DVA325 Formal Languages, Automata and Models of Computation (FABER)

Transcription

1 DVA325 Formal Languages, Automata and Models of Computation (FABER) Lecture 1 - Introduction School of Innovation, Design and Engineering Mälardalen University 11 November 2014 Abu Naser Masud FABER November 11, / 87

2 CONTENTS 1 Adminstrivia 2 Basic Set Theory 3 Countable Sets 4 Uncountable sets 5 Cardinality of Sets Abu Naser Masud FABER November 11, / 87

3 ACKNOWLEDGEMENTS The lecture materials in the slides are either directly copied or influenced from the lecture material of Prof. Gordana Dodig-Crnkovic. Some parts of the lecture are taken from Prof. Johnnie Baker s slides. Abu Naser Masud FABER November 11, / 87

4 COURSE HOME PAGE URL: Visit home page regularly! Abu Naser Masud FABER November 11, / 87

5 JFLAP Abu Naser Masud FABER November 11, / 87

6 WORK HOUR Abu Naser Masud FABER November 11, / 87

7 REFERENCE BOOK Abu Naser Masud FABER November 11, / 87

8 WHY THEORY OF COMPUTATION 1 A real computer can be modelled by a mathematical object: a theoretical computer. 2 A formal language is a set of strings, and can represent a computational problem. 3 A formal language can be described in many different ways that ultimately prove to be identical. 4 Simulation: the relative power of computing models can be based on the ease with which one model can simulate another. Abu Naser Masud FABER November 11, / 87

9 WHY THEORY OF COMPUTATION 1 A real computer can be modelled by a mathematical object: a theoretical computer. 2 A formal language is a set of strings, and can represent a computational problem. 3 A formal language can be described in many different ways that ultimately prove to be identical. 4 Simulation: the relative power of computing models can be based on the ease with which one model can simulate another. Abu Naser Masud FABER November 11, / 87

10 WHY THEORY OF COMPUTATION 1 A real computer can be modelled by a mathematical object: a theoretical computer. 2 A formal language is a set of strings, and can represent a computational problem. 3 A formal language can be described in many different ways that ultimately prove to be identical. 4 Simulation: the relative power of computing models can be based on the ease with which one model can simulate another. Abu Naser Masud FABER November 11, / 87

11 WHY THEORY OF COMPUTATION 1 A real computer can be modelled by a mathematical object: a theoretical computer. 2 A formal language is a set of strings, and can represent a computational problem. 3 A formal language can be described in many different ways that ultimately prove to be identical. 4 Simulation: the relative power of computing models can be based on the ease with which one model can simulate another. Abu Naser Masud FABER November 11, / 87

12 WHY THEORY OF COMPUTATION 5 Robustness of a general computational model. 6 The Church-Turing thesis: everything algorithmically computable is computable by a Turing machine. 7 Study of non-determinism: languages can be described by the existence or non-existence of computational paths. 8 Understanding unsolvability: for some computational problems there is no corresponding algorithm that will unerringly solve them. Abu Naser Masud FABER November 11, / 87

13 PRACTICAL APPLICATIONS 1 Efficient compilation of computer languages 2 String search 3 Investigation of the limits of computation, recognizing difficult/unsolvable problems 4 Applications to other areas: circuit verification economics and game theory (finite automata as strategy models in decision-making); theoretical biology (L-systems as models of organism growth) computer graphics (L-systems) linguistics (modelling by grammars) Abu Naser Masud FABER November 11, / 87

14 Basic Set Theory Abu Naser Masud FABER November 11, / 87

15 SET A set is an unordered collection of objects referred to as elements. A set is said to contain its elements. Different ways of describing a set (Set Representation). 1 Explicitly: listing the elements of a set {1, 2, 3} is the set containing 1, 2, and 3. The members or elements of the set are included inside the braces. {1, 1, 2, 3, 3} = {1, 2, 3} since repetition is irrelevant. {1, 2, 3} = {3, 2, 1} since sets are unordered. Abu Naser Masud FABER November 11, / 87

16 SET 1 Explicitly: listing the elements of a set {1, 2, 3,..., 99} is the set of positive integers less than 100; {1, 2, 3,...} is a way we denote an infinite set (in this case, the natural numbers). = {} is the empty set, or the set containing no elements. Abu Naser Masud FABER November 11, / 87

17 SET 1 Explicitly: listing the elements of a set {1, 2, 3,..., 99} is the set of positive integers less than 100; {1, 2, 3,...} is a way we denote an infinite set (in this case, the natural numbers). = {} is the empty set, or the set containing no elements. Note { } Abu Naser Masud FABER November 11, / 87

