September 22

# September 22 - CSCI 2670 Introduction to Theory of...

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CSCI 2670 Introduction to Theory of Computing September 22, 2005

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September 22, 2005 Agenda Yesterday Pushdown automata Today Equivalence of pushdown automata and CFG’s Pumping lemma for CFG’s
September 22, 2005 Announcements Matrix Reloaded tonight! 6:30 in Boyd 328 Free popcorn 25 cent Coke Pizza will be ordered at cost if people want

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September 22, 2005 Equivalence of PDA’s and CFG’s Theorem: A language is context free if and only if some pushdown automaton recognizes it Proved in two lemmas – one for the “if” direction and one for the “only if” direction We will only do the “only if” step – i.e., show that every context-free language has an associated PDA
September 22, 2005 CFG’s are recognized by PDA’s Lemma: If a language is context free, then some pushdown automaton recognizes it Proof idea: Construct a PDA following CFG rules

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September 22, 2005 CFG’s are recognized by PDA’s Format of the new PDA q q q ε , ε S\$ ε , \$ ε a,a ε ε ,A w Start by pushing the start variable and stack bottom marker Have a transition for each rule replacing the variable with its right hand side Have a transition that allows us to read each alphabet symbol if it is at the top of the stack Finish only if the stack is empty
September 22, 2005 Constructing the PDA You can read any symbol in Σ when that symbol is at the top of the stack Transitions of the form a,a ε The rules will be pushed onto the stack – when a variable A is on top of the stack and there is a rule A w, you pop A and push w You can go to the accept state only if the stack is empty

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Idea of PDA construction for A xBz State control a b A t State control a b x B z t
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September 22 - CSCI 2670 Introduction to Theory of...

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