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1 Single Final State for NFAs and DFAs

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2 Observation Any Finite Automaton (NFA or DFA) can be converted to an equivalent NFA with a single final state

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3 NFA Equivalent NFA Example

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4 NFA In General Equivalent NFA Single final state

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5 Extreme Case NFA without final state Add a final state Without transitions

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6 Some Properties of Regular Languages

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7 Properties Concatenation:Star: Union: Are regular Languages For regular languages and we will prove that:

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8 We Say: Regular languages are closed under Concatenation:Star: Union:

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9 Regular language Single final state NFA Single final state Regular language NFA

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10 Example

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11 Union NFA for

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12 Example NFA for

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13 Concatenation NFA for

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14 Example NFA for

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15 Star Operation NFA for

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16 Example NFA for

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17 Regular Expressions

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18 Regular Expressions Regular expressions describe regular languages Example: describes the language

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19 Recursive Definition Are regular expressions Primitive regular expressions: Given regular expressions and

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20 Examples A regular expression: Not a regular expression:

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21 Languages of Regular Expressions : language of regular expression Example

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22 Definition For primitive regular expressions:

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23 Definition (continued) For regular expressions and

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24 Example Regular expression:

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25 Example Regular expression

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26 Example Regular expression

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27 Example Regular expression = { all strings with at least two consecutive 0 }

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28 Example Regular expression = { all strings without two consecutive 0 }

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29 Equivalent Regular Expressions Definition: Regular expressions and are equivalent if

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30 Example = { all strings with at least two consecutive 0 } and are equivalent regular expr.

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31 Regular Expressions and Regular Languages

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32 Theorem Languages Generated by Regular Expressions Regular Languages

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33 Theorem - Part 1 1. For any regular expression the language is regular Languages Generated by Regular Expressions Regular Languages

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34 Theorem - Part 2 Languages Generated by Regular Expressions Regular Languages 2. For any regular language there is a regular expression with

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35 Proof - Part 1 1. For any regular expression the language is regular Proof by induction on the size of

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36 Induction Basis Primitive Regular Expressions: NFAs regular languages

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37 Inductive Hypothesis Assume for regular expressions and that and are regular languages

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38 Inductive Step We will prove: Are regular Languages

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39 By definition of regular expressions:

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40 By inductive hypothesis we know: and are regular languages Regular languages are closed under union concatenation star We also know:

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41 Therefore: Are regular languages

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42 And trivially: is a regular language

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43 Proof – Part 2 2. For any regular language there is a regular expression with Proof by construction of regular expression

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44 Since is regular take the NFA that accepts it Single final state

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45 From construct the equivalent Generalized Transition Graph transition labels are regular expressions Example:

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46 Another Example:

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47 Reducing the states:

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48 Resulting Regular Expression:

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49 In General Removing states:

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50 Obtaining the final regular expression:

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