18 SET 1 Explicitly: listing the elements of a set {1, 2, 3,..., 99} is the set of positive integers less than 100; {1, 2, 3,...} is a way we denote an infinite set (in this case, the natural numbers). = {} is the empty set, or the set containing no elements. Note Sometimes we give a name to a set. For example A = {1, 2, 3}, here, A is the set containing the elements 1, 2, and 3. Abu Naser Masud FABER November 11, / 87

19 SET 1 Explicitly: listing the elements of a set {1, 2, 3,..., 99} is the set of positive integers less than 100; {1, 2, 3,...} is a way we denote an infinite set (in this case, the natural numbers). = {} is the empty set, or the set containing no elements. 2 Implicitly: by using a set builder notations S = {j j > 0, and j = 2k for some k > 0}. S = {j j >= 0, and j =< 100}. Abu Naser Masud FABER November 11, / 87

20 SET 1 Explicitly: listing the elements of a set {1, 2, 3,..., 99} is the set of positive integers less than 100; {1, 2, 3,...} is a way we denote an infinite set (in this case, the natural numbers). = {} is the empty set, or the set containing no elements. 2 Implicitly: by using a set builder notations S = {j j > 0, and j = 2k for some k > 0}. S = {j j >= 0, and j =< 100}. should be read as such that Abu Naser Masud FABER November 11, / 87

21 SOME IMPORTANT SETS Natural Numbers ( non negative integers) N = {1, 2, 3...} Integers Z = {..., 2, 1, 0, 1, 2, 3,...} Positive Integers Z + = {1, 2, 3,...} Rational Numbers Q = {p/q p Z, q Z, and q 0} Real Numbers R x S means x is an element of set S x S means x is not an element of set S Abu Naser Masud FABER November 11, / 87

22 UNIVERSAL SET Abu Naser Masud FABER November 11, / 87

23 SET OPERATIONS Suppose A = {1, 2, 3} and B = {2, 3, 4, 5} Union A B = {1, 2, 3, 4, 5} Intersection A B = {2, 3} Difference A \ B = {1} B \ A = {4, 5} sometimes we write - instead of \ A B A B A B A B A B A \ B Abu Naser Masud FABER November 11, / 87

24 SET OPERATIONS Complement Suppose Universal set is {1,..., 7} and A = {1, 2, 3} A = U A = A = {4, 5, 6, 7} Abu Naser Masud FABER November 11, / 87

25 SET OPERATIONS Complement Suppose Universal set is {1,..., 7} and A = {1, 2, 3} A = U A = A = {4, 5, 6, 7} Note A = A Abu Naser Masud FABER November 11, / 87

26 SET OPERATIONS Abu Naser Masud FABER November 11, / 87

27 SET OPERATIONS A B means A is a subset of B. Equivalently, B contains A. Every element of A is also in B. x((x A) (x B)). Abu Naser Masud FABER November 11, / 87

28 SET OPERATIONS A B means A is a subset of B. A B means B is a superset of A. A = B if and only if A and B have exactly the same elements. Equivalently iff, A B and B A iff, x((x A) (x B)). So in order to show equality of sets A and B, show: A B B A Abu Naser Masud FABER November 11, / 87

29 SET OPERATIONS A B means A is a proper subset of B. That means, A B, and A B. Abu Naser Masud FABER November 11, / 87

30 EXAMPLES - SET OPERATIONS Quick examples: {1, 2, 3} {1, 2, 3, 4, 5} {1, 2, 3} {1, 2, 3, 4, 5} Is {1, 2, 3}? Abu Naser Masud FABER November 11, / 87

31 EXAMPLES - SET OPERATIONS Quick examples: {1, 2, 3} {1, 2, 3, 4, 5} {1, 2, 3} {1, 2, 3, 4, 5} Is {1, 2, 3}? Yes! Abu Naser Masud FABER November 11, / 87

32 EXAMPLES - SET OPERATIONS Quick examples: {1, 2, 3} {1, 2, 3, 4, 5} {1, 2, 3} {1, 2, 3, 4, 5} Is {1, 2, 3}? Yes! Is {1, 2, 3}? Abu Naser Masud FABER November 11, / 87

33 EXAMPLES - SET OPERATIONS Quick examples: {1, 2, 3} {1, 2, 3, 4, 5} {1, 2, 3} {1, 2, 3, 4, 5} Is {1, 2, 3}? Yes! Is {1, 2, 3}? No! Abu Naser Masud FABER November 11, / 87

34 EXAMPLES - SET OPERATIONS Quick examples: {1, 2, 3} {1, 2, 3, 4, 5} {1, 2, 3} {1, 2, 3, 4, 5} Is {1, 2, 3}? Yes! Is {1, 2, 3}? No! Is {, 1, 2, 3}? Abu Naser Masud FABER November 11, / 87

