1. CFG => PDA
Sipser 2 (pages )

2. Formally… A pushdown automaton is a sextuple
M = (Q, Σ, Γ, δ, q0, F), where
Q is a finite set of states
Σ is a finite alphabet (the input symbols)
Γ is a finite alphabet (the stack symbols)
δ: (Q × Σε ×Γε) → P(Q × Γε)
is the transition function
q0∈Q is the initial state, and
F⊆Q is the set of accept states

3. Balanced brackets Let M = (Q, Σ, Γ, δ, q0, F), where Q = {q1, q2, q3}
Σ = {[, ]}
Γ = {[, $}
q0 = q1
F = {q1, q3}
δ is given by the transition diagram:

4. Finite automata and Pushdown automata?
Pushdown languages
Regular languages

5. Regular => Pushdown
Proposition: Every finite automaton can be viewed as a pushdown automaton that never operates on its stack.
Proof:
Let M = (Q, Σ, δ, q0, F) be a finite automaton.
Define M' = (Q, Σ, Γ, δ', q0, F), where…

6. Finite automata and Pushdown automata?
Pushdown languages
Regular languages

7. Pushdown automata are nondeterministic
Build a machine to recognize
PAL = {w | w∈{0, 1}* and w =wR} 1 1 1 1 1
reading head
stack or
pushdown
store 1
Finite control $

8. Recognizing context-free languages
Lemma 2.21: If a language is context-free, then some pushdown automaton recognizes it.
Proof:

9. Shorthand for…

10. For example… Let’s use this construction on: G = (V, Σ, R, S), where
V = {S}
Σ = {[, ]}
R = { S →G ε,
S →G SS,
S →G [S]}