# Finite Automata. Dr. Nadeem Akhtar. Assistant Professor Department of Computer Science & IT The Islamia University of Bahawalpur

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1 Finite Automata Dr. Nadeem Akhtar Assistant Professor Department of Computer Science & IT The Islamia University of Bahawalpur PhD Laboratory IRISA-UBS University of South Brittany European University of Brittany, Bretagne - France 1

2 Automata Theory Automata theory is the study of abstract computing devices, or machines. (Computational model idealized computer) A manageable mathematical theory can be set up directly on a Computational model. The simplest computational model is called the finite state machine or finite automaton. 2

3 Automata Theory In 1930 s A. Turing studied an abstract machine that had all the capabilities of today s computers. Turing goal was to describe precisely the boundary between what a computing machine could do and what it could not do; his conclusions apply not only to his abstract Turing machines, but to today s real machines. 3

4 Automata Theory In the 1940 s and 1950 s, simpler kinds of machines, which we today call finite automata were studied by a number of researchers. 4

5 Automata Theory In the late 1950 s, N. Chomsky began the study of formal grammars These grammars have close relationships to abstract automata and serve today as the basis of some important software components, including parts of compilers. 5

6 Automata Theory In 1969, S. Cook extended Turing s study of what could and what could not be computed. Cook was able to separate those problems that can be solved efficiently by computer from those problems that can in principle be solved, but in practice take so much time that computers are useless for all but very small instances of the problem. These class of problems is called intractable or NP-hard. 6

7 Automata Theory It is highly unlikely that the exponential improvement in computing speed that computer hardware has been following ( Moore s Law ) will have significant impact on our ability to solve large instances of intractable problems. 7

8 Automata Theory What computer scientists do today depend on the theoretical developments. Finite automata and certain kinds of formal grammars, are used in the design and construction of important kinds of software. Turing machine help us understand what we can expect from our software. 8

9 Finite Automata 1- Software for designing and checking the behavior of digital circuits. 2- The lexical analyzer of a compiler, that is compiler component that breaks the input text into logical units, such as identifiers, keywords and punctuation. 3- Software for scanning large bodies of text, such as collection of Web pages, to find occurrences of words, phrases, or other patterns. 4- Software for verifying systems of all types that have finite number of distinct states, such as communications protocols or protocols for secure exchange of information. 9

10 Formal definition of a Finite Automaton A finite automaton is a 5-tuple (Q, Σ, δ, q o, F) where 1. Q is a finite set called the states, 2. Σ is a finite set called the alphabet, 3. δ: Qx Σ Q is the transition function, 4. q o Є Q is the start state, and 5. F Q is the set of accept states. 10

11 The finite automaton M 1 (1/2) 11

12 The finite automaton M 1 (2/2) A = { w wcontains at least one 1 and an even number of 0s follow the last 1}. Then L(M 1 ) = A, or equivalently, M 1 recognizes A. 12

13 The finite automaton M 2 13

14 The finite automaton M 2 14

15 The finite automaton M 3 15

16 The finite automaton M 3 16

17 The finite automaton M 4 17

18 The finite automaton M 4 Machine M 4 has two accept states q 1 and r 1. Alphabet Σ = {a, b} Accept strings are a, b, aa, bb, abs, bab M 4 accepts all strings that start or end with a or that start or end with b. M 4 accepts all strings that start or end with the same symbol. 18

19 The finite automaton M 5 Alphabet Σ = {0, 1, 2, <RESET>} Machine M 5 keeps a running count of the sum of the numerical input symbols it reads, modulo 3. Every time it receives <RESET> symbol it resets the count to 0. It accepts if the sum is 0, modulo 3, or in other words if the sum is a multiple of 3. 19

20 Designing Finite Automata (1) Reader as Automaton method You are given some language and want to design a finite automaton that recognizes it. Pretending to be the automaton, you receive an input string and must determine whether it isamemberofthelanguagetheautomaton is supposed to recognize. 20

21 Designing Finite Automata (2) Yougettoseethesymbolsinthestringoneby one. After each symbol you must decide whether the string seen so far is in the language. The reason is that you, like the machine, don t know when the end of the string is coming so you must always be ready with the answer. 21

22 Designing Finite Automata (3) Finite automaton is a type of machine having only finite number of states, which means a finite memory. 22

23 Designing Finite Automata (4) Example: Alphabetis{0,1}and The language consists of all strings with an odd number of 1s. Construct a finite automaton E 1 to recognize this language 23

24 Designing Finite Automata (5) Example: Alphabetis{0,1}and Thelanguageconsistsofallstringswithanoddnumberof 1s. ConstructafiniteautomatonE 1 torecognizethislanguage 24

25 Designing Finite Automata (6) Example: Design a finite automaton E 2 to recognize the regular language of all strings that contain the string 001 as a substring. For example, 0010, 1001, 001, and are all language, but11and0000arenot. So there are four possibilities: 1. haven't just seen any symbols of the pattern, 2. havejustseena0, 3. havejustseen00,or 4. have seen the entire pattern 001. Assignthestatesq,q 0,q 00,andq 001 tothesepossibilities. 25

26 Designing Finite Automata (7) Accepts strings containing

27 Nondeterministic Finite Automaton N 1 27

28 Deterministic and NonDeterministicComputations with an accepting branch 28

29 The computation of N 1 on input

30 Nondeterministic Finite Automaton N 2 30

31 Nondeterministic Finite Automaton N 2 Let A be the language consisting of all strings over {0, 1} containing a 1 in the third position from the end e.g , 00111,

32 NFA N 3 NFA N 3 that has an input alphabet {0} consisting of a single symbol. An alphabet containing only one symbol is called a unary alphabet. 32