35 EXAMPLES - SET OPERATIONS Quick examples: {1, 2, 3} {1, 2, 3, 4, 5} {1, 2, 3} {1, 2, 3, 4, 5} Is {1, 2, 3}? Yes! Is {1, 2, 3}? No! Is {, 1, 2, 3}? Yes! Abu Naser Masud FABER November 11, / 87

36 EXAMPLES - SET OPERATIONS Quick examples: {1, 2, 3} {1, 2, 3, 4, 5} {1, 2, 3} {1, 2, 3, 4, 5} Is {1, 2, 3}? Yes! Is {1, 2, 3}? No! Is {, 1, 2, 3}? Yes! Is {a} {a}? Abu Naser Masud FABER November 11, / 87

37 EXAMPLES - SET OPERATIONS Quick examples: {1, 2, 3} {1, 2, 3, 4, 5} {1, 2, 3} {1, 2, 3, 4, 5} Is {1, 2, 3}? Yes! Is {1, 2, 3}? No! Is {, 1, 2, 3}? Yes! Is {a} {a}? No! Abu Naser Masud FABER November 11, / 87

38 EXAMPLES - SET OPERATIONS Quick examples: {1, 2, 3} {1, 2, 3, 4, 5} {1, 2, 3} {1, 2, 3, 4, 5} Is {1, 2, 3}? Yes! Is {1, 2, 3}? No! Is {, 1, 2, 3}? Yes! Is {a} {a}? No! Is {a} {a}? Abu Naser Masud FABER November 11, / 87

39 EXAMPLES - SET OPERATIONS Quick examples: {1, 2, 3} {1, 2, 3, 4, 5} {1, 2, 3} {1, 2, 3, 4, 5} More About Empty or Null Set ( ) Is {1, 2, 3}? Yes! S = S Is {1, 2, 3}? No! S = Is {, 1, 2, 3}? Yes! S \ = S Is {a} {a}? No! \ S = Is {a} {a}? No! = Universal Set Abu Naser Masud FABER November 11, / 87

40 DEMORGAN S LAWS 1 A B = A B 2 A B = A B Abu Naser Masud FABER November 11, / 87

41 DEMORGAN S LAWS 1 A B = A B 2 A B = A B Proof of First Law: Suppose x A B Suppose x A B = x A B = x A and x B = x A and x B = x A and x B = x A and x B = x A B = x A B = x A B Abu Naser Masud FABER November 11, / 87

42 DEMORGAN S LAWS 1 A B = A B 2 A B = A B Proof of First Law: Suppose x A B = x A B = x A and x B = x A and x B = x A B Try to prove the second law. Suppose x A B = x A and x B = x A and x B = x A B = x A B Abu Naser Masud FABER November 11, / 87

43 CARDINALITY OF SETS Definition The cardinality of any finite set S is represented as S, and is the number of distinct elements in S. Examples: S = {1, 2, 3}. Thus S = 3. S = {1, 2, 3, 4, 4, 4, 5}. Thus S = 5. S =. Thus S = 0. S = {, { }}. Thus S = 2. S = {0, 1, 2, 3,...}. Thus S =infinite. Abu Naser Masud FABER November 11, / 87

44 POWER SETS Definition The power set of any set S is the set of all the subsets of S. The power set of any set S is represented by 2 S or P(S) Examples S = {a}, 2 S = {, {a}} S = {a, b}, 2 S = {, {a}, {b}, {a, b}} S =, 2 S = { } Fact: If S is finite, 2 S = 2 S. Abu Naser Masud FABER November 11, / 87

45 CARTESIAN PRODUCT Abu Naser Masud FABER November 11, / 87

46 INJECTIVE, SURJECTIVE, AND BIJECTIVE FUNCTIONS Let f : A B be a function. Here domain of f is set A, and range of f is set B. Definitions 1 If for any a 1, a 2 A, f (a 1 ) = f (a 2 ) = a 1 = a 2, f is said to be injective or one-to-one. 2 If for any b B, there exists a A such that f (a) = b, f is said to be surjective or onto. 3 If a function f is both injective and surjective, it is called bijective or one-to-one correspondence. Abu Naser Masud FABER November 11, / 87

47 INJECTIVE, SURJECTIVE, AND BIJECTIVE FUNCTIONS Abu Naser Masud FABER November 11, / 87

48 PROOF TECHNIQUES 1 Proof by Construction 2 Proof by Induction 3 Proof by Contradiction Abu Naser Masud FABER November 11, / 87

49 PROOF BY CONSTRUCTION ( AN EXAMPLE) We define a graph to be k-regular if every node in the graph has degree k. Theorem For each even number n > 2 there exists 3-regular graph with n nodes. Abu Naser Masud FABER November 11, / 87

