Automaten und Formale Sprachen alias Theoretische Informatik. Sommersemester 2014

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1 Automaten und Formale Sprachen alias Theoretische Informatik Sommersemester 2014 Dr. Sander Bruggink Übungsleitung: Jan Stückrath Sander Bruggink Automaten und Formale Sprachen 1

2 Who are we? Teacher: Dr. Sander Bruggink Roomm LF Teaching assistent: Jan Stückrath Room LF Tutors: Lars Stoltenow / Martin Weber Lars Stoltenow: lars.stoltenow@stud.uni-due.de Martin Weber: martin.weber.x@stud.uni-duisburg-essen.de Sander Bruggink Automaten und Formale Sprachen 2

3 Introduction Who are you? BAI ISE Others Website Moodle-Site Sander Bruggink Automaten und Formale Sprachen 3

4 Appointments Lecture: Tuesday, 12pm 2pm, room LB 131 Exercise groups: Group ISE: Tuesday, 8am 10am, Room LE 120 (English) Martin Weber Group BAI-1: Wednesday, 12pm 2pm Uhr, Room LE 120 Lars Stoltenow Group BAI-2: Thursday, 12pm 2pm, Room LF 125 Lars Stoltenow Group BAI-3: Thursday, 4pm 6pm, Room LC 137 Martin Weber Group BAI-4: Friday, 10am 12pm, Room LC 137 Jan Stückrath Sander Bruggink Automaten und Formale Sprachen 4

5 Advice about the exercises Please try to split evenly among the exercise groups. Visit the exercise groups and do the homework. The material of this lecture can only be mastered by frequent practice. Memorizing doesn t help much. The exercise groups start in the third week of the semester. Thus, the first exercise group take place from 22 to 25 April. Sander Bruggink Automaten und Formale Sprachen 5

6 Advice about the exercises The exercise sheet is put online every week on Tuesday at the latest. The written exercises must be handed in on Tuesday, 8am of the following week. In this week the exercise sheet is discussed in the exercise groups. Handing in: in the letter box adjacent to room LF 259. online through Moodle. Plase write clearly your name, student number and group number on your exercise. Also write down the name of the lecture. Sander Bruggink Automaten und Formale Sprachen 6

7 Exam Oral exam in the module Theoretische Informatik ( Automaten und formale Sprachen together with Berechenbarkeit und Komplexität ) For: BAI Students, who started this summer semester, may choose to do the two oral exams of the module separately. Klausur Für: BAI (PO 2012), ISE, Nebenfach Sander Bruggink Automaten und Formale Sprachen 7

8 Bonus points Bonus points: During the semester the will be 12 (or 11) exercise sheets of 20 points each. If you receive 50% of the points, you will recieve one grade level higher (for example 2,0 instead of 2,3) for your exam. You can obtain 10 extra bonus points by publically presenting the answer to an exercise in the exercise group (this is possible only once). For the oral exam of the module Theoretische Informatik you must obtain the bonus in both Automaten und Formale Sprachen and in Berechenbarkeit und Komplexität. Sander Bruggink Automaten und Formale Sprachen 8

9 Literature We use the following book: Uwe Schöning: Theoretische Informatik kurzgefaßt. Spektrum, (5. Auflage) Other relevant literature: Neuauflage eines alten Klassikers: Hopcroft, Motwani, Ullman: Introduction to Automata Theory, Languages, and Computation. Addison-Wesley, Sander Bruggink Automaten und Formale Sprachen 9

10 Literatur Sander Bruggink Automaten und Formale Sprachen 10

11 Motivation / Introduction Informal Overview Automata Finite representations of languages finite automata, pushdown automata, (Turing machines),... Other method to finitely represent languages: grammars, regular expressions and formal languages Language = set of finite sequences of symbols (= words) For example: Set of arithmetical expressions Set of syntactically correct Java programs Set of all German sentences Set of satisfiable logical formulas Sander Bruggink Automaten und Formale Sprachen 11

12 Motivation / Introduction Motivation: Vending Machine Bild: Wikipedia 50 Cent 20 Cent 10 Cent Paying 70 cent with 50, 20 und 10 cent coins. Sander Bruggink Automaten und Formale Sprachen 12

13 Motivation / Introduction Motivation: Vending Machine Automaton: Language: Sequences of coints (from 10, 20, 50 cent), that are wearth 70 cents. Sander Bruggink Automaten und Formale Sprachen 13

14 Motivation / Introduction Adventure-Problem Warming up: we consider the adventure problem, in which an adventurer searches a path through an adventure. (Later we will find out, what this has to do with formal languages.) Sander Bruggink Automaten und Formale Sprachen 14

