|1

L₂ ={0ⁱ1^j|i,j >0}

Examples

Example II

What language this grammar denes?

S →aSBC|abC CB →BC bB →bb bC →bc cC →cc

Examples

Solution II

I S →aSBC|abC,CB →BC,bB→bb,bC →bc,cC →cc I L={aⁿbⁿcⁿ|n >0}

I Not every derivation can be nished:

S →aS BC →aaS BCBC →aaabC BCBC → aaabcBCBC →aaabcBBCC

I But:

S →aS BC →aaS BCBC →aaabCBCBC → aaabBC CBC →aaabBCBCC →aaabBBCCC → aaabbBCCC →aaabbbC CC →aaabbbcC C → aaabbbccC →aaabbbccc

Examples

Examples III

1. Construct a grammar for palindromes! (Σ ={a,b})

2. Construct a grammar for a language where the number of as and the number of bs are the same!

3. Is there a grammar for all languages that contain nite number of words?

4. Is there a grammar for every language?

5. Construct a grammar for even length words!

6. Construct a grammar for the correct parentheses!

7. Construct a grammar for L={a^kbⁿc^m |n=k+m}

Examples

Solutions III

1. Construct a grammar for palindromes! (Σ ={a,b}) S →aSa|bSb|a|b|ε

2. Construct a grammar for a language where the number of as and the number of bs are the same!

S →aSbS|bSaS|ε

3. Is there a grammar for all languages that contain nite number of words?

Yes, there is. You can create a rule for all words.

4. Is there a grammar for every language?

No, the number of languages are greater than the number of possible grammars.

Examples

Solutions III

5. Construct a grammar for even length words!

S →SS|aa|ab|ba|bb|ε

6. Construct a grammar for the correct parentheses!

S →SS|(S)|ε

7. Construct a grammar for L={a^kbⁿc^m |n=k+m} S →S₁S₂

S₁ →aS₁b|ε S₂ →bS₂c|ε