50 PROOF BY CONSTRUCTION We can construct a graph G = (V, E) such that V = n and n > 2 nodes. V = {0, 1,..., n 1} We make two kind of edges: 1 E 1 = {(i, i + 1) 0 i n 1} {(n 1, 0)} Edges between adjacent nodes 2 E 2 = {(i, i + n/2) 0 i n/2 1} Cross edges between nodes in two halves E = E 1 E 2 End of Proof Abu Naser Masud FABER November 11, / 87

51 PROOF BY INDUCTION We have statements P 1, P 2, P 3,... If we know for some k that P 1, P 2,..., P k are true for any n k that P 1, P 2,..., P n imply P n+1 Then Every P i is true Abu Naser Masud FABER November 11, / 87

52 PROOF BY INDUCTION (PROCEDURE) Inductive Basis or Base Case Prove that P 1 is true Inductive Hypothesis (IH) Let s assume P 1, P 2,..., P n are true, for any n >= 1 Inductive Step Prove that P n+1 is true from the IH. Abu Naser Masud FABER November 11, / 87

53 PROOF BY INDUCTION (AN EXAMPLE) Theorem A binary tree of height n has at most 2 n leaves. Proof let L(i) be the number of leaves at level i Abu Naser Masud FABER November 11, / 87

54 PROOF We want to show: L(i) 2 i Inductive basis L(0) = 1 (the root node) Inductive hypothesis Let s assume L(i) 2 i for all i = 0, 1,..., n Induction step we need to show that L(n + 1) 2 n+1 Abu Naser Masud FABER November 11, / 87

55 PROOF - INDUCTIVE STEP Abu Naser Masud FABER November 11, / 87

56 PROOF - INDUCTIVE STEP Abu Naser Masud FABER November 11, / 87

57 PROOF BY INDUCTION (ANOTHER EXAMPLE) Theorem - Cardinality of Power Set Let S be a finite set with S = n (S has n elements). Then P(S) = 2 n elements. In other words, S has 2 n subsets. This statement can be proved by induction. We can see that it is true for n=0,1,2,3 by examination. = { } 2 0 = 1 {1} = {, {1}} 2 1 = 2 {1, 2} = {, {1}, {2}, {1, 2}} 2 2 = 4 {1, 2, 3} = {, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}} 2 3 = 8 Abu Naser Masud FABER November 11, / 87

58 INDUCTIVE PROOF BASE CASE The simplest case is the set with 0 elements (there is just one such set), that has 2 0 = 1 subsets. INDUCTIVE HYPOTHESIS (IH) Assume that the claim is valid for all sets with k elements, i.e. that every set with k elements has 2 k subsets. Abu Naser Masud FABER November 11, / 87

59 INDUCTIVE PROOF BASE CASE The simplest case is the set with 0 elements (there is just one such set), that has 2 0 = 1 subsets. INDUCTIVE HYPOTHESIS (IH) Assume that the claim is valid for all sets with k elements, i.e. that every set with k elements has 2 k subsets. Prove the Inductive step from IH Show that IH implies that the proposition is true for every set with k + 1 element, that is show that each set with k + 1 elements has 2 k+1 subsets. Abu Naser Masud FABER November 11, / 87

60 INDUCTIVE STEP Let x S. Consider the set S = S \ {x} where x is any element of S. S = k + 1 implies S = k. In counting the number of subsets of S, we have 1 The number of subsets of S (subsets of S excluding x) 2 All the subsets of S with x adjoined to them (subsets of S including x). From the induction hypothesis, there are 2 k subsets of S. Abu Naser Masud FABER November 11, / 87

61 INDUCTIVE STEP (CONTD.) Now adjoin x to all the subsets of S. This does not change its number. So, there are 2 k subsets of S including x in each of them. So, in total we have 2 k + 2 k = 2 2 k = 2 k+1 subsets of S. Thus P(k) = P(k + 1) Therefore: n N S = n = P(S) = 2 n Abu Naser Masud FABER November 11, / 87

62 PROOF BY CONTRADICTION We want to prove that a statement P is true. 1 We assume that P is false. 2 Then we arrive at a conclusion that contradicts our assumptions. 3 Therefore, statement P must be true. Abu Naser Masud FABER November 11, / 87

63 PROOF BY CONTRADICTION (AN EXAMPLE) Theorem 2 is not rational (irrational) Proof: Assume by contradiction that it is rational 2 = n/m, where n and m have no common factors We will show that this is impossible Abu Naser Masud FABER November 11, / 87

64 PROOF BY CONTRADICTION Abu Naser Masud FABER November 11, / 87

65 COUNTABLE SETS* Definition (At most countable) A countable set is a set with the same cardinality (number of elements) as some subset of the set of natural numbers. Definition (formal) A set S is called countable if there exists an injective function f from S to the natural numbers N = {0, 1, 2, 3,...} (f : S N). (*) The term is due to Georg Cantor ( ) Abu Naser Masud FABER November 11, / 87

