Chapter 2: Context-Free Languages

For Your Enjoyment

• A 24 minute Coursera Introduction to Pushdown Automata.

Chapter 2.2, Pushdown Automata

A pushdown automaton (PDA) is a finite automaton that also has access to a stack with infinite memory.

Factoids

• Context-free grammars generate context-free languages (CFLs).
• Pushdown automata recognize context-free languages.
• Context-free grammars and pushdown automata are equivalent. (Monday's reading will show how to prove this.)
• Unlike regular languages, the set of non-deterministic CFLs is larger than the set of deterministic CFLs.

Pushdown Automaton 6-Tuple

A pushdown automaton is a 6-tuple (Q, Σ, Γ, δ, q₀, F), where Q, Σ, Γ, and F are all finite sets, and

1. Q is the set of states,
2. Σ is the input alphabet,
3. Γ is the stack alphabet,
4. δ: Q × Σ_ε × Γ_ε → P(Q × Γ_ε) is the transition function,
5. q₀ is the start state, and
6. F ⊆ Q is the set of accept states.

Lecture Problem

• Design a PDA that recognizes {ww^R | w ∈ {1,2}^*}.
• What is each element of the 6-tuple?
• Show the sequence of states that the PDA goes through to accept the string 2112.
• Show the context-free grammar that generates this language.

Active Learning Problem

• Design a PDA that recognizes {1ⁿ2^k3ⁿ | w ∈ {1, 2, 3}^*}.
• What is each element of the 6-tuple?
• Show the sequence of states that the PDA goes through to accept the string 1122233.
• Show the context-free grammar that generates this language.

Acceptance

PDA M accepts input w (w₁...w_m where each w_i ∈ Σ_ε) if there is a sequence of states r₀, ..., r_m ∈ Q and strings s₀, ..., s_m ∈ Γ^* such that

1. r₀ = q₀ and s₀ = ε
2. (r_i+1, b) ∈ δ(r_i, w_i+1, a) for i = 0, 1, ..., m-1 where s_i = at and s_i+1 = bt for some a, b ∈ Γ_ε and t ∈ Γ^*
3. r_m ∈ F

In-Class Notes