Chapter 2: Context-Free Languages

For Your Enjoyment

• An 18 minute Coursera Introduction to showing that PDAs and CFGs are equivalent.

Chapter 2.2, Pushdown Automata

Theorem 2.20

A language is context free if and only if some pushdown automaton recognizes it.

CFG → PDA Construction

Step One: Create a 3-State PDA

• The states are q_start, q_loop and q_accept.
• q_accept is an accept state.
• q_start transitions to q_loop with the rule ε, ε → S$
• q_loop transitions to itself with the rules ε, A → w (for each rule A → w) and a, a → ε (for each terminal a).
• q_loop transitions to q_accept with the rule ε, $ → ε

Step Two: Add States to Push One Symbol at a Time

For example, instead of q_start transitioning to q_loop directly on rule ε, ε → S$, it transitions to an intermediate state on rule ε, ε → $ and the intermediate state transitions to q_loop on rule ε, ε → S.

Active Learning Problem

Convert the following CFG to a PDA using the procedure above.

  E → E + T | T
  T → T * F | F
  F → (E) | a

PDA → CFG Construction

Preliminary Construction

Convert the original PDA to a PDA with the following characteristics.

• The PDA should have a single accept state.
• The PDA should empty its stack before accepting.
• Each transition should either push a symbol onto the stack or pop one symbol off. It should not do both at the same time.

Active Learning

Convert the following transition δ(r, 0, 0) → (s, 1) to meet the criteria above.

Construction

Construct G from P = (Q, Σ, Γ, δ, q₀, {q_accept}). The variables of G are {A_pq | p, q ∈ Q} and the start variable is A_{q0, qaccept}.

• For each p, q, r, s ∈ Q, u ∈ Γ and a,b ∈ Σ_ε, if δ(p, a, ε) contains (r, u) and δ(s, b, u) contains (q, ε), add the following rule to G: A_pq → aA_rsb
• For each p, q, r, s ∈ Q, add the following rule to G: A_pq → A_prA_rq
• For each p ∈ Q, add the following rule to G: A_pp → ε

Active Learning

• Consider the PDA in Figure 2.18. Show at least one resulting rule for each of the three bullets above.

In-Class Notes