66 COUNTABLE SETS 2 n n =< 0 f (n) = 2n 1 n >= 1 Abu Naser Masud FABER November 11, / 87

67 COUNTABLE AND UNCOUNTABLE SETS Abu Naser Masud FABER November 11, / 87

68 CARDINALITY OF INFINITE SETS We started with the natural numbers N = {0, 1, 2, 3,...}. Then Add infinitely many negative whole numbers to get the integers Z = {0, 1, 1, 2, 2, 3, 3,...} Then add infinitely many rational fractions to get the rationals R = {0, 1/1, 1/2, 1/3,...} Each infinite addition seem to increase cardinality: N < Z < Q But is this true? Abu Naser Masud FABER November 11, / 87

69 CARDINALITY OF INFINITE SETS We started with the natural numbers N = {0, 1, 2, 3,...}. Then Add infinitely many negative whole numbers to get the integers Z = {0, 1, 1, 2, 2, 3, 3,...} Then add infinitely many rational fractions to get the rationals R = {0, 1/1, 1/2, 1/3,...} Each infinite addition seem to increase cardinality: N < Z < Q But is this true? NO! Abu Naser Masud FABER November 11, / 87

70 RATIONAL NUMBERS Theorem The set of positive rational numbers is countable. Positive Rational Numbers: 1 2, 3 4, 7 8,... Abu Naser Masud FABER November 11, / 87

71 NAIVE IDEA Abu Naser Masud FABER November 11, / 87

72 BETTER APPROACH Abu Naser Masud FABER November 11, / 87

73 BETTER APPROACH Abu Naser Masud FABER November 11, / 87

74 RATIONAL NUMBERS We proved: the set of rational numbers is countable by describing an enumeration procedure Abu Naser Masud FABER November 11, / 87

75 LANGUAGE Language A language is a set of strings that are constructed by some specific rules related with the language. Strings A string is a sequence of symbols defined over an alphabet Σ. Alphabet An alphabet is a finite nonempty set Σ of symbols. Example Σ = {a, b} Strings = {a, b, ab, ba} Abu Naser Masud FABER November 11, / 87

76 Let S be any set. An enumeration procedure for S is an algorithm that generates all elements of S one by one. A set is countable if there is an enumeration procedure for it. Abu Naser Masud FABER November 11, / 87

77 COUNTABLE FINITE STRINGS Theorem The set of all finite strings of symbols {a, b, c} + is countable. Proof: We will describe the enumeration procedure (Constructive proof). Abu Naser Masud FABER November 11, / 87

78 COUNTABLE FINITE STRINGS Abu Naser Masud FABER November 11, / 87

79 BETTER APPROACH Assume a better enumeration order of strings: 1 Produce all strings of length 1 2 Produce all strings of length 2 3 Produce all strings of length 3 4 Produce all strings of length 4 5 and so on... Abu Naser Masud FABER November 11, / 87

80 COUNTABLE FINITE STRINGS Abu Naser Masud FABER November 11, / 87

81 COUNTABLE FINITE STRINGS (CONTD.) Theorem The set of all finite strings is countable. Proof: Any finite string can be encoded with a binary string of 0 s and 1 s. Find an enumeration procedure for the set of finite strings. Abu Naser Masud FABER November 11, / 87

82 COUNTABLE FINITE STRINGS (CONTD.) Abu Naser Masud FABER November 11, / 87

83 Abu Naser Masud FABER November 11, / 87

84 UNCOUNTABLE SETS Definition A set is uncountable if it is not countable. If a set is not countable, there is no enumeration procedure to enumerate all the elements of the set. Why? Abu Naser Masud FABER November 11, / 87

85 UNCOUNTABLE SETS Theorem The set of all infinite strings is uncountable. Proof (by contradiction): We assume we have an enumeration procedure for the set of infinite strings. Abu Naser Masud FABER November 11, / 87

86 CANTOR S DIAGONAL ARGUMENT Abu Naser Masud FABER November 11, / 87

87 CANTOR S DIAGONAL ARGUMENT We can construct a new string W that is missing in our enumeration! Conclusion The set of all infinite strings is uncountable! An infinite string can be seen as a function f : N N (nth output is nth bit in the string) Corollary There are some integer functions that cannot be described by finite strings (programs/algorithms). Abu Naser Masud FABER November 11, / 87

88 EXAMPLE OF UNCOUNTABLE INFINITE SETS Theorem Let S be an countably infinite set. The powerset 2 S of S is uncountable. Proof We show 2 S is uncountably infinite by showing that 2 N is uncountably infinite. There exists a bijection between 2 N and 2 S. As the bijection exists between N to S. Thus it is sufficient to show that 2 N is uncountably infinite. Abu Naser Masud FABER November 11, / 87

