Chapter 7: Pushdown Automata

Chapter 7: Pushdown Automata Quan Thanh Tho qttho@cse.hcmut.edu.vn 2 Pushdown Automata • There are context-free languages that are not regular. • Finite automata cannot recognize all context-free languages. 3 Example 7.1 • {a, b}* is regular. • {akbk | k is a constant} is regular. • {anbn | n ≥ 0} is not regular. 4 Pushdown Automata Control unit q 0 Input file yes/no Stack 5 Non-deterministic Pushdown Automata (NPDA) M = (Q, ∑, Γ, δ, q 0 , 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 q 0 ∈ Q: initial state z ∈ Γ: stack start symbol F ⊆ Q: set of final states 6 Transition Function δ: Q × (∑∪{λ}) × Γ → finite subsets of Q × Γ* stack top stack top replacement 7 Example 7.2 δ(q 1 , a, b) = {(q 2 , cd), (q 3 , λ)} b d c q 1 q 2 q 3 8 Example 7.3 M = (Q, ∑, Γ, δ, q 0 , z, F) Q = {q 0 , q 1 , q 2 , q 3 } δ(q 0 , a, 0) = {(q 1 , 10), (q 3 , λ)} ∑ = {a, b} δ(q 0 , λ, 0) = {(q 3 , λ)} Γ = {0, 1} δ(q 1 , a, 1) = {(q 1 , 11)} z = 0 δ(q 1 , b, 1) = {(q 2 , λ)} F = {q 3 } δ(q 2 , b, 1) = {(q 2 , λ)} δ(q 2 , λ, 0) = {(q 3 , λ)} 9 Instantaneous Description (q, w, u) current state unread part of stack contents input string 10 Instantaneous Description Move move: (q 1 , aw, bx) | (q 2 , w, yx) iff (q 2 , y) ∈ δ(q 1 , a, b) [...]... S, P) T=Σ V = {(q Aq ) | A ∈ Γ} i j S = (q zq ) 0 f P: (q Aq ) → a iff (q , λ) ∈ δ(q , a, A) i j j i (q Aq ) → a(q Bq )(q Cq ) iff (q , BC) ∈ δ(q , a, A) i m i m m j j i 31 Deterministic Pushdown Automata A DPDA is a pushdown automaton that never has a choice in its move: 1 2 δ(q, a, b) contains at most one element if δ(q, λ, b) is not empty, then δ(q, a, b) must be empty for every a ∈ Σ 32 Deterministic . Chapter 7: Pushdown Automata Quan Thanh Tho qttho@cse.hcmut.edu.vn 2 Pushdown Automata • There are context-free languages that are not regular. • Finite automata cannot recognize all. is regular. • {anbn | n ≥ 0} is not regular. 4 Pushdown Automata Control unit q 0 Input file yes/no Stack 5 Non-deterministic Pushdown Automata (NPDA) M = (Q, ∑, Γ, δ, q 0 , z, F) Q: finite set