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1 CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 2 School of Innovation, Design and Engineering Mälardalen University 2011

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2 Content Languages, Alphabets and Strings Strings & String Operations Languages & Language Operations Regular Expressions Finite Automata, FA Deterministic Finite Automata, DFA

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3 Languages, Alphabets and Strings

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4 defined over an alphabet: Languages A language is a set of strings A String is a sequence of letters An alphabet is a set of symbols

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5 Alphabets and Strings We will use small alphabets: Strings

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6 Operations on Strings

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7 String Operations m n bbbv aaaw y bbbaaa x abba Concatenation (sammanfogning) xy abbabbbaaa

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8 Reverse (reversering) Example: Longest odd length palindrome in a natural language: saippuakauppias (Finnish: soap sailsman)

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9 String Length Length: Examples:

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10 Recursive Definition of Length For any letter: For any string : Example:

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11 Length of Concatenation vuuv aababaabuv 5, 3, vabaabv uaabu Example:

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12 Proof of Concatenation Length Claim: Proof: By induction on the length Induction basis: From definition of length:

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13 Inductive hypothesis: Inductive step: we will prove for

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14 Inductive Step Write, where From definition of length: From inductive hypothesis: Thus: END OF PROOF

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15 Empty String A string with no letters: (Also denoted as ) Observations:

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16 Substring (delsträng) Substring of a string: a subsequence of consecutive characters String Substring

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17 Prefix and Suffix Suffixes prefix suffix Prefixes

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18 Repetition Example: Definition: n n } (String repeated n times)

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19 The (Kleene* star) Operation the set of all possible strings from alphabet [* Kleene is pronounced "clay-knee“]

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20 The + (Kleene plus) Operation : the set of all possible strings from the alphabet except ,ba ,,,,,,,,,*aabaaabbbaabaaba

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21 Example *, oj, fy, usch, ojoj, fyfy,uschusch, ojfy, ojusch * ,fyoj , usch oj, fy, usch, ojoj, fyfy,uschusch, ojfy, ojusch

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22 Operations on Languages

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23 Language A language is any subset of Example: Languages: ,,,,,,,,*, aaabbbaabaaba ba },,,,,{,, aaaaaaabaababaabba aabaaa

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24 Example An infinite language

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25 Operations on Languages The usual set operations ,,,,,,,,,*aabaaabbbaabaaba Complement:

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26 Reverse Examples: Definition:

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27 Concatenation Definition: Example

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28 Repeat Definition: Special case:

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29 Example

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30 Star-Closure (Kleene *) Definition: Example:

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31 Positive Closure Definition *L 2 1 L LL

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32 Regular Expressions

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33 Regular Expressions: Recursive Definition are Regular Expressions Primitive regular expressions: Given regular expressions and

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34 Examples A regular expression: Not a regular expression:

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35 Zero or more. a* means "zero or more a's." To say "zero or more ab's," that is, {, ab, abab, ababab,...}, you need to say (ab)*. ab* denotes {a, ab, abb, abbb, abbbb,...}. Building Regular Expressions

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36 One or more. Since a* means "zero or more a's", you can use aa* (or equivalently, a*a) to mean "one or more a's.“ Similarly, to describe "one or more ab's," that is, {ab, abab, ababab,...}, you can use ab(ab)*. Building Regular Expressions

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37 Any string at all. To describe any string at all (with = {a, b, c}), you can use (a+b+c)*. Any nonempty string. This can be written as any character from followed by any string at all: (a+b+c)(a+b+c)*. Building Regular Expressions

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38 Any string not containing.... To describe any string at all that doesn't contain an a (with = {a, b, c}), you can use (b+c)*. Any string containing exactly one... To describe any string that contains exactly one a, put "any string not containing an a," on either side of the a, like this: (b+c)*a(b+c)*. Building Regular Expressions

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39 Languages of Regular Expressions Example language of regular expression

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40 Definition For primitive regular expressions:

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41 Definition (continued) For regular expressions and

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42 Example Regular expression:

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43 Example Regular expression

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44 Example Regular expression

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

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46 Example Regular expression (consists of repeating 1’s and 01’s). = { all strings without two consecutive 0 }

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47 Example = { all strings without two consecutive 0 } (In order not to get 00 in a string, after each 0 there must be an 1, which means that strings of the form are repeated. That is the first parenthesis. To take into account strings that end with 0, and those consisting of 1’s solely, the rest of the expression is added.) Equivalent solution:

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

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49 Example = { all strings without two consecutive 0 } and are equivalent regular expressions.

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Lennart Salling’s Video Resources Lennart Salling Introduktion: Program, strings, integers and integer functions Vad handlar kursen om? Kan alla problem lösas av program? Vad har stora och små oändligheter med saken att göra? Vad har språk och beräkningar med varandra att göra? Reguljära språk, vad är det? Vilka automater är specialiserade på reguljära språk? Varför icke-determinism? Hur ser problem ut som inte kan lösas av program? Hur kan man visa att ett problem inte kan lösas av program? 50

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Lennart Salling’s Video Video Resources ecidable_and_decidable_problems.mov ates.mov 51

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More Video’s 1 SubsetConstruction Infinities Rice'sTheorem Finite Automata Strings and Languages Regular Languages Accept and decide (TM) 52

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More Video’s 2 Regular Expression to NFA 2 - Convert Regular Expression to Finite-State Automaton Convert Regular Expression to DFA Great Principles of Computing - Peter J. Denning Stephen Wolfram: Computing a theory of everything Computing Beyond Turing - Jeff Hawkins 53

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