33 NFA N 3 This machine N 3 accepts all strings of the form 0 k where kis a multiple of 2 or 3. (Note: Superscript denotes repetition, not numerical exponentiation) N 3 accepts the strings, 00, 000, 0000, and , but not 0 or

34 NFA N 4 34

35 NFA N 4 This machine N 4 accepts the strings ε, a, baba, and baa, It does not accept the strings b, bb, and babba. 35

36 Formal definition of a Nondeterministic Finite Automaton (NFA) A nondeterministic finite automaton is a 5-tuple (Q, Σ, δ, q o, F) where 1. Q is a finite set called the states, 2. Σ is a finite set called the alphabet, 3. δ: Qx Σ ε P(Q) is the transition function, 4. q o Є Q is the start state, and 5. F Q is the set of accept states. Note: Σ ε =ΣU{ε} 36

37 NFA N 1 37

38 NFA N 1 38

39 Formal definition of computation for an NFA 39

40 Equivalence of NFAS and DFAS Deterministic and nondeterministic finite automata recognize the same class of languages. (Regular languages) Equivalence of NFAs and DFAs is useful because describing an NFA for a given language sometimes is much easier than describing a DFA for that language. Two machines are equivalent if they recognize the same language. 40

41 PROOF (1) Theorem-1 Every nondeterministic finite automaton has an equivalent deterministic finite automaton. PROOF IDEA: If a language is recognized by an NFA, then we must show the existence of a DFA that also recognizes it. The idea is to convert the NFA into an equivalent DFA that simulates the NFA. 41

42 PROOF (2) If k is the number of states of the NFA, it has 2 k subsets of states. Each subset corresponds to one of the possibilities that the DFA must remember, so the DFA simulating the NFA will have2 k states. Now we need to figure out which will be the start state and accept states of the DFA, and what will be its transition function. 42

43 PROOF (3) 43

44 PROOF (4) 44

45 PROOF (5) 45

46 PROOF (6) 46

47 47

48 PROOF Example (1) NFA N 4 Thus in the formal description of N 4 = (Q, {a,b}, δ, 1, {1}), the set of states Q is {I, 2, 3} as shown in the following figure. 48

49 PROOF Example -(2) 49

50 PROOF Example -(3) 50

51 PROOF Example -(4) 51

52 PROOF Example -(5) 52

53 PROOF Example -(6) 53

54 PROOF Example -(7) 54

55 Regular language A language is called a regular language if some finite automaton recognizes it. 55

56 Regular Language Formal definition The collection of regular languages over an alphabet Σ is defined recursively as follows: The empty language Ø, and the empty string language {ε} are regular languages. For each a Σ (a belongs to Σ), the singleton language {a} is a regular language. If A and B are regular languages, then A B (union), A B (concatenation), and A*(Kleene star) are regular languages. No other languages over Σ are regular. 56

57 Regular Operations (1) In arithmetic, the basic objects are numbers and the tools are operations for manipulating them,suchas+andx. In the theory of computation the objects are languages and the tools include operations specifically designed for manipulating them. 57

58 Regular Operations (2) 58

59 Regular Operations (3) The star operation applies to a single language rather than to two different languages. That is, the star operation is a unary operation instead of a binary operation. It works by attaching any number of strings in A together to get a string in the new language. Because "any number" includes 0 as a possibility, the empty stringεis always a member of A*, no matterwhatais. 59

60 Regular Operations (4) Example: 60

61 Regular Operations (5) Let N = {I, 2, 3,... } be the set of natural numbers. When we say that N is closed under multiplication we mean that, for any x and y in N, the product x x y also is in N. In contrast N is not closed under division, as 1 and 2 are in N but 1/2 is not. Generally speaking, a collection of objects is closed under some operation if applying that operation to members of the collection returns an object still in the collection. The collection of regular languages is closed under all three of the regular operations. 61

62 Closure under the Regular Operations The class of regular languages is closed under the union operation. And The class of regular languages is closed under the concatenation operation. 62

63 Theorem-A (1) The class of regular languages is closed under the union operation. PROOF IDEA: We have regular languages A 1 and A 2 and want toprovethata 1 UA 2 isregular. The idea is to take two NFAs, N 1 and N 2 for A 1 and A 2, and combine them into one new NFA, N. 63

64 Theorem-A (2) Machine N must accept its input ifeither N 1 and N 2 acceptsthisinput. The new machine has a new start state that branches to the start states of the old machines with ε arrows. In this way the new machine nondeterministically guesses which of the two machines accepts the input. If one of them acceptstheinput,nwillacceptit,too. 64

65 Theorem-A (3) Construction of an NFA N to recognize A 1 U A 2 65

66 Theorem-A (4) 66

67 Theorem-A (5) 67

68 Theorem-B (1) The class of regular languages is closed under the concatenation operation. PROOF IDEA: We have regular languages A 1 and A 2 and want to provethata 1 oa 2 isregular. The idea is to take two NFAs, N 1 and N 2 for A 1 and A 2, and combine them into a new NFA N as shown in Figure. 68

69 Theorem-B (2) Construction of N to recognize A 1 oa 2 69

70 Theorem-B (3) 70

71 Theorem-B (4) 71

72 Theorem-C (1) The class of regular languages is closed under the star operation. PROOF IDEA: We have a regular language A 1 and want to prove that A 1 * alsoisregular. We take an NFA N 1 for A 1 and modify it to recognize A 1 *, as shown in the following figure. The resulting NFA N will accept its input whenever it can be broken into several pieces and N 1 accepts each piece. 72

73 Theorem-C (2) ConstructionofNtorecognizeA 1 * 73

74 Theorem-C (3) 74

75 Theorem-C (3) 75

76 Exercise 76

77 Exercise 77

78 Exercise 78

79 Exercise 79

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