Chapter 7: Pushdown Automata

Chapter 7: Pushdown AutomataQuan Thanh Tho qttho@cse.hcmut.edu.vn... Pushdown Automata• There are context-free languages that are not regular.. • Finite automata cannot recognize all con

Chapter 7: Pushdown Automata

Quan Thanh Tho qttho@cse.hcmut.edu.vn

Pushdown Automata

• There are context-free languages that are not regular.

• Finite automata cannot recognize all context-free languages.

Example 7.1

• {a, b}* is regular.

• {akbk | k is a constant} is regular.

• {anbn | n ≥ 0} is not regular.

Non-deterministic Pushdown

Automata (NPDA)

M = (Q, ∑ , Γ , δ , q0, z , F)

Q: finite set of internal states

∑: finite set of symbols - input alphabet

Γ: finite set of symbols - stack alphabet

δ: Q × (∑∪{λ}) ×Γ → finite subsets of Q ×Γ* transition function

q0 ∈ Q: initial state

z ∈ Γ: stack start symbol

F ⊆ Q: set of final states

Transition Function

δ : Q × ( ∑∪ { λ }) × Γ → finite subsets of Q × Γ *

replacement

Example 7.2

δ (q1, a, b) = {(q2, cd), (q3, λ )}

b

d c

q

q 2

q

Instantaneous Description

(q, w, u)

input string

Instantaneous Description Move

move: (q1, a w, b x) | (q2, w, y x)

iff (q2, y) ∈ δ (q1, a, b)

Instantaneous Description Transition

(q1, x, y) |∗Μ (q2, u, v) (q1, x, y) |+Μ (q2, u, v)

Language accepted by NPDA

Let M = (Q, ∑ , Γ , δ , q0, z, F) be an NPDA.

L(M) = {w ∈ Σ * | (q0, w, z) |∗Μ (qf, λ , u), qf ∈ F, u ∈ Γ *}

NPDA and Context-Free Languages

Theorem 7.1

For any context-free language L not containing λ , there exists

an NPDA M such that L = L(M).

Context-Free Grammars for NPDA

M = (Q, ∑ , Γ , δ , q0, z, F)

G = (V, T, S, P) L(G) = L(M)

Context-Free Grammars for NPDA

M = (Q, ∑ , Γ , δ , q0, z, F)

M^: Single final state

Final state entered iff the stack is empty

δ (qi, a, A) = {c1, c2, , cn}

ci = (qj, λ ) ci = (qj, BC)

Context-Free Grammars for NPDA

δ (qi, a, A) = {(qj, λ ), }

At qi erase A and move to qj if receiving a

(qiAqj) → a

Context-Free Grammars for NPDA

δ (qi, a, A) = {(qj, BC), } ?

At qi erase A and move to qk if receiving a and

at qj erase BC and move to qk

At qi erase A and move to qk if receiving a and

at qj erase B and move to qm and

at qm erase C and move to qk

Context-Free Grammars for NPDA

δ (qi, a, A) = {(qj, BC), }

At qi erase A and move to qk if receiving a and

at qj erase B and move to qm and

at qm erase C and move to qk

Context-Free Grammars for NPDA

Start symbol: (q0zqf)

Theorem 7.2

If L = L(M) for some NPDA M, then L is a context-free

language.

Deterministic Pushdown Automata

A DPDA is a pushdown automaton that never has a

choice in its move:

1 δ (q, a, b) contains at most one element.

2 if δ (q, λ , b) is not empty, then δ (q, a, b) must be empty for

every a ∈ Σ

Deterministic Context-Free

Language

A language L is said to be a DCFL iff there exists a

DPDA M such that L = L(M).

• Exercises: 5, 10, 13 of Section 7.1 - Linz’s book.

• Exercises: 1, 2, 3, 4, 5, 6, 7, 8, 12 of Section 7.2 - Linz’s book.