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  Deterministic Pushdown Automata (DPDA), Finite Automata with External Storage

Deterministic pushdown automata (DPDA)

PDA: FSM controls the stack. Un-bounded memory, limited last-in-first-out (or LIFO) access.

Stack operations: push, pop, test empty. Abbreviation: Aε = A ∪{ ε }, Bε = B ∪ {ε} .

Definition DPDA:   M= (Q, A, B, f, q0, F) or M= (Q, A, B, f, q0) in the version ‘accept by the empty stack’

Q: set of states, q0: initial state, F ⊆ Q: final or the accepting states
A: input alphabet; B: stack alphabet (convention: comprise a for all a ∈ A , and bottom of stack symbol ¢)

Transition fct f: Q x Aε x Bε -> Q x B*, transitions (q, a, b) -> (q’, v),  a ∈ Aε , b ∈ Bε , v ∈ B*.

If M is shown in a diagram, then this transition is drawn as arrow from q to q’ labeled  a, b -> v  or  a, b/v.

Reading the stack symbol implies ‘pop’. When the stack is not consulted, read ε from stack.

Example: Parenthesis expressions including 2 pairs of parentheses,( ) and [ ] are recognized by a 1-state controller that accepts by the empty stack (that is, additional error or trap state and error transitions are not shown).

2272_DPDA.jpg

Transition function f: current state, input symbol, pop top of stack -> next state, push stack

q, (, ε -> q, (;   q, [, ε -> q, [ ;   q, ), (  -> q, ε ;    q, ], [  -> q, ε

Example: M accepts (through empty stack) palindromes with center marker c. L = {w c wreverse | w ∈ {0, 1}*}.

2091_example of DPDA.jpg

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