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2001Lec3a

2001Lec3a - CSE 2001 Introduction to Theory of Computation...

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1/31/2006 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/ ) and from Nathaly Verwaal ( cpsc.ucalgary.ca/~verwaal/313/F2005 /) 1/31/2006 CSE 2001, Winter 2006 2 Non-regular Languages §1.4 Which languages cannot be recognized by finite automata? Examples: L= {0 n 1 n | n N } PAL = {w | w = w R } • ‘Playing around’ with FA convinces you that the ‘finiteness’ of FA is problematic for “all n N • The problem occurs between the 0 n and the 1 n • Informal: cannot count arbitrary integers using a fixed number of states ; the memory of a FA is limited by the the number of states |Q| 1/31/2006 CSE 2001, Winter 2006 3 Proving non-regularity Simpler technique (not in the text) -- less general Pumping Lemma (Sec 1.4) – More difficult – More general 1/31/2006 CSE 2001, Winter 2006 4 Distinguishing strings wrt L Define L/x = {z ∈Σ * | xz L} x,y distinguishable with respect to L if L/x L/y z xz L, yz L E.g. L = {w| w ends in 10} x = 01, y = 00. z = 0 distinguishes x,y 1/31/2006 CSE 2001, Winter 2006 5 Theorem Suppose that for L Σ * and for some positive integer n, there are n strings in Σ * such that any two of them are distinguishable with respect to (wrt) L. Then every DFA recognizing L must have at least n states. Notation: δ *(q 0 ,x 1 ) – generalization of transition function. 1/31/2006 CSE 2001, Winter 2006 6 Proof of Theorem Suppose x 1 , …, x n are n strings, any two of which are distinguishable wrt L. If M is a DFA with fewer than n states, then by the Pigeonhole Principle , the states δ *(q 0 ,x 1 ), …, δ *(q 0 ,x n ) are not all distinct, and so for some i j, δ *(q 0 ,x i ) = δ *(q 0 ,x j ). Since x i , x j are distinguishable wrt L, it follows that M cannot recognize L.

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1/31/2006 CSE 2001, Winter 2006 7 Applying this Theorem Show that there is an infinite set of strings , any two of which are distinguishable wrt L. So any DFA recognizing L cannot have a finite number of states – i.e., it does not exist. 1/31/2006 CSE 2001, Winter 2006 8 Example: PAL is not regular Any two distinct strings x,y ∈Σ * can be distinguished wrt PAL. If |x| = |y|, we can choose z = x R so that xz = x x R PAL and yz = y x R PAL. Else, w.l.o.g., |x| < |y|. Let y=y 1 y 2 , |y 1 | = |x|. Then let z = ww R x R , |w| = |y 2 |, w y 2 . So xz= xww R x R PAL, yz = y 1 y 2 ww R x R PAL, because w y 2 . 1/31/2006 CSE 2001, Winter 2006 9 Next: The Pumping Lemma 1/31/2006 CSE 2001, Winter 2006 10 Repeating DFA Paths q 1 q k q j Consider an accepting DFA M with size |Q| On a string of length p, p+1 states get visited For p |Q|, there must be a j such that the computational path looks like: q 1 ,…,q j ,…,q j ,…,q k 1/31/2006 CSE 2001, Winter 2006 11 Repeating DFA Paths q 1 q k q j The action of the DFA in q j is always the same.
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2001Lec3a - CSE 2001 Introduction to Theory of Computation...

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