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# 2001Lec7 - CSE 2001 Introduction to Theory of Computation...

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2/28/2006 CSE 2001, Winter 2006 1 CSE 2001: Introduction to Theory of Computation Winter 2006 Suprakash Datta [email protected] Office: CSEB 3043 Phone: 416-736-2100 ext 77875 Course page: http://www.cs.yorku.ca/course/2001 Some of these slides are adapted from Wim van Dam’s slides ( www.cs.berkeley.edu/~vandam/CS172/ ) 2/28/2006 CSE 2001, Winter 2006 2 Next Equivalence of PDA’s, CFL’s Non-CF languages CFL pumping lemma Closure properties of CFLs 2/28/2006 CSE 2001, Winter 2006 3 PDAs and CFL Theorem 2.12 : A language L is context-free if and only of there is a pushdown automata M that recognizes L. Two step proof : 1) Given a CFG G, construct a PDA M G 2) Given a PDA M, make a CFG G M 2/28/2006 CSE 2001, Winter 2006 4 Converting a CFL to a PDA Basic idea Start with the start symbol Non-deterministically apply rules Stop when a string of terminals is left. Compare with input – accept if match. Detail: How do you store the intermediate string on the stack? 2/28/2006 CSE 2001, Winter 2006 5 Converting a CFL to a PDA - 2 In q start , initialize the stack to contain S\$ and make an ε -transition to q loop . In q loop , if top of stack contains a variable A, replace it with a terminal w for some rule A w Else, if top of stack contains a terminal w, compare with next input symbol. If unequal, reject. Else if the top of the stack is \$ (empty stack), make an ε -transition to q accept . 2/28/2006 CSE 2001, Winter 2006 6 Converting a PDA to a CFG Basic idea: For each pair of states p,q use a variable A pq which generates all strings that take the PDA from p to q. The start variable is A q 0 q accept Details: Assume that the PDA – Has a single accept state – Empties stack before accepting – Each transition either pops or pushes.

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2/28/2006 CSE 2001, Winter 2006 7 Converting a PDA to a CFG - 2 p,q,r,s Q, t ∈ Γ , a,b ∈ Σ ε , if δ (p,a, ε ) contains (r,t) and if δ (s,b,t) contains (q, ε ) add the rule A pq aA rs b p,q,r Q, add the rule A pq A pr A rq p Q, add the rule A pp ε 2/28/2006 CSE 2001, Winter 2006 8 Converting a PDA to a CFG - 3 Correctness - Easier direction: If A pq generates x, then x brings PDA P from p with empty stack to q with empty stack.
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2001Lec7 - CSE 2001 Introduction to Theory of Computation...

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