15 Motivation / Introduction Sander Bruggink Automaten und Formale Sprachen 15

16 Motivation / Introduction Adventure Problem (Level 1) Rules of the Adventure Problem: The Treasure Rule You must find at least two treasures. The Door Rule You can only go through a door, when you found a key before. (The key can be used arbitrarily many times.) Sander Bruggink Automaten und Formale Sprachen 16

17 Motivation / Introduction Adventure Problem (Level 1) The Dragon Rule Immediately after the encounter with a dragon, you must jump into a river, because the dragon will otherwise ignite you. This is not the case anymore, if you have previously found a sword, because then you can kill the dragon. Remark: Dragons, treasures and keys are refilled after you left the according field. We are look for a path from a start to an end state, which satisfies all of the above conditions. Sander Bruggink Automaten und Formale Sprachen 17

18 Motivation / Introduction Adventure Problem (Level 1) Question (Level 1) Is there a solution in the example? Adventure Yes! The shortest solution is: 1, 2, 3, 1, 2, 4, 10, 4, 5, 6, 4, 5, 6, 4, 11, 12 (length 16). Is there a general solving procedure which given an adventure in the form of a graph can always determine whether there is a solution? Yes! We will see this procedure in the lecture. In order to be able to implement this procedure, we need also formal description of the rules (door rule, dragon rule, treasure rule). Sander Bruggink Automaten und Formale Sprachen 18

19 Motivation / Introduction Adventure Problem (Level 2) New Door Rule The keys are magical and disappear immediately after being used to open a door. As soon as you go through a door, the door is locked again. However, you can carry more than one key. Sander Bruggink Automaten und Formale Sprachen 19

20 Motivation / Introduction Adventure Problem (Level 2) Questions (Level 2) Is there a solution in the example? Adventure Yes! The shortest solution is: 1, 2, 3, 1, 2, 4, 10, 4, 7, 8, 9, 4, 7, 8, 9, 4, 11, 12. (length 18) Is there a general solving procedure? Yes! We will see this procedure in the lecture. Why is the new problem harder? We have to count the keys. Sander Bruggink Automaten und Formale Sprachen 20

21 Motivation / Introduction Adventure Problem (Expert Level) New Dragon Rule Swords become unusable by the dragon s blood, as soon as one has killed a dragon. However, dragons are replaced after being killed. Key Regel A magic gate can only be passed, when you don t own a key. Sword Rule A river can only be passed, when you don t have a sword (otherwise, you ll drown). Sander Bruggink Automaten und Formale Sprachen 21

22 Motivation / Introduction Adventure Problem (Expert Level) Questions (Expert Level) Is there a solution in the example? Adventure Yes! The shortest solution is: Ja! Die kürzeste Lösung ist 1, 2, 3, 1, 2, 4, 10, 4, 7, 8, 9, 4, 10, 4, 5, 6, 4, 11, 12. (Länge 19) Is there a general solving procedure? No! It is a so-called undecidable problem. This is not discussed in this lecture, but in Berechenbarkeit und Komplexität. Sander Bruggink Automaten und Formale Sprachen 22

23 Motivation / Introduction Adventure-Problem und Formale Sprachen Automata: Adventure instances Automaton that accepts possible solutions Languages: Possible object sequences of an adventure instance Object sequences that satisy the treasure rule Object sequences that satisy the dragon rule Object sequences that satisy the door rule Sander Bruggink Automaten und Formale Sprachen 23

24 Motivation / Introduction Formal Languages Questions Typical questions here are: Is a language L empty or does it contain (at least) one word? L =? Is a word w in the language? w L? Are two languages included in one another? L 1 L 2? Depending on the language (or languages) these question are either decidable (there is a general procedure to solve the problem) or undecidable Sander Bruggink Automaten und Formale Sprachen 24

25 Motivation / Introduction Adventure Problem and Formal Languages The single levels of the adventure belong to the following language classes: Level 1 regular languages Level 2 context free languages Expert level Chomsky-0 languages (semi-decidable languages) These are discussed in Berechenbarkeit & Komplexität. Sander Bruggink Automaten und Formale Sprachen 25

26 Contents of the Lecture For theoretical computer science How can infinite structures be represented by finite descriptions (automata, grammars)? There are numerous applications for example in the following areas: searching in texts (regular expressions) syntax of (programming) languages and compiler construction modelling system behaviour verification of systems Sander Bruggink Automaten und Formale Sprachen 26

27 Contents of the Lecture Contents of the lecture Automata and formal languages Mathematical foundations and formal proofs Languages, grammars and automata Chomsky Hierarchy (different language classes) Regular languages and context free languages How can we show that a language is not of a certain class? Decision procedures Closure properties (is the intersection of two regular languages also regular?) Sander Bruggink Automaten und Formale Sprachen 27

