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Recap: Nondeterministic Finite Automaton (NFA)

Recap: Nondeterministic Finite Automaton (NFA). A deterministic finite automaton (NFA) is a 5-tuple (Q, ,,s,F) where: Q is a finite set of elements called states  is a finite input alphabet s  Q called the start state F  Q called the favorable states.

deborah-mosley
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Recap: Nondeterministic Finite Automaton (NFA)

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  1. Recap: Nondeterministic Finite Automaton (NFA) • A deterministic finite automaton (NFA) is a 5-tuple (Q,,,s,F) where: • Q is a finite set of elements called states •  is a finite input alphabet • s Q called the start state • F  Q called the favorable states • : (Q × (  {e}))  (Q) Constant! … a1 a2

  2. p Suppose that while processing a w from state r only by going to state p, a favorable state can be reached. Then, the oracle will choose p a r e q Acceptance of a Word in an NFA A • Acceptance: w is accepted by A if at least one sequence of transformations yields the empty word and end up in an accepting state • The oracle explanation of NFAs: there is an oracle that always makes the right choice Oracle will find a path to an accepting state that process all characters only if there is one such path

  3. Pushdown Automaton stack a’1 a’1 … • A pushdown automaton is a 6-tuple (Q,,, ,s,F) where: • Q is a finite set of elements called states •  is a finite input alphabet •  is a set of stack symbols • s Q called the start state • F  Q called the favorable states • : (Q × (  {e}) × (  {e})) (Q × (  {e})) Constant! … a1 a2

  4. Computation in Pushdown Automata PA = (Q,,, ,s,F) where: • :(Q × (  {e}) × (  {e})) (Q × (  {e})) Let (q’,))  ((q, ,)), it is compute as follows: • if: • the automaton is in an state q • the next character is  • The word  is on top of the stack • Then: • pop  from top of the stack • Jump to the state q’ • Push the word  on top of the stack

  5. String Accepted by Pushdown Automaton • Given a pushdown automaton A, and a string w *, we say that w is accepted by A if there is a sequence of computations that: • begin from the starting state, the word w, and the empty stack • terminates in a favorable state and process all characters in w • Note: the stack does not have to be empty at the end The set of all words accepted by an automaton A form the language L(A) accepted or recognized by the automaton.

  6. Example Pushdown automaton recognizing {anbn : n = 0, 1, 2, …}

  7. Example Pushdown automaton recognizing L = {wwR : w is in *}

  8. Homework for Friday 12. • 2.3 (n), (o) • 2.4 (c) • 2.6 (b) • 2.21 • Construct a pushdown automaton for the language of exercise 2.21

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