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l6-handout - Outline Pumping Lemma Lecture 6 Pumping Lemma...

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Lecture 6: Pumping Lemma and Closure Properties David Dill Department of Computer Science 1 Outline Pumping Lemma Closure Properties 2 Pumping Lemma Primary application: Proving that certain languages are not regular. Lemma: For every regular language L , there exists a positive constant n such that for every string w in L such that | w | ≥ n , there exist strings x, y, z such that w = xyz with the following properties: y 6 = • | xy | ≤ n xy k z L for all k 0 . 3 Understanding the Pumping Lemma The pumping lemma is a little difficult to understand because of all the ”alternating quantifiers” ( L n w x, y, z . . . ) With such theorems, it is often helpful to think of a game with two players. Player 1: Choose a regular language L Player 2: Chooses the constant n . Player 1: Chooses w L such that | w | ≥ n . Player 2: Chooses xyz = w such that y 6 = and | xy | ≤ n . Player 1: Chooses k . If either player “cheats” (e.g., player 1 chooses a non-regular language), the other player immediately wins.

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