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CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 2

CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 2 School of Innovation, Design and Engineering Mälardalen University 2011. Content Languages, Alphabets and Strings Strings & String Operations Languages & Language Operations Regular Expressions

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CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 2

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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

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

  3. Languages, Alphabets and Strings

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

  5. Alphabets and Strings • We will use small alphabets: Strings

  6. Operations on Strings

  7. = w a a a L x abba 1 2 n = v b b b y  bbbaaa L 1 2 m String Operations Concatenation (sammanfogning) xy abbabbbaaa

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

  9. Length: String Length Examples:

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

  11. = = u aab , u 3 = = v abaab , v 5 = = uv aababaab 8 = + = + = uv u v 3 5 8 Length of Concatenation Example:

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

  13. Inductive hypothesis: for • Inductive step: we will prove for

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

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

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

  17. prefix suffix Prefix and Suffix • Suffixes Prefixes

  18. (String repeated n times) Repetition n = • Example: • Definition: w ww... w } n

  19. The (Kleene* star) Operation • the set of all possible strings from alphabet [* Kleene is pronounced "clay-knee“] http://en.wikipedia.org/wiki/Kleene_star

  20. { } S = l * , a , b , aa , ab , ba , bb , aaa , aab , K The + (Kleene plus) Operation :the set of all possible strings from the alphabet except { } S = a , b

  21. } { S = oj , fy , usch { S = l, oj, fy, usch, ojoj, fyfy,uschusch, ojfy, ojusch * } + + S S K Example = S - l * { = oj, fy, usch, ojoj, fyfy,uschusch, ojfy, ojusch } K

  22. Operations on Languages

  23. Language • A language is any subset of • Example: • Languages: { } S = a , b { } S = l * , a , b , aa , ab , ba , bb , aaa , K { } l { } a , aa , aab l { , abba , baba , aa , ab , aaaaaa }

  24. Example • An infinite language

  25. { } S = l * , a , b , aa , ab , ba , bb , aaa , aab , K Complement: Operations on Languages • The usual set operations

  26. Reverse Definition: Examples:

  27. Concatenation • Definition: Example

  28. Repeat • Definition: • Special case:

  29. Example

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

  31. Positive Closure • Definition + 1 2 = L L U L U L { } = - l L *

  32. Regular Expressions

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

  34. A regular expression: Not a regular expression: Examples

  35. Building Regular Expressions • 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, ...}.

  36. Building Regular Expressions • 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)*.

  37. Building Regular Expressions • 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)*.

  38. Building Regular Expressions • 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)*.

  39. Languages of Regular Expressions language of regular expression Example

  40. Definition • For primitive regular expressions:

  41. Definition (continued) • For regular expressions and

  42. Example Regular expression:

  43. Example • Regular expression

  44. Example • Regular expression

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

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

  47. Example = { all strings without two consecutive 0 } Equivalent solution: (In order not to get 00 in a string, after each 0 there must be an 1, which means that strings of the form 1....101....1 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.)

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

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

  50. Lennart Salling’s Video Resources • http://www.math.uu.se/~salling/Lennart Salling • http://www.math.uu.se/~salling/AUTOMATA_DV/index.html • Introduktion: http://www2.math.uu.se/~salling/Movies/Intro%20to%20Automata.mov • Program, strings, integers and integerfunctions • http://www2.math.uu.se/~salling/Movies/StringsNumbersAndFunctions.mov • http://www.youtube.com/embed/EuhbL0MPryU Vad handlar kursen om? • http://www.youtube.com/embed/VM5SUcyY4sI?hl=en&fs=1 Kan alla problem lösas av program? • http://www.youtube.com/embed/acsUNSkzUgg?hl=en&fs=1 Vad har stora och små oändligheter med saken att göra? • http://www.youtube.com/embed/eQoP_kuNgXU?hl=sv&fs=1 Vad har språk och beräkningar med varandra att göra? • http://www.youtube.com/embed/-4K72wsQmPI?hl=sv&fs=1 Reguljära språk, vad är det? • http://www.youtube.com/embed/R2bHYnBXWFs?hl=sv&fs=1 Vilka automater är specialiserade på reguljära språk? • http://www.youtube.com/embed/2PNyEWl1AI0?hl=sv&fs=1 Varför icke-determinism? • http://www.youtube.com/embed/2abHjjS8Tqc?hl=sv&fs=1 Hur ser problem ut som inte kan lösas av program? • http://www.youtube.com/embed/WMN_wz-b3K0?hl=sv&fs=1 • http://www.youtube.com/embed/y-zMnV3G9pg?hl=sv&fs=1 Hur kan man visa att ett problem inte kan lösas av program?

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