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Pushdown Automaton (PDA)

Pushdown Automaton (PDA). A Pushdown Automaton is a nondeterministic finite state automaton (NFA) that permits ε -transitions and a stack. Pushdown Automaton (PDA). Q : A finite set of states. S : A finite set of input symbols. G : A finite stack alphabet.

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Pushdown Automaton (PDA)

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  1. Pushdown Automaton (PDA) A Pushdown Automaton is a nondeterministic finite state automaton (NFA) that permits ε-transitions and a stack.

  2. Pushdown Automaton (PDA) • Q: A finite set of states. • S : A finite set of input symbols. • G: A finite stack alphabet. • d: The transition function with input: • qi is a state in Q. • a is a symbol in S or a = e (the empty string). • tm is a stack symbol, tmÎG. • and the output is a finite set of pairs: • qk the new state. • tn is the string of stack symbols that replaces tm at the top of the stack. If tn = e, then the stack is popped. • q0: The start state. • t0 : Initially, the PDA’s stack consists this symbol and nothing else. • F : The set of accepting states.

  3. PDA Example: The language, Lwwr, is the even-length palindromes over alphabet {0,1}. Lwwr is a Context-Free Language (CFL) generated by the grammar: One PDA for Lwwr is given on the following slide...

  4. PDA for Lwwr

  5. A Graphical Notation for PDA’s • The nodes correspond to the states of the PDA. • An arrow labeled Start indicates the unique start state. • Doubly circled states are accepting states. • Edges correspond to transitions in the PDA as follows: • An edge labeled (ai, tm)/tn from state q to state p means that d(q, ai, tm) contains the pair (p, tn), perhaps among other pairs.

  6. q0 q1 q2 q3 Graphical Notation for PDA of Lwwr (0, 0)/00 (0, 1)/01 (1, 0)/10 (1, 1)/11 (0,0)/ε (1,1)/ε (0, t0)/0t0 (1, t0)/1t0 start (ε,t0) / t0 (ε, t0) / t0 (ε, 0) / 0 (ε, 1) / 1 (EOF,t0) / t0 All possibilities that do not have explicit edges, have implicit edges that go to an implicit reject state. • This is a nondeterministic machine. • Think of the machine as following all possible paths. • Kill a path if it leads to a reject state. • If any path leads to an accept state, then the machine accepts.

  7. Exercise 1 Design a PDA that recognizes legal sequences of ‘if’ and ‘else’ statements in a C program. In the PDA, let ‘i’ stands for ‘if’ and ‘e’ stands for ‘else’. Hint: There is a problem whenever the number of ‘else’ statements in any prefix exceeds the number of ‘if’ statements in that prefix.

  8. Exercise 2 Design a PDA to accept the language:

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