89 PROOF (CONTD.) Assume that the set 2 N is countably infinite (proof by contradiction). So, there is a one-to-one function f : 2 N N. (Counts the number of elements of 2 N ) Equivalently, there is a onto function f : N 2 N (it enumerates the elements of the power set 2 N ) We can enumerate the elements of 2 N as follows: Make a binary table T such that 1 if j f (i) T ij = 0 otherwise Abu Naser Masud FABER November 11, / 87

90 PROOF (CONTD.) Make a binary table T such that 1 if j f (i) T ij = 0 otherwise f(0) f(1) f(2) f(3) f(4) Abu Naser Masud FABER November 11, / 87

91 PROOF (CONTD.) Make a binary table T such that 1 if j f (i) T ij = 0 otherwise Consider the set X N corresponding to the flipped diagonal. X = {j T jj = 0} X = {j j f (j)} f(0) f(1) f(2) f(3) f(4) Abu Naser Masud FABER November 11, / 87

92 PROOF (CONTD.) Make a binary table T such that 1 if j f (i) T ij = 0 otherwise Consider the set X N corresponding to the flipped diagonal. X = {j T jj = 0} X = {j j f (j)} f(0) f(1) f(2) f(3) f(4) Such X will never appear as a row in this table. And so f is not onto. Abu Naser Masud FABER November 11, / 87

93 Thus we have an element X N such that j X implies j f (j) = j X, a contradiction. Conclusion Thus 2 N and hence 2 S is uncountably infinite. Abu Naser Masud FABER November 11, / 87

94 APPLICATION (LANGUAGES) Example Alphabet: Σ = {a, b} The set of all finite strings: S = {a, b} = {λ, a, b, aa, ab, ba, bb, aaa, aab,...} Infinite and Countable The powerset of S contains all languages: P(S) = {{λ}, {a}, {a, b}, {aa, ab, aab},...} Uncountable and Infinite Abu Naser Masud FABER November 11, / 87

95 APPLICATION Finite strings (algorithms): countable Languages (power set of strings): uncountable There are infinitely many more languages than finite strings. Conclusion: There are some languages that cannot be described by finite strings (algorithms). Abu Naser Masud FABER November 11, / 87

96 NUMBER SETS Venn Diagram of Number Sets ( N Z Q A R R C Abu Naser Masud FABER November 11, / 87

97 MORE ON CARDINALITY Cardinality is a measure of the size of a set. Cardinality of a finite set is simply its number of elements. Two finite sets have the same cardinality if they have the same number of elements. Abu Naser Masud FABER November 11, / 87

98 MORE ON CARDINALITY Cardinality is a measure of the size of a set. Cardinality of a finite set is simply its number of elements. Two finite sets have the same cardinality if they have the same number of elements. What about the cardinality of an infinite set? Abu Naser Masud FABER November 11, / 87

99 MORE ON CARDINALITY Cardinality is a measure of the size of a set. Cardinality of a finite set is simply its number of elements. Two finite sets have the same cardinality if they have the same number of elements. What about the cardinality of an infinite set? We cannot count the number of elements of an infinite set. An infinite set has infinite cardinality. Abu Naser Masud FABER November 11, / 87

100 MORE ON CARDINALITY Cardinality is a measure of the size of a set. Cardinality of a finite set is simply its number of elements. Two finite sets have the same cardinality if they have the same number of elements. What about the cardinality of an infinite set? We cannot count the number of elements of an infinite set. An infinite set has infinite cardinality. Are all infinities the same? Abu Naser Masud FABER November 11, / 87

101 MORE ON CARDINALITY Cardinality is a measure of the size of a set. Cardinality of a finite set is simply its number of elements. Two finite sets have the same cardinality if they have the same number of elements. What about the cardinality of an infinite set? We cannot count the number of elements of an infinite set. An infinite set has infinite cardinality. Are all infinities the same? No Abu Naser Masud FABER November 11, / 87

102 MORE ON CARDINALITY Cardinality is a measure of the size of a set. Cardinality of a finite set is simply its number of elements. Two finite sets have the same cardinality if they have the same number of elements. What about the cardinality of an infinite set? We cannot count the number of elements of an infinite set. An infinite set has infinite cardinality. Are all infinities the same? No How can we compare the cardinality of two infinite sets? Abu Naser Masud FABER November 11, / 87

103 MORE ON CARDINALITY (CONTD...) Equal cardinality If we can establish a one-to-one correspondence between two sets, then we have proved that the two sets have the same cardinality. This is true for both finite and infinite sets. Are the cardinality of the two sets {1, 2, 3} and {1/4, 1/3, 1/2} equal? Yes, we can simply count the number of elements. Another way to check the equal cardinality is the one-to-one correspondence 1 1/4 2 1/3 3 1/2 Abu Naser Masud FABER November 11, / 87

