COMS W3261
Computer Science Theory
Lecture 12: October 19, 2009
Pushdown Automata

1. Outline

A pushdown automaton is an ε-NFA with a pushdown stack (last-in, first-out stack).
Pushdown automata define exactly the context-free languages.
There are seven components to a PDA P = (Q, Σ, Γ, δ, q₀, Z₀, F):

Suppose δ(q, a, X) contains (p, γ). Then whenever P is in state q, looking at the input symbol a with X on top of the stack, P may go into state p, move to the next input symbol, and replace X on top of the stack by the string γ.
The second component, a, may be ε in which case P makes the move without looking at the input symbol and does not move to the next input symbol.
Note that P is nondeterministic so there may be more than one pair in δ(q, a, X).

We can represent a configuration of the PDA P above by a triple (q, w, γ) where:

q is the state of the finite-state control.
w is the string of remaining input symbols.
γ is the string of symbols on the stack. If γ = XYZ, then X is on top of the stack.

Suppose δ(q, a, X) contains (p, α). Then to represent a single move of P we write

(q, aw, Xβ) |– (p, w, αβ)

for all strings w in Σ* and β in Γ*.

Note that a may be empty.

A PDA P = (Q, Σ, Γ, δ, q₀, Z₀, F) can define a language two ways.
Acceptance by final state: P can accept an input string w by reading all of it and entering a final state:

Formally, we define L(P), the language accepted by P by final state, to be the set of input strings w such that P can go from its initial ID (q₀, w, Z₀) in a sequence of zero or more moves to an accepting ID of the form (q, ε, α) where q is a final state and α is any stack string (perhaps empty).

Acceptance by empty (null) stack: P can accept an input string by reading all of it and emptying its stack empty:

Formally, we define N(P), the language accepted by P by empty stack, to be the set of input strings w such that P can go from its initial ID (q₀, w, Z₀) in a sequence of zero or more moves to an accepting ID of the form (q, ε, ε) for any state q.

These two modes of acceptance are equivalent. That is, L has a PDA that accepts it by final state iff L has a PDA that accepts it by empty stack.

Construct a PDA that accepts { wcw^R | w is any string of a's and b's } by final state.
Construct a PDA that accepts { wcw^R | w is any string of a's and b's } by empty stack.
Construct a PDA that accepts { ww^R | w is any string of a's and b's } by final state.
Construct a PDA that accepts { ww^R | w is any string of a's and b's } by empty stack.