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1 More Properties of Regular Languages

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2 We have proven Regular languages are closed under: Union Concatenation Star operation Reverse

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3 Namely, for regular languages and : Union Concatenation Star operation Reverse Regular Languages

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4 We will prove Regular languages are closed under: Complement Intersection

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5 Namely, for regular languages and : Complement Intersection Regular Languages

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6 Complement Theorem: For regular language the complement is regular Proof: Take DFA that accepts and make nonfinal states final final states nonfinal Resulting DFA accepts

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7 Example:

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8 Intersection Theorem: For regular languages and the intersection is regular Proof:Apply DeMorgan’s Law:

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9 regular

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10 Standard Representations of Regular Languages

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11 Standard Representations of Regular Languages Regular Languages DFAs NFAs Regular Expressions Regular Grammars

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12 When we say: We are given a Regular Language We mean:Language is in a standard representation

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13 Elementary Questions about Regular Languages

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14 Membership Question Question:Given regular language and string how can we check if ? Answer:Take the DFA that accepts and check if is accepted

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15 DFA

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16 Given regular language how can we check if is empty: ? Take the DFA that accepts Check if there is a path from the initial state to a final state Question: Answer:

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17 DFA

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18 Given regular language how can we check if is finite? Take the DFA that accepts Check if there is a walk with cycle from the initial state to a final state Question: Answer:

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19 DFA is infinite DFA is finite

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20 Given regular languages and how can we check if ? Question: Find if Answer:

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21 and

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22 or

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23 Non-regular languages

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24 Regular languages Non-regular languages

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25 How can we prove that a language is not regular? Prove that there is no DFA that accepts Problem: this is not easy to prove Solution: the Pumping Lemma !!!

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26 The Pigeonhole Principle

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27 pigeons pigeonholes

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28 A pigeonhole must contain at least two pigeons

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29........... pigeons pigeonholes

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30 The Pigeonhole Principle........... pigeons pigeonholes There is a pigeonhole with at least 2 pigeons

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31 The Pigeonhole Principle and DFAs

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32 DFA with states

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33 In walks of strings:no state is repeated

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34 In walks of strings:a state is repeated

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35 If the walk of string has length then a state is repeated

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36 If in a walk of a string transitions states of DFA then a state is repeated Pigeonhole principle for any DFA:

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37 In other words for a string : transitions are pigeons states are pigeonholes

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38 In general: A string has length number of states A state must be repeated in the walk...... walk of

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39 The Pumping Lemma

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40 Take an infinite regular language DFA that accepts states

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41 Take string with There is a walk with label :......... walk

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42 If string has lengthnumber of states then, from the pigeonhole principle: a state is repeated in the walk...... walk

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43 Write......

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44...... Observations:length number of states length

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45 The string is acceptedObservation:......

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46 The string is accepted Observation:......

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47 The string is accepted Observation:......

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48 The string is accepted In General:......

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49 In other words, we described: The Pumping Lemma !!!

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50 The Pumping Lemma: Given a infinite regular language there exists an integer for any string with length we can write with and such that:

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51 Applications of the Pumping Lemma

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52 Theorem: The language is not regular Proof: Use the Pumping Lemma

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53 Assume for contradiction that is a regular language Since is infinite we can apply the Pumping Lemma

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54 Let be the integer in the Pumping Lemma Pick a string such that: length Example: pick

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55 Write: it must be that: length From the Pumping Lemma Therefore:

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56 From the Pumping Lemma: Thus: We have:

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57 Therefore: BUT: CONTRADICTION!!!

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58 Our assumption that is a regular language is not true Conclusion: is not a regular language Therefore:

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59 Regular languages Non-regular languages

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