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1 Let’s Recapitulate
2 Regular Languages DFAs NFAs Regular Expressions Regular Grammars
3 A standard representation of a regular language : A DFA that accepts A NFA that accepts A regular expression that generates A regular grammar that generates
4 When we say: “We are given a Regular Language “ We mean: Language in a standard representation
5 Elementary Questions about Regular Languages
6 Question: Given regular language how can we check if a string ?
7 Question: Given regular language how can we check if a string ? Answer: Take the DFA that accepts and check if is accepted
8 Question: Given regular language how can we check if is empty, finite, infinite ? Answer: Take the DFA that accepts Then check the DFA
9 If there is a walk from the start state to a final state then: is not empty If the walk contains a cycle then: is infinite Otherwise finite Otherwise empty
10 Question: Given regular languages and how can we check if ?
11 Question: Given regular languages and how can we check if ? Answer: take And find if
12 Question: Given language how can we check if is not a regular language ?
13 Question: Given language how can we check if is not a regular language ? Answer: The answer is not obvious We need the Pumping Lemma
14 The Pigeonhole Principle
15 4 pigeons 3 pigeonholes
16 A pigeonhole must have two pigeons
17........... pigeons pigeonholes
18 The Pigeonhole Principle........... pigeons pigeonholes There is a pigeonhole with at least 2 pigeons
19 The Pigeonhole Principle and DFAs
20 DFA with states
21 In walks of strings: no state is repeated
22 In walks of strings: a state is repeated
23 If the walk of string has length Then a state is repeated
24 If in a walk: transitions states Then: A state is repeated The pigeonhole principle:
25 In other words: transitions are pigeons states are pigeonholes
26 In general: A string has length number of states A state must be repeated in the walk......
27 The Pumping Lemma
28 Take an infinite regular language DFA that accepts states
29 Take string with There is a walk with label :.........
30 If string has lengthnumber of states Then, from the pigeonhole principle: A state is repeated in the walk......
32...... Observations : length number of states length
33 The string is accepted Observation:......
34 The string is accepted Observation:......
35 The string is accepted Observation:......
36 The string is accepted In General:......
37 In other words, we described: The Pumping Lemma
38 The Pumping Lemma: 1. Given a infinite regular language 2. There exists an integer 3. For any string with length 4. We can write 5. With and 6. Such that: string
39 Applications of the Pumping Lemma
40 Claim: The language is not regular Proof: Use the Pumping Lemma
41 Proof: Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma
42 Let be the integer in the Pumping Lemma Pick a string such that: length Example: pick
43 Write it must be that length From the Pumping Lemma Therefore:
44 From the Pumping Lemma: Thus:
45 Therefore, BUT: and CONTRADICTION!!!
46 Our assumption that is a regular language cannot be true CONCLUSION: is not a regular language Therefore:
47 Claim: The language is not regular Proof: Use the Pumping Lemma
48 Proof: Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma
49 Let be the integer in the Pumping Lemma Pick a string such that: length Example: pick
50 Write it must be that length From the Pumping Lemma Therefore:
51 From the Pumping Lemma: Thus:
52 Therefore, BUT: and CONTRADICTION!!!
53 Our assumption that is a regular language cannot be true CONCLUSION: is not a regular language Therefore:
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