Talen en Automaten Lecture 1: Regular Languages Herman Geuvers 1/19

Transcription

1 Talen en Automaten Lecture 1: Regular Languages Herman Geuvers 1/19

2 Outline Organisation Regular Languages 2/19

3 About this course I Lectures Teachers: Jurriaan Rot (except today: Herman Geuvers) Weekly, 2 hours, on Tuesdays 8:45 10:30 Presence not compulsory... but active, polite attitude expected, when present The lectures follow: these slides, available via the web Languages and Automata by Alexandra Silva (LnA) Course URL for all info (slides, exercises, schedule etc.): (Link exists in blackboard, under Announcements ). Check there first, before you dare to ask/mail a question! 3/19

4 About this course II Exercises There are weekly exercises; the ones marked with points are to be handed in. Handing in is compulsory: To receive a grade for the course, you have to hand in every week. Exercises must be done individually Weekly exercise classes, on Fridays, 8:45 10:30 (for most of you), 10:45-12:30 (for Science students), and one class 15:45 17:30. Presence not compulsory Answers (for old exercises) & Questions (for new ones) Schedule: New exercises on the web: Tuesday afternoon Next exercise meeting (Friday) you can ask questions Hand-in: Tuesday before 8:45, handwritten or typed, in the delivery boxes or by to the assistant of your group. 4/19

5 About this course III Exercise Classes Michiel de Bondt 8:45 10:30 Demian Janssen 8:45 10:30 Alexis Linard 8:45 10:30 David Venhoek 8:45 10:30 Tom van Bussel 8:45 10:30 Leon Gondelmans 8:45 10:30 Ties Robroek 10:45 10:30 (Science) Bas Steeg 10:45 10:30 (Science) Jan Martens 15:45 17:30 Nienke Wessel 15:45 17:30 see for locations Please register for an exercise group via blackboard by Tuesday November 14, 17:00 You will be assigned to an exercise class by me Each assistant has a blue delivery box on the ground floor of the Mercator 1 building 5/19

6 About this course IV Examination There is a half-way test (Tuesday December 12, 8:30-10:30) and a final test (Wednesday January 22, 8:30-11:30 ). The final grade is composed of the grade of your half-way test, h, the grade of your final test, f, the average grade of your exercises, a, Your final grade is min(10, f+h 2 + a 10 ) Additional requirements (everything at least 5) in study guide won t be applied The re-exam is a full 3hrs exam about the whole course. You keep the (average) grade of the exercises. If you fail again, you must start all over next year (including re-doing new exercises, and additional requirements) 6/19

7 Languages and Automata Let s start! 7/19

8 Overview Topics Languages: Automata: Grammars: regular finite regular context-free push-down context-free [natural languages] [bounded Turing machine] [context-sensitive] [enumerable] [Turing machine] [unrestricted] Automata: Grammars: accept words of a language given a word, compute if it is in the language generate words of a language produce all correct words in the language 8/19

9 Languages An alphabet A is a (finite) set of symbols Examples A 1 = {a} A 2 = {0, 1} A 3 = {A, C, G, T } A 4 = {a, b, c, d,..., x, y, z} A 5 = {s s is an ascii symbol} A 6 = { } Japanese alphabet: 2 52 signs A 7 = {,... } Chinese alphabet: signs A 8 = {0, 1, +,, x 0, x 1, x 2,...} mathematical alphabet, countably infinite A 9 = {0, 1, +,, x 0, x 1, x 2,...} {c r r R} mathematical alphabet, uncountably infinite 9/19

10 Words A word (string) over alphabet A is a finite sequence of elements from A The set A consists of all words over A Inductive definition of the set of words, A 1. λ A (λ denotes the empty word). 2. If a A and v A, then a v A. Note that a λ is just a Note the difference between a A and a A Think of a word as a chain of letters on a necklace: λ = Eva = E v a The difference between a and a is clear 10/19

11 Operation on words Inductive definition of the set of words, A 1. λ A (λ denotes the empty word). 2. If a A and v A, then a v A. Operations on words v A, u A v u A, concatenation v A, n N v n A, repetition v A v R A, reverse Inductive definitions of concatenation, repetition and reverse λ u = u (av) u = a(v u) v 0 = λ λ R = λ v k+1 = v v k (av) R = (v R ) a We write concatenation v u as v u 11/19

12 Operation on words; Language A language over A is a subset of A, notation L A Examples (with A = {a, b}) L 1 = {w {a, b} abba is a substring of w} L 2 = {w {a, b} w = w R } 12/19

13 Examples of languages Let A = {a, b, c}. 1. L 1 = {a n n N is even} 2. L 2 = {a n b n n N} 3. L 3 = {a n b n c n n 2} 4. L 4 = {a n n N is prime} Over other alphabets: 1. L 5 = {n n denotes an integer number} 2. L 6 = {e e is a well-formed arithmetical expression} 3. L 7 = {P P is a syntactically correct Java program} 4. L 8 = {S S is a grammatically correct English sentence} 13/19

14 Operations on languages Given languages L 1, L 2, L A we can define new languages: L 1 L 2 L 1 L 2 L L 1 L 2 L L 1 L 2 = {w w L 1 or w L 2 } L 1 L 2 = {w w L 1 and w L 2 } L = {w A w / L} L 1 L 2 = {w 1 w 2 w 1 L 1 and w 2 L 2 } L 0 = {λ} L n+1 = L L n L = n N L n = L 0 L 1 L 2... {w n w L, n N} 14/19

15 Regular expressions Regular expressions are a way to describe languages. Really important concept in (theoretical) computer science Used a lot in text processing: search (efficiently!) for specific patterns Example Let A = {a, b} be the alphabet. Then a(ba) bb is a regular expression denoting L = {a(ba) n bb n N} = {abb, ababb, abababb, ababababb,..., a(ba) n bb,...} 15/19

16 Regular expressions and languages over an alphabet A For general A the regular expressions over alphabet A are generated by rexp A ::= 0 1 s (rexp A rexp A ) (rexp A + rexp A ) (rexp A ) with s A This means 0 rexp A, 1 rexp A, and s rexp A for s A and e 1, e 2 rexp A (e 1 + e 2 ) rexp A e 1, e 2 rexp A (e 1 e 2 ) rexp A e rexp A (e) rexp A For example (abb) (a + 1) is a regular expression 16/19

17 We economize on brackets rexp A ::= 0 1 s (rexp A rexp A ) (rexp A + rexp A ) (rexp A ) We omit the outermost brackets, binds strongest, + binds weakest. So a + ba denotes ((a + (b(a) ))). This denotes the language of either just a or b followed by a finite (possibly 0) number of a s. 17/19

18 Regular languages For a regular expression e over alphabet A we define the language L(e): L(0) = L(1) = {λ} L(s) = {s} L(e 1 e 2 ) = L(e 1 )L(e 2 ) L(e 1 + e 2 ) = L(e 1 ) L(e 2 ) L(e ) = (L(e)) A language L is called regular if L = L(e) for some e rexp 18/19

19 Examples Let A = {a, b}. Also L = {w w begins with bb} is regular L = L(bb(a + b) ) L = {w bb occurs in w} is regular L = L((a + b) bb(a + b) ) L = {w w b 2} is regular NB. w denotes the length of w, w b denotes the number of b s in w L = L(a (ba b + b + 1)a ) 19/19