1 CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 2 School of Innovation, Design and Engineering Mälardalen University 2012

2 Content Languages, Alphabets and Strings Strings & String Operations Languages & Language Operations Regular Languages and their Representations Regular Expressions

3 Languages, Alphabets and Strings Based on C Busch, RPI, Models of Computation

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

5 Alphabets and Strings We will use small alphabets: Strings

6 Operations on Strings

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

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

9 String Length Length: Examples:

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

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

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

13 Inductive hypothesis: 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 and Suffix Suffixes prefix suffix Prefixes

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

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

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

21 Example    *, oj, fy, usch, ojoj, fyfy,uschusch, ojfy, ojusch     *  ,fyoj , usch    oj, fy, usch, ojoj, fyfy,uschusch, ojfy, ojusch   

22 Operations on Languages

23 Language A language is any subset of Example: Languages:   ,,,,,,,,*, aaabbbaabaaba ba     },,,,,{,, aaaaaaabaababaabba aabaaa

24 Example An infinite language

25 Operations on Languages The usual set operations  ,,,,,,,,,*aabaaabbbaabaaba  Complement:

26 Reverse Examples: Definition:

27 Concatenation Definition: Example

28 Repeat Definition: Special case:

29 Example

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

31 Positive Closure Definition   *L    2 1 L LL 

32 Regular Languages

Definition of Set of Regular Languages Basis Clause: where    are regular languages. Inductive Clause: If L r and L s are regular languages, then L r  L s, L r L s and L r * are regular languages. Nothing is a regular language unless it is obtained from the above two clauses. 33

34 Standard Representations of Regular Languages

35 Regular Language Representations DFA NFA Regular Expression Regular Grammar Regular Language

36 Regular Expressions

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

38 Examples A regular expression: Not a regular expression:

39 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

40 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

41 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

42 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

43 Languages of Regular Expressions Example language of regular expression

44 Definition For primitive regular expressions:

45 Definition (continued) For regular expressions and

46 Example Regular expression:

47 Example Regular expression

48 Example Regular expression

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

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

51 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:

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

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

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? 54

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

More Video’s 1 SubsetConstruction Infinities Rice'sTheorem Finite Automata Strings and Languages Regular Languages Accept and decide (TM) 56

More Video’s 2 Regular Expression to NFA 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 57