Pushdown Automata

CS 3240 – Chapter 7

Language Machine Grammar

Regular Finite Automaton Regular Expression,Regular Grammar

Context-Free Pushdown Automaton

Context-Free Grammar

Recursively Enumerable

Turing Machine Unrestricted Phrase-Structure Grammar

2CS 3240 - Introduction

Take an FA and add a stack A restricted form of unbounded memory

Operations: Pop Push Ignore (λ) (Pop always precedes push)

3CS 3240 - Pushdown Automata

Three items per transition edge <read>,<pop>,<push> any or all could be λ

A PDA accepts when: An accepting state is reached, and The stack is empty

▪ Different from book!

▪ The machine crashes when no move is possible▪ implicit jail


A deterministic PDA (no choices; λ is okay sometimes). The stack alphabet (Γ = {X}) can differ from the input alphabet (Σ = {a, b}) .

Trace aabb…


q0 aabb λq0 abb Xq0 bb XXq1 b Xq1 λ λ


Traditional notation (see book):

(q0,aabb, λ) ⊢ (q0,abb,X) ⊢ (q0,bb,XX) ⊢ (q1,b,X) ⊢ (q1,λ, λ)

Also: (q0,aabb,λ) ⊢* (q1,λ, λ) (⇒ aabb ∈ L)


How would you do a2nbn?

CS 3240 - Pushdown Automata 9

Each b must pop 2 X’s

Is this deterministic?

Trace the string abaacaaba.


Input: (state-1, input character, pop character)

Output: (state-2, push character(s))

Example: The edge (a,λ,a) on previous slide:

δ(q0,a,λ) = (q0,a)

Exercise: Rewrite wcwR in functional form


Is this deterministic?Trace the string abaaaaaba.Trace the string abaabaaba.Trace the string abaaaaba.



Trace aaabbbb

The grammar:

S → aSb | aSbb | λ


Trace ababbaab


Try abababbbb.How would you do Twicea?

Reasoning: There must be 2 b’s for every a. When we read an a, there are three possibilities:

1)Its 2 b’s have already been read2)1 b has been read and 1 remains3)Both b’s are yet to be read

Q: set of states∑: input alphabetΓ: stack alphabetδ: Q x (∑∪λ) x (Γ∪λ) → finite subsets

of Q x Γ*

q0: start stateF ⊆ Q: final states

(z ⋲ Γ: optional stack start symbol)CS 3240 - Pushdown Automata 20

Not strictly necessary Can serve as a “bottom marker” to allow

detection of an empty stack A convenience

Can make things easier to design/understand

Sometimes handy when seeking an associated CFG

Can eliminate some lambda pops▪ Handy later


Pushdown Automata

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