104 MORE ON CARDINALITY (CONTD...) Are the cardinality of the two sets {1, 2, 3} and {4, 5, 6, 7} equal? Abu Naser Masud FABER November 11, / 87

105 MORE ON CARDINALITY (CONTD...) Are the cardinality of the two sets {1, 2, 3} and {4, 5, 6, 7} equal? No, there is no one-to-one correspondence between the two sets. Abu Naser Masud FABER November 11, / 87

106 MORE ON CARDINALITY (CONTD...) Are the cardinality of the two sets {1, 2, 3} and {4, 5, 6, 7} equal? No, there is no one-to-one correspondence between the two sets. Are the cardinality of the set of natural numbers N and the set of whole numbers Z equal? Abu Naser Masud FABER November 11, / 87

107 MORE ON CARDINALITY (CONTD...) Are the cardinality of the two sets {1, 2, 3} and {4, 5, 6, 7} equal? No, there is no one-to-one correspondence between the two sets. Are the cardinality of the set of natural numbers N and the set of whole numbers Z equal? Yes. Abu Naser Masud FABER November 11, / 87

108 MORE ON CARDINALITY (CONTD...) Are the cardinality of the two sets {1, 2, 3} and {4, 5, 6, 7} equal? No, there is no one-to-one correspondence between the two sets. Are the cardinality of the set of natural numbers N and the set of whole numbers Z equal? We can make the following one-to-one correspondence between the two sets Abu Naser Masud FABER November 11, / 87

109 MORE ON CARDINALITY (CONTD...) In order to show that this is indeed the one-to-one correspondence, we need to show that 1 There is a REGULAR PATTERN here that we can continue FOREVER. 2 If we continue this pattern, we will never run out of numbers to pair up from each set, and we will never skip any numbers in either set. Does the previous one-to-one correspondence satisfy condition 1 and 2 above? Abu Naser Masud FABER November 11, / 87

110 MORE ON CARDINALITY (CONTD...) ℵ 0 (Aleph-null) The cardinality of the set of natural numbers is an infinite quantity. We call it ℵ 0 (Aleph-null). We have proved that N = Z = ℵ 0. Abu Naser Masud FABER November 11, / 87

111 MORE ON CARDINALITY (CONTD...) ℵ 0 (Aleph-null) The cardinality of the set of natural numbers is an infinite quantity. We call it ℵ 0 (Aleph-null). We have proved that N = Z = ℵ 0. But the set of whole numbers Z contain one more element 0 than the natural numbers. So, we can conclude that ℵ 0 = ℵ Abu Naser Masud FABER November 11, / 87

112 MORE ON CARDINALITY (CONTD...) ℵ 0 (Aleph-null) The cardinality of the set of natural numbers is an infinite quantity. We call it ℵ 0 (Aleph-null). We have proved that N = Z = ℵ 0. But the set of whole numbers Z contain one more element 0 than the natural numbers. So, we can conclude that ℵ 0 = ℵ This is not true for any regular finite number. For any regular number n, n n + 1. But ℵ 0 is not a regular number. Abu Naser Masud FABER November 11, / 87

113 MORE ON CARDINALITY (CONTD...) What is the consequences of ℵ 0 = ℵ 0 + 1? We get ℵ 0 = ℵ = ℵ = ℵ = ℵ = ℵ = ℵ 0 + n, for any finite regular number n Abu Naser Masud FABER November 11, / 87

114 MORE ON CARDINALITY (CONTD...) There is a one-to-one correspondence between the set of integers Z and the set of natural numbers N. Can you find that correspondence? Abu Naser Masud FABER November 11, / 87

115 MORE ON CARDINALITY (CONTD...) There is a one-to-one correspondence between the set of integers Z and the set of natural numbers N. Can you find that correspondence? So, the cardinality Z is Z = ℵ 0. Abu Naser Masud FABER November 11, / 87

116 MORE ON CARDINALITY (CONTD...) There is a one-to-one correspondence between the set of integers Z and the set of natural numbers N. Can you find that correspondence? So, the cardinality Z is Z = ℵ 0. But the number of integers must be twice the number of natural numbers. So, we have the following consequences: 2ℵ = ℵ 0 2ℵ = ℵ ℵ 0 = ℵ 0 4ℵ 0 = 2ℵ 0 = ℵ 0 Abu Naser Masud FABER November 11, / 87

117 MORE ON CARDINALITY (CONTD...) Also, since 2ℵ 0 3ℵ 0 4ℵ 0. We have the following implication ℵ 0 3ℵ 0 ℵ 0 = 3ℵ 0 = ℵ 0 If we continue this argument, we get nℵ 0 = ℵ 0 for any regular finite number n. Abu Naser Masud FABER November 11, / 87

