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Pushdown Automata

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”

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Pushdown Automata

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  1. Pushdown Automata Hopcroft, Motawi, Ullman, Chap 6

  2. 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?

  3. PDA definition • A pushdown automaton M is a tupleM = (Q, , , , q0, Z0, 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 • q0 is the start state of M • Z0 is the starting stack symbol (initially on the stack) • F  Q is the set of accepting states or final states of M

  4. PDA transitions • A transition between states qx and qy can be carried out in a PDA when a given next input symbol a is read and a given stack symbol X is on top of the stack • When the transition to qy is carried out, the symbol X is replaced with a string w of stack symbols • (qx,a, X) contains the pair (qy ,w) • How are the stack operations push, pop, inspect top, and check if empty carried out? • push Y on input a: (qx,a, X) -> (qy ,YX) • pop Y: (qx,a, X) -> (qy ,є) • How about inspect top and check if empty?

  5. Example • L = {anbn: n >= 0} • PDA M =({q0,q1,q2}, {a,b}, {Z0,X}, , q0, Z0, {q2}) •  defined (informally) as follows: • On input a (from q0) push X onto the stack • On input b (from q0 and q1), pop X and transition to q1 (note: cannot transition if X is not on top of the stack) • From q0 and q1, transition to final state q2 if stack is “empty” (Z0 is on top)

  6. Transition function example • (q0,a, Z0) = {(q0 ,XZ0)} • (q0,a, X) = {(q0 ,XX)} • (q0,b, X) = {(q1 ,є)} • (q1,b, X) = {(q1 ,є)} • (q0,є, Z0) = {(q2 ,Z0)} • (q1,є, Z0) = {(q2 ,Z0)} • (q0,b, Z0) = (q1,b, Z0) = (q1,a, X) and many others are all equal to {} push X onto the stack on input a pop X from the stack on input b go to final stateif Z0 is on top

  7. PDA diagram a, Z0, XZ0 a, X, XX b, X, є b, X, є q1 q0 є, Z0, Z0 є, Z0, Z0 q2

  8. Alternative diagram b, pop X a, push X b, pop X q1 q0 є, empty є, empty q2

  9. 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 (Z0 has been popped out), regardless of what state it is currently in • These two “kinds” of PDAs are equivalent (proofs in section 6.2)

  10. 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

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