28 Mathematical Foundations and Formal Proofs Sets Set A set M of elements is denoted as enumerations M = {0, 2, 4, 6, 8,... } or a a set of elements with a certain property General format: M = {n n N and n even} M = {x P(x)} (M is the set of all elements x, which satisfy property P.) Sander Bruggink Automaten und Formale Sprachen 28

29 Mathematical Foundations and Formal Proofs Sets Remarks: The elements of a set a unordered, that is, their order is not important. For example: {1, 2, 3} = {1, 3, 2} = {2, 1, 3} = {2, 3, 1} = {3, 1, 2} = {3, 2, 1} An element cannot occur in a set more than once. It is either in the set, or not. For example: {1, 2, 3} {1, 2, 3, 4} = {1, 2, 3, 4, 4} Sander Bruggink Automaten und Formale Sprachen 29

30 Mathematical Foundations and Formal Proofs Sets Element of a set We write a M, when an element a is contained in the set M. Number of elements of a set For a set M the number of elements of M is denoted by M. Empty set The empty set (set without elements) is denoted by. Subset We write A B when every element of A is also an element of B. A is then called a subset of B. The relation is also called inclusion. Sander Bruggink Automaten und Formale Sprachen 30

31 Mathematical Foundations and Formal Proofs Sets Example: 2 {1, 2, 3}? 2 {1, 2, 3}? {2} {1, 2, 3} {1, 2} {1, 2, 3}? {1, 2} {1, 2, 3}? {A, B, C}? {A, B, C}? Sets can also contain sets: 1 {{1}, {3, 4}}? {1} {{1}, {3, 4}? {1} {{1}, {3, 4}? {{1}} {{1}, {3, 4}? Wichtig: a {a} Sander Bruggink Automaten und Formale Sprachen 31

32 Mathematical Foundations and Formal Proofs Venn-Diagrams Venn-Diagrams are graphical representation of sets and the relationships between them. A B B A Sander Bruggink Automaten und Formale Sprachen 32

33 Mathematical Foundations and Formal Proofs Set operations Union: A B = {e e A oder e B} Intersection: A B = {e e A und e B} Difference: A \ B = {e e A und e / B} A B A B A \ B A B A B A B Sander Bruggink Automaten und Formale Sprachen 33

34 Mathematical Foundations and Formal Proofs Power set Power set Let M be a set. The set P(M) is the set of all subsets of M. P(M) = {A A M} We have: P(M) = 2 M (for a finite set M). Sander Bruggink Automaten und Formale Sprachen 34

35 Mathematical Foundations and Formal Proofs Tuples Tuple Besides sets we also use tuples, which are written with (round) parenthesis: (a 1,..., a n ) In a tuple the elements are ordered. For example: (1, 2, 3) (1, 3, 2) An element can occur multiple times in a tuple. Tuples of different size are always unequal. For example (1, 2, 3, 4) (1, 2, 3, 4, 4) A tuple (a 1,..., a n ) consisting of n elements is called n-tuple. A 2-tupel is also called a pair. Sander Bruggink Automaten und Formale Sprachen 35

36 Mathematical Foundations and Formal Proofs Cross Product Cross product (or cartesian product) Let A, B be two sets. The set A B is the set of all pairs (a, b), where a is an element of A and b an element of B. A B = {(a, b) a A, b B} We have: A B = A B (for finite sets A, B). Sander Bruggink Automaten und Formale Sprachen 36

37 Mathematical Foundations and Formal Proofs Relations Binary relation Let A, B be two sets. A binary relation between A and B is a set of pairs R A B. A B f g A = {f, g, h} B = {1, 2, 3, 4} R = {(f, 1), (f, 2), h R 4 (g, 4), (h, 2)} Sander Bruggink Automaten und Formale Sprachen 37

38 Mathematical Foundations and Formal Proofs The sets A and B can also be equal. Relation over A Let A be a set. A (binary) relation over A is a set of pairs R A A. 2 A A = {1, 2, 3, 4, 5, 6} R = {(1, 3), (1, 2), (2, 6), (3, 6), (4, 4), (5, 1), (5, 5), (5, 6)} Sander Bruggink Automaten und Formale Sprachen 38

39 Mathematical Foundations and Formal Proofs Properties of Relations Let R A A be a elation from A to A. R is reflexive, when for all x A: x R x. a R is irreflexive, when for all x A: not x R x. a Es gibt Relationen die nicht reflexiv aber auch nicht irreflexiv sind. Sander Bruggink Automaten und Formale Sprachen 39

40 Mathematical Foundations and Formal Proofs Properties of Relations (Continued) Let R A A be a elation from A to A. R is symmetric, when for all x, y A it holds, that when x R y, then y R x. a b R is antisymmetric, when for all x, y A it holds, that when x R y and y R x, then x = y. a b (wobei a b) R is transitive, when for x, y, z A it holds that when x R y and y R z, then x R z. a b c Sander Bruggink Automaten und Formale Sprachen 40

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