118 MORE ON CARDINALITY (CONTD...) Also, since 2ℵ 0 3ℵ 0 4ℵ 0. We have the following implication ℵ 0 3ℵ 0 ℵ 0 = 3ℵ 0 = ℵ 0 If we continue this argument, we get nℵ 0 = ℵ 0 for any regular finite number n. Try to prove that the cardinality of the set of rational numbers R = ℵ 0 and ℵ n 0 = ℵ 0 Abu Naser Masud FABER November 11, / 87

119 MORE ON CARDINALITY (CONTD...) Is the cardinality of the set of real numbers R = ℵ 0? Abu Naser Masud FABER November 11, / 87

120 MORE ON CARDINALITY (CONTD...) Is the cardinality of the set of real numbers R = ℵ 0? No! Suppose we have the following correspondence: Abu Naser Masud FABER November 11, / 87

121 MORE ON CARDINALITY (CONTD...) Is the cardinality of the set of real numbers R = ℵ 0? No! Suppose we have the following correspondence: We can use the Cantor s diagonal argument to show that there is no one-to-one correspondence. Abu Naser Masud FABER November 11, / 87

122 MORE ON CARDINALITY (CONTD...) So, the cardinality of the set of real numbers is a different infinite quantity. Let s call it ℵ 1 (or c). Abu Naser Masud FABER November 11, / 87

123 MORE ON CARDINALITY (CONTD...) So, the cardinality of the set of real numbers is a different infinite quantity. Let s call it ℵ 1 (or c). Interpretation of ℵ 0 and ℵ 1 : (Figure captured from: Abu Naser Masud FABER November 11, / 87

124 MORE ON CARDINALITY (CONTD...) It has been conjectured that 2 ℵ 0 = ℵ 1 = c. However, the set theory can neither prove nor disprove the hypothesis (undecidable problem). Abu Naser Masud FABER November 11, / 87

125 MORE ON CARDINALITY (CONTD...) It has been conjectured that 2 ℵ 0 = ℵ 1 = c. We also have ℵ 0 < ℵ 1 However, the set theory can neither prove nor disprove the hypothesis (undecidable problem). Abu Naser Masud FABER November 11, / 87

126 MORE ON CARDINALITY (CONTD...) It has been conjectured that 2 ℵ 0 = ℵ 1 = c. We also have ℵ 0 < ℵ 1 The Continuum Hypothesis The continuum hypothesis states that there are no other cardinalities between ℵ 0 and ℵ 1. However, the set theory can neither prove nor disprove the hypothesis (undecidable problem). Abu Naser Masud FABER November 11, / 87

127 MORE ON CARDINALITY (CONTD...) It has been conjectured that 2 ℵ 0 = ℵ 1 = c. We also have ℵ 0 < ℵ 1 The Continuum Hypothesis The continuum hypothesis states that there are no other cardinalities between ℵ 0 and ℵ 1. However, the set theory can neither prove nor disprove the hypothesis (undecidable problem). Generalized version of the continuum hypothesis (Hausdorff in 1908) which is also undecidable: 2 ℵn = ℵ n+1 Abu Naser Masud FABER November 11, / 87

128 MORE ON CARDINALITY (CONTD...) Cantor s Theorem For every set S, the cardinality of S is strictly less than the cardinality of its power set: S < P(S). Abu Naser Masud FABER November 11, / 87

129 MORE ON CARDINALITY (CONTD...) Cantor s Theorem For every set S, the cardinality of S is strictly less than the cardinality of its power set: S < P(S). So, by Cantor s theorem, we have ℵ 0 = N < P(N) < P(P(N)) < P(P(P(N))) <... These increasing cardinalities are in one-to-one correspondence with the natural numbers N. Thus we have a countably infinite number of infinities! Abu Naser Masud FABER November 11, / 87

130 Seems Abulike Naser Masud we have another countably FABER infinite set of infinite November 11, / 87 MORE ON CARDINALITY (CONTD...) Cantor s Theorem For every set S, the cardinality of S is strictly less than the cardinality of its power set: S < P(S). So, by Cantor s theorem, we have ℵ 0 = N < P(N) < P(P(N)) < P(P(P(N))) <... These increasing cardinalities are in one-to-one correspondence with the natural numbers N. Thus we have a countably infinite number of infinities! By using the cardinality of real numbers, we obtain ℵ 1 = R < P(R) < P(P(R)) < P(P(P(R))) <...

131 MORE ON CARDINALITY (CONTD...) Basically, we really have only one countably infinite set of infinite cardinalities. ℵ 0 = N < P(N) = ℵ 1 = R < P(P(R)) < P(P(P(R))) <... Abu Naser Masud FABER November 11, / 87

132 READING MATERIALS 1 An Introduction to Formal Languages and Automata, Peter Linz. Section 1.1, Infinite Sets: Abu Naser Masud FABER November 11, / 87