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### Lecture#14-15

Unit-2

Grammar

- A grammar G = (V, T, S, P) consists of the following quadruple:
- a set V of variables (non-terminal symbols), including a starting symbol S NT
- a set T of terminals (same as an alphabet, )
- A start symbol S V
- a set P of production rules

- Example:
S aS | A

A bA | λ

Unit-2

Derivation

- Strings are “derived” from a grammar
- Example of a derivation
S aS aaS aaA aabA aab

- At each step, a nonterminal is replaced by the sentential form on the right-hand side of a rule (a sentential form can contain nonterminals and/or terminals)
- Automata recognize languages; grammars generate languages

Unit-2

Regular grammar

- A grammar is said to be right-linear if all productions are of the form AxB or Ax, where A and B are variables and x is a string of terminals. (This means that if there is a variable on the right side of the production rule, then it is the rightmost element in the rule.)
- A grammar is said to be left-linear if all productions are of the form ABx or Ax
- A regular grammar is either right-linear or left-linear.

Unit-2

Linear grammar

- A grammar can be linear without being right- or left-linear.
- A linear grammar is a grammar in which at most one variable can occur on the right side of any production rule, without any restriction on the position of the variable.
- Example:
S aS | A

A Ab | λ

Unit-2

Another formalism for regular languages

- Every regular grammar generates a regular language, and every regular language can be generated by a regular grammar.
- A regular grammar is a simpler, special-case of a context-free grammar
- The regular languages are a proper subset of the context-free languages

Unit-2

Exercises

- Find a regular grammar that generates the language on = {a,b} consisting of all strings with no more than three a’s.

Unit-2

Exercises

- Find a regular grammar that generates the language on = {a,b} consisting of all strings with no more than three a’s
S bS | aA | λ

A bA | aB | λ

B bB | aC | λ

C bC | λ

Unit-2

Exercises

- Find a regular grammar that generates the language consisting of even-length strings over {a,b}.

Unit-2

Exercises

- Find a regular grammar that generates the language consisting of even-length strings over {a,b}.
S aaS | abS | baS | bbS | λ

Unit-2

Programming languages

- Programming languages are context-free, but not regular
- Programming languages have the following features that require infinite “stack memory”
- matching parentheses in algebraic expressions
- nested if .. then .. else statements, and nested loops
- block structure

Unit-2

Exercise

- Given a grammar, you should be able to say what language it generates
- Use set notation to define the language generated by the following grammars
1) S aaSB | λ

B bB | b

2) S aSbb | A

A cA | c

Unit-2

Exercise

S aaSB | λ

B bB | b

It helps to list some of the strings that can be formed:

S aaSB aaB aab

S aaSB aaB aabB aabb

S aaSB aaB aabB aabbB aabbb

S aaSB aaB aabB aabbB aabbbB aabbbb

S aaSB aaaaSBB aaaaBB aaaaBb aaaabb

S aaSB aaaaSBB aaaaBB aaaaBbB aaaaBbb aaaabbb

What is the pattern?

L = {(aa)nbnb*}

Unit-2

Exercise

- Given a language, you should be able give a grammar that generates it.
- For example, give a regular (right-linear) grammar for the language consisting of all strings over {a, b, c} that begin with a, contain exactly two b’s, and end with cc.

Unit-2

Exercise

- Give a regular (right-linear) grammar for the language consisting of all strings over {a, b, c} that begin with a, contain exactly two b’s, and end with cc:
S aA

A bB | aA | cA

B bC | aB | cB

C aC | cC | cD

D c

Unit-2

Derivation

Given the grammar,

S aaSB | λ

B bB | b

the string aab can be derived in two different ways:

S aaSB aaB aab

S aaSB aaSb aab

Unit-2

Leftmost (rightmost) derivation

In a leftmost derivation, the leftmost nonterminal is replaced at each step.

In a rightmost derivation, the rightmost nonterminal is replaced at each step.

Many derivations are neither leftmost nor rightmost.

