Pushdown Automata

Hopcroft, Motawi, Ullman, Chap 6

PDAs: Pushdown Automata

   Same as Finite Automata (e.g., NFAs), but with the addition of a stack:  On each transition, a stack is manipulated and monitored (operations: push, pop, inspect top, check for empty)  Allows the automaton to “count” Definition will need to incorporate stack events When is a string acceptable?

PDA definition

 A pushdown automaton M is a tuple M = (Q,  ,  ,  , q 0 , Z 0 , F), where:  Q is a set of states        is the input alphabet  is the stack alphabet  : Q   +є    (Q   *)* is the state transition function mapping a (state, symbol, stack symbol) triple to a set of (state, string) pairs q 0 is the start state of M Z 0 is the starting stack symbol (initially on the stack) F  Q is the set of accepting states or final states of M

PDA transitions

  A transition between states q stack x and q y can be carried read and a given stack symbol X is on top of the When the transition to q y is carried out, the symbol X    (q x , a, X) contains the pair (q y , w) How are the stack operations push, pop, inspect top, and check if empty carried out?

   push Y on input a:  (q x , a, X) -> (q y , YX) pop Y:  (q x , a, X) -> (q y , є) How about inspect top and check if empty?

Example

   L = {a n b n : n >= 0} PDA M = ({q 0 ,q 1 ,q 2 }, {a,b}, {Z 0 ,X},  , q 0 , Z 0 , {q 2 })  defined (informally) as follows:   On input a (from q 0 ) push X onto the stack On input b (from q 0 the stack) and q 1 ), pop X and transition to q1 (note: cannot transition if X is not on top of  From q0 and q1, transition to final state q 2 is “empty” (Z 0 is on top) if stack

Transition function example

          (q (q (q 0 0 0 , a, Z , є, Z 0 0 ) = {(q , b, X) = {(q ) = {(q 1 0 2 , XZ , є)}  (q 1 , b, X) = {(q 1 , є)} , Z 0 0  (q 0 , a, X) = {(q 0 , XX)} )}  (q 1 , є, Z 0 ) = {(q 2 , Z 0 )} )} push X onto the stack on input a pop X from the stack on input b go to final state if Z 0 is on top  (q 0 , b, Z 0 ) =  (q 1 , b, Z 0 ) =  (q 1 , a, X) and many others are all equal to {}

PDA diagram

a, Z 0 , XZ 0 a, X, XX q 0 є , Z 0 , Z 0 b, X, є q 2 є , Z 0 , Z 0 b, X, є q 1

Alternative diagram

a, push X q 0 є , empty b, pop X b, pop X є , empty q 1 q 2

Accepting strings in a PDA

   By final state, just like in regular final automata Or by empty stack: string is acceptable once the stack is empty (Z is currently in 0 has been popped out), regardless of what state it These two “kinds” of PDAs are equivalent (proofs in section 6.2)

Equivalence of PDAs and CFGs

  PDAs and CFGs describe the same set of languages (section 6.3) Proof requires:   Construction from a grammar to an equivalent PDA Construction from a PDA to an equivalent grammar