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Pushdown Automata

[Section 7.1]. Pushdown Automata. like NFA-  but also has a stack transition takes the current state, the current input symbol, and the top-of-the-stack symbol (which is removed from the stack) and returns a state and a string to place in the stack

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Pushdown Automata

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  1. [Section 7.1] Pushdown Automata • like NFA- but also has a stack • transition takes the current state, the current input symbol, and the top-of-the-stack symbol (which is removed from the stack) and returns a state and a string to place in the stack • the automaton starts in the initial state, reading the first symbol on the tape, and having a special initial symbol on the stack • automaton accepts a string x if, after reading all of x, it can get to an accepting state (the stack content does not matter) • if the stack is empty, the automaton is stuck

  2. [Section 7.1] Pushdown Automata Example: L = { akbk | k ¸ 0 }

  3. [Section 7.1] Pushdown Automata Example: L = { w 2 {a,b}* | w is an even-length palindrome }

  4. [Section 7.1] Pushdown Automata • Def: A pushdown automaton (PDA) is a 7-tuple (Q,,,q0,Z0,A,) where • Q is a finite set of states •  is a finite input alphabet •  is a finite stack alphabet • q02 Q is the initial state • Z02 is the initial stack symbol • A µ Q is the set of accepting states •  is the transition function from ______________ to finite subsets of Q £*

  5. [Section 7.1] Pushdown Automata Def: Let M = (Q,,,q0,Z0,A,) be a PDA. A configuration is a triple (p,x,) where p is the current state, x is the rest of the input (the yet-unread part), and  is the content of the stack (the top is on the left). We write (p,x,) `M (q,y,) if we can get from the configuration (p,x,) to the configuration (q,y,) using one transition of M. Similarly we write (p,x,) `M* (q,y,) if we can get from (p,x,) to (q,y,) through a finite sequence of transitions of M. The language accepted by M : L(M) = { x 2* | ________________________ }

  6. [Section 7.2] Deterministic Pushdown Automata Def: Let M = (Q,,,q0,Z0,A,) be a PDA. M is deterministic (DPDA) if A language L is a deterministic context-free language (DCFL) if there exists a DPDA accepting L.

  7. [Section 7.2] Deterministic Pushdown Automata Example: Give a DPDA for L = { akbk | k ¸ 1 }. Can you give a DPDA for L2 = { akbk | k ¸ 0 } ?

  8. [Section 7.2] Determinism vs. Nondeterminism For finite automata : For PDA : There exists a language which can be accepted by a PDA but not by a DPDA !

  9. [Section 7.2] Determinism vs. Nondeterminism Example: Is the language given by S  SS | (S) |  a DCFL ?

  10. [Section 7.2] Determinism vs. Nondeterminism Example: { x 2 {a,b}* | number of a’s in x = number of b’s in x }

  11. [Section 7.3] CFG vs. PDA It is possible to show that every language generated by a CFG can be accepted by a PDA and vice versa. Thm: Let G = (V,G,S,P) be a CFG. Then there exists a PDA M = (Q,M,,q0,Z0,A,) such that L(M) = L(G). Thm: Let M = (Q,M,,q0,Z0,A,) be a PDA. Then there exists a CFG G = (V,G,S,P) such that L(G) = L(M). Which one is easier to prove ?

  12. [Section 7.3] CFG vs. PDA Thm: Let G = (V,G,S,P) be a CFG. Then there exists a PDA M = (Q,M,,q0,Z0,A,) such that L(M) = L(G). Example: S  S + T | T T  T * R | R R  (S) | x

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