Unit-2

Theorem 3.3

- Every language generated by a right-linear grammar is regular.
- Proof:
- Specify a procedure for automatically constructing an NFA that mimics the derivations of a right-linear grammar.

Unit-2

Theorem 3.3

- Justification:
The sentential forms produced by a right linear grammar have exactly one variable, which occurs as the rightmost symbol.

Assume that our grammar has a production rule

D dE

and that, during the derivation of a string, there is a step wcD wcdE

We can construct an NFA which has states D and E, and an arc labeled d from D to E.

NFAs can be converted to DFAs.

All languages accepted by DFAs are regular.

Unit-2

Theorem 3.3

Construction:

For each variable Vi in the grammar there will be a state in the automaton labeled Vi.

The initial state of the automaton will be labeled V0 and will correspond to the S variable in the grammar.

For each production rule Vi a1a2…amVj the automaton will have transitions such that

δ*(Vi, a1a2…am) = Vj

For each production rule Vi a1a2…am the automaton will have transitions such that

δ*(Vi, a1a2…am) = Vfinal

Unit-2

Theorem 3.3

Construct an NFA that accepts the language generated by the grammar:

S aA convert to: V0 aV1

A abS | b V1 abV0 | b

a

b

V0

V1

Vf

b

a

Unit-2

Theorem 3.4

- Every regular language can be generated by a right-linear grammar.
- Proof:
- Generate a DFA for the language.
- Specify a procedure for automatically constructing a right-linear grammar from the DFA.

Unit-2

Theorem 3.4

- Given a regular language L, let M = (Q, , δ, q0, F) be a DFA that accepts L. Let Q = {q0, q1, …, qn} and = {a1, a2, …, am}.
- Construct the grammar G = (V, T, S, P) with:
V = {q0, q1, …, qn}

T = {a1, a2, …, am}

S = q0.

P = {} initially.

- P, the set of production rules, is constructed as follows:

Unit-2

Theorem 3.4

- For each transition
δ(qi, aj) = qk

in the transition table of M, add to P the production:

qi ajqk

- If qk is in F, then add to P the production:
qk λ

Unit-2

Example

- Construct a right-linear grammar for the language L = L(aab*a)
- First, build an NFA for L:

a

a

a

q0

q1

q2

qf

b

Unit-2

Example, cont.

a

a

a

q0

q1

q2

qf

b

P = {} initially.

Add to P a rule for each transition in the NFA:

q0 aq1

q1 aq2

q2 bq2

q2 aqf

Since qf is in F, add to P the production:

qf λ

Unit-2

Example

a

a

a

q0

q1

q2

qf

b

Now P = You can convert to normal grammar notation:

{q0 aq1 S aA

q1 aq2 A aB

q2 bq2 B bB

q2 aqf B aC

qf λ } C λ

Unit-2

Theorem 3.5

A language L is regular if and only if there exists a left-linear grammar G such that L = L(G).

Proof:

The strategy here is a little tricky. We describe an algorithm to construct a right-linear grammar that generates the reverse of all the strings generated by the left-linear grammar.

Unit-2

Theorem 3.5

Given any left-linear grammar we can construct from it an right-linear grammar G’ by replacing productions of the form:

A Bv with A vRB

and

A v with A vR

Since L(G’) is generated by a right-linear grammar, it is regular.

It can be demonstrated that L(G) = (L(G’))R.

It can be proven that the reverse of any regular language is also regular (see exercise 12, section 2.3 in the Linz text).

Hence, L is regular.

Unit-2

Theorem 3.6

A language L is regular if and only if there exists a regular grammar G such that L = L(G).

Proof:

Combine our definition of regular grammars, which includes the statement, “A regular grammar is either right-linear or left-linear”, with theorems 3.4 and 3.5

Unit-2

3 ways of specifying regular languages

Regular expressions

DFA NFA

Regular grammars

describe

accept

Regular languages

generate

Unit-2

Unit-2

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