INF210 Datamaskinteori (Models of Computation)

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INF210 Datamaskinteori (Models of Computation) Textbook: John Martin, Introduction to Languages

Fomin Datamaskinteori 1 Datamaskinteori 2 Some parts from John Savage, Models of Computation,

Booklets with selected parts from Savage's book will be available at Studia for approximately 80

1 INF210 Datamaskinteori (Models of Computation) Textbook: John Martin, Introduction to Languages and the Theory of Computation, McGraw Hill, Third Edition. Fedor V. Fomin Datamaskinteori 1 Datamaskinteori 2 Some parts from John Savage, Models of Computation, Addison-Wesley, will be used. Booklets with selected parts from Savage's book will be available at Studia for approximately 80 NOK For more information see Datamaskinteori 3 Datamaskinteori 4 What this course is about? What this course is about? Model of computation Datamaskinteori 5 Datamaskinteori 6

kinds of machines finite automata were studied by a number of researchers.

2 What this course is about? Surprisingly, the question is much older than computers are. What a computing machine can do and what it could not do? 1936: an abstract machine that has all the capabilities of today s computers Alan Turing ( ) Datamaskinteori 7 Datamaskinteori 8 In 1940 s and 1950 s simpler kinds of machines finite automata were studied by a number of researchers. Study of formal grammars Naom Chomsky (1928-) Datamaskinteori 9 Datamaskinteori 10 What could and what could not be computed Type of machines Very restricted (finite automata) More complicated (push-down automata) Turing machine Logical circuits Stephen Cook Datamaskinteori 11 Datamaskinteori 12

3 Subject to study What these machines can do and what they can not. Part I. Mathematical Notations and Techniques Sets operations Datamaskinteori 13 Datamaskinteori 14 Venn diagram, example Symmetric difference De Morgan laws Datamaskinteori 15 Datamaskinteori 16 Power set The set of all subsets of the set A: 2 A Cartesian product (A cross B) A B If A has n elements then 2 A has 2 n elements (proof by induction) Datamaskinteori 17 Datamaskinteori 18

4 Logic Proposition, compound proposition, logical connectives: Logic Conditional connective: if p then q: Datamaskinteori 19 Datamaskinteori 20 Logical Implication and Equivalence Quantifiers Datamaskinteori 21 Datamaskinteori 22 Question: What is that? Answer: Graph 3-coloring Datamaskinteori 23 Datamaskinteori 24

5 Functions Functions Onto (surjection) One-to-one (injection) Bijection Compositions an Inverses Datamaskinteori 25 Datamaskinteori 26 Relations A relation on a set A is a subset of A A Relations 1. Reflexive a A (ara) 2. Symmetric a,b A (arb -> bra) 3. Transitive a,b,c A (arb brc -> arc) Equivalence relation Datamaskinteori 27 Datamaskinteori 28 Examples Relatives Subsets Languages Alphabet Σ A string over an alphabet The length of a string x over Σ, x Null string Λ Σ * Datamaskinteori 29 Datamaskinteori 30

6 Examples of languages (maybe not very sophisticated) Language L over alphabet Σ is a subset of Σ * The set of strings of 0 s and 1 s with an equal number of each: {Λ, 01,10, 0011, 0101, 1001, } The set of binary numbers whose value is even and positive {10, 100,1000, } {Λ} Datamaskinteori 31 Datamaskinteori 32 Strings Concatenation Substring Concatenation Languages {para,huma}{normal,noid}={paranoral, paranoid, humanormal, humanoid} L{Λ}=L Datamaskinteori 33 Datamaskinteori 34 Power a k =aaλ a, (a Σ) x k =xx Λ x, (x Σ * ) Σ k =ΣΣΛΣ={x Σ * x =k} L k =LL Λ L (L Σ * ) Important a 0 =x 0 =Λ Σ 0 =Λ 0 ={Λ} Datamaskinteori 35 Datamaskinteori 36

7 Example Example alphabet Set of strings Datamaskinteori 37 Datamaskinteori 38 Kleene star How to describe language??? Stephen Cole Kleene Datamaskinteori 39 Datamaskinteori 40 Problem Example Primality test: given a string of 0 s and 1 s to say yes if the string is a binary representation of a prime or say no if not. Datamaskinteori 41 Datamaskinteori 42

8 If it is hard to decide whether a given string s of the language L x of valid strings in programming language X Compiling programs in X is hard If it is hard to decide whether a given string s of the language L x of valid strings in programming language X Compiling programs in X is hard Proof: If it is easy to generate code we can run translator, and conclude that the s is a valid member of L x when translator producing object code Datamaskinteori 43 Datamaskinteori 44 Main problems Generate languages Recognize languages Summary An alphabet is any finite set of symbols A string is a finite-length sequence of symbols A language is a set (possibly infinite) set of strings, all of which choose their symbols from some one alphabet. Datamaskinteori 45 Datamaskinteori 46

Introduction to Automata

Introduction to Automata Seungjin Choi Department of Computer Science and Engineering Pohang University of Science and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjin@postech.ac.kr 1 /