Theory of Computation: Context-Free Grammars, Lecture notes of Theory of Computation

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Theory of Computation

(Context-Free Grammars)

Pramod Ganapathi

Department of Computer Science State University of New York at Stony Brook

January 24, 2021

Contents

Contents Context-Free Grammars (CFG) Context-Free Languages Pushdown Automata (PDA) Transformations Pumping Lemma

Computer program compilation

C++ program:

1. #include
2. using namespace std;
3. int main()
4. {
5. if (true)
6. {
7. cout << "Hi 1";
8. else
9. cout << "Hi 2";
10. }
11. return 0;
12. }

C++ program:

1. #include
2. using namespace std;
4. int main()
5. {
6. if (true)
7. cout << "Hi 1";
8. else
9. cout << "Hi 2";
11. return 0;
12. }

Computer program compilation

C++ program:

1. #include
2. using namespace std;
3. int main()
4. {
5. if (true)
6. {
7. cout << "Hi 1";
8. else
9. cout << "Hi 2";
10. }
11. return 0;
12. }

C++ program:

1. #include
2. using namespace std;
4. int main()
5. {
6. if (true)
7. cout << "Hi 1";
8. else
9. cout << "Hi 2";
11. return 0;
12. }

Output: error: expected ‘}’ before ‘else’

Output: Hi 1

Construct CFG for L = { a n b n^ | n ≥ 0 }

Problem Construct a CFG that accepts all strings from the language L = {a n b n^ | n ≥ 0 }

Construct CFG for L = { a n b n^ | n ≥ 0 }

Problem Construct a CFG that accepts all strings from the language L = {a n b n^ | n ≥ 0 } Solution Language L = {, ab, aabb, aaabbb, aaaabbbb,.. .} CFG G. S → aSb S → 

Construct CFGs

Problems Construct CFGs to accept all strings from the following languages: R = a∗ R = a+ R = a∗b∗ R = a+b+ R = a∗^ ∪ b∗ R = (a ∪ b)∗ R = a∗b∗c∗

Construct CFG for palindromes over { a, b }

Problem Construct a CFG that accepts all strings from the language L = {w | w = w R^ and Σ = {a, b}}

Construct CFG for palindromes over { a, b }

Solution (continued) CFG G. S → aSa | bSb | a | b |  Accepting . S ⇒  B 1 step Accepting a. S ⇒ a Accepting b. S ⇒ b Accepting aa. S ⇒ aSa ⇒ aa B 2 steps Accepting bb. S ⇒ bSb ⇒ bb Accepting aaa. S ⇒ aSa ⇒ aaa B 2 steps Accepting aba. S ⇒ aSa ⇒ aba Accepting bab. S ⇒ bSb ⇒ bab Accepting bbb. S ⇒ bSb ⇒ bbb Accepting aaaa. S ⇒ aSa ⇒ aaSaa ⇒ aaaa B 3 steps Accepting abba. S ⇒ aSa ⇒ abSba ⇒ abba Accepting baab. S ⇒ bSb ⇒ baSab ⇒ baab Accepting bbbb. S ⇒ bSb ⇒ bbSbb ⇒ bbbb

Construct CFG for non-palindromes over { a, b }

Problem Construct a CFG that accepts all strings from the language L = {w | w 6 = w R^ and Σ = {a, b}}

Construct CFG for non-palindromes over { a, b }

Solution (continued) CFG G. S → aSa | bSb | aAb | bAa A → Aa | Ab |  Accepting abbbbaaba. B 7-step derivation S ⇒ aSa ⇒ abSba ⇒ abbAaba ⇒ abbAaaba ⇒ abbAbaaba ⇒ abbAbbaaba ⇒ abbbbaaba

What is a context-free grammar (CFG)?

Grammar = A set of rules for a language Context-free = LHS of productions have only 1 nonterminal

Definition A context-free grammar (CFG) G is a 4-tuple G = (N, Σ, S, P ), where,

1. N : A finite set (set of nonterminals/variables).
2. Σ: A finite set (set of terminals).
3. P : A finite set of productions/rules of the form A → α, A ∈ N, α ∈ (N ∪ Σ)∗. B Time (computation) B Space (computer memory)
4. S: The start nonterminal (belongs to N ).

What is a context-free language (CFL)?

Definition If G = (N, Σ, S, P ) is a CFG, the language generated by G is L(G) = {w ∈ Σ∗^ | S ⇒∗ G w} A language L is a context-free language (CFL) if there is a CFG G with L = L(G).

Construct CFG for L = { w | n a( w ) = n b( w )}

Problem Construct a CFG that accepts all strings from the language L = {w | n a (w) = n b (w)}