sliek môn máy tự động chương 7 pushdown automata

1 Pushdown Automata • There are context-free languages that are not regular. • Finite automata cannot recognize all context-free languages. 2 Pushdown Automata • {a, b}* is regular. • {a k b k | k is a constant} is regular. • {a n b n | n ≥ 0} is not regular. 3 Finite Automata Control unit q 0 Input file yes/no 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 Non-deterministic Pushdown Automata (NPDA) δ: Q × (∑∪{λ}) × Γ → finite subsets of Q × Γ* stack top stack top replacement 7 Example δ(q 1 , a, b) = {(q 2 , cd), (q 3 , λ)} b d c q 1 q 2 q 3 8 Example 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: (q 1 , aw, bx) | (q 2 , w, yx) iff (q 2 , y) ∈ δ(q 1 , a, b) [...]... z)} NPDA and Context-Free Languages Greibach NF δ(q0, λ, z) = {(q1, Sz)} S → aSA | a δ(q1, a, S) = {(q1, SA), (q0, λ)} A → bB δ(q1, b, A) = {(q1, B)} B→b δ(q1, b, B) = {(q1, λ)} δ(q1, λ, z) = {(q2, λ)} 17 Theorem For any context-free language L not containing λ, there exists an NPDA M such that L = L(M) 18 Theorem Proof: G = (V, T, S, P) M = ({q0, q1, qf}, T, V∪{z}, δ, q0, z, {qf}) z ∉ V δ(q0, λ, z) =... 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 27 Context-Free Grammars for NPDA δ(qi, a, A) = {(qj, λ), } At qi erase A and move to qj if receiving a (qiAqj) → a 28 Context-Free Grammars for NPDA δ(qi, a, A) = {(qj, BC), } At qi erase A and move . 1 Pushdown Automata • There are context-free languages that are not regular. • Finite automata cannot recognize all context-free languages. 2 Pushdown Automata • {a, b}* is. 0} is not regular. 3 Finite Automata Control unit q 0 Input file yes/no 4 Pushdown Automata Control unit q 0 Input file yes/no Stack 5 Non-deterministic Pushdown Automata (NPDA) M = (Q, ∑, Γ,. Q: set of final states 6 Non-deterministic Pushdown Automata (NPDA) δ: Q × (∑∪{λ}) × Γ → finite subsets of Q × Γ* stack top stack top replacement 7 Example δ(q 1 , a, b) = {(q 2 , cd), (q 3 ,