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Erik Jonsson School of Engineering and Computer Science
FEARLESS Engineering
CS 5349 – 001 CS 4384 – 001
Automata Theory
http://www.utdallas.edu/~pervin
Thursday: EXAMINATION 1
Tuesday: Context-free LanguagesTuesday 9-30-14
Extra AssignmentOnly for CS 5349 Students!
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The Pumping Lemma Game
We play against an opponent. Our goal is to win the game by establishing a contradiction to the PL, while the opponent tries to foil us. There are four moves in the game.
1) The opponent picks p.2) Given p we pick a string s(p) in L of length ≥ p. 3) The opponent chooses the decomposition xyz subject
to |xy| ≤ p, |y| ≥ 1. We have to assume that the opponent makes the choice that will make it harder for us to win the game.
4) We pick i so that the pumped string is not in L.
5Don't forget! It will be on the comprehensive final exam!
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In class I pointed out that intersecting with the regular language a* makes the problem slightly easier because one would have to pick the s_p we used above.
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Theorem: Let M be a DFA with p states.
(i) L(M) is not empty iff M accepts a string z with |z| < p.
(ii) L(M) has an infinite number of members iff M accepts a string z with p <= |z| < 2p.
In each case we used the Pumping Lemma to pump “down” to show that the smallest member of the language cannot be of length (i) greater or equal to p; (ii) greater or equal to 2p.
Decision Procedures
11Slightly modified
12M&S P. 84 #2.20(4)
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13M&S Problem 2.21
See M&S P. 85 #2.26-2.28
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16Martin, P.120 #6.8b
17Linz, P. 89 #9b
18Du, P. 53, Example 9.3
19M&S, P. 64
a*b[(b + ab*a)a*b]*
BOOK:
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In class, on the board, I considered the language L = {ww^R | w \in {a,b}*}. (Where w^R is the word w reversed.) I suggested s(p) = a^pbba^p would work for the Pumping Lemma since the two b’s must be and the end of w and the beginning of w^R so our opponent must choose y consisting only of a’s and the b’s would still indicate the middle of the pumped string.
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Example: Sudkamp 2-22The set of strings over {a,b} with an even number of a’s and an even number of b’s.
Slide #16 from first class!
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Left-Linear Grammars26
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