Theory of Languages and Automata

Transcription

1 Theory of Languages and Automata Chapter 1- Regular Languages & Finite State Automaton Sharif University of Technology

2 Finite State Automaton We begin with the simplest model of Computation, called finite state machine or finite automaton. are good models for computers with an extremely limited amount of memory. Embedded Systems Markov Chains are the probabilistic counterpart of Finite Automata Theory of Languages and Automata Prof. Movaghar 2

3 Simple Example Automatic door Door Theory of Languages and Automata Prof. Movaghar 3

4 Simple Example (cont.) State Diagram State Transition Table Neither Front Rear Both Closed Closed pen Closed Closed pen Closed pen pen pen Theory of Languages and Automata Prof. Movaghar 4

5 Formal Definition A finite automaton is a 5-tuple (Q,Σ,δ,q 0, F), where 1. Q is a finite set called states, 2. Σ is a finite set called the alphabet, 3. δ : Q Σ Q is the transition function, 4. q 0 Q is the start state, and 5. F Q is the set of accept states. Theory of Languages and Automata Prof. Movaghar 5

6 Example M 1 = (Q, Σ, δ, q 0, F), where 1. Q = {q 1, q 2, q 3 }, 2. Σ = {0,1}, 3. δ is described as q 1 q 1 q 2 q 2 q 3 q 2 q 3 q 2 q 2 1. q 1 is the start state, and 2. F = {q 2 }. 0 1 Theory of Languages and Automata Prof. Movaghar 6

7 Language of a Finite machine If A is the set of all strings that machine M accepts, we say that A is the language of machine M and write: L(M) = A. We say that M recognizes A or that M accepts A. Theory of Languages and Automata Prof. Movaghar 7

8 Example L(M 1 ) = {w w contains at least one 1 and even number of 0s follow the last 1}. Theory of Languages and Automata Prof. Movaghar 8

9 Example M 4 accepts all strings that start and end with a or with b. Theory of Languages and Automata Prof. Movaghar 9

10 Formal Definition M = (Q, Σ, δ, q 0, F) w = w 1 w 2 w n i, w i Σ M accepts w 0 r, 1 r,, n r i, i r Q 1. r 0 = q 0, 2. δ(r i, w i+1 ) = i+1 r, for i = 0,, n-1, 3. n r F. Theory of Languages and Automata Prof. Movaghar 10

11 Regular Language A language is called a regular language if some finite automaton recognizes it. Theory of Languages and Automata Prof. Movaghar 11

12 Example L (M 5 ) = {w the sum of the symbols in w is 0 modulo 3, except that <RESET> resets the count to 0}. As M 5 recognizes this language, it is a regular language. Theory of Languages and Automata Prof. Movaghar 12

13 Designing Finite Automata Put yourself in the place of the machine and then see how you would go about performing the machine s task. Design a finite automaton to recognize the regular language of all strings that contain the string 001 as a substring. Theory of Languages and Automata Prof. Movaghar 13

14 Designing Finite Automata (cont.) There are four possibilities: You 1. haven t just seen any symbols of the pattern, 2. have just seen a 0, 3. have just seen 00, or 4. have seen the entire pattern 001. Theory of Languages and Automata Prof. Movaghar 14

15 The Regular perations Let A and B be languages. We define the regular operations union, concatenation, and star as follows. Union: A B = {x x A or x B}. Concatenation: A B = {xy x A and y B }. Star: A * = {x 1 x 2 x k k 0 and each x i A }. Theory of Languages and Automata Prof. Movaghar 15

16 Closure Under Union THEREM The class of regular languages is closed under the union operation. Theory of Languages and Automata Prof. Movaghar 16

17 Proof Let M 1 = (Q 1, Σ 1, δ 1, q 1, F 1 ) recognize A 1, and M 2 = (Q 2, Σ 2, δ 2, q 2, F 2 ) recognize A 2. Construct M = (Q, Σ, δ, q 0, F) to recognize A 1 A Q = Q 1 Q 2 2. Σ = Σ 1 Σ 2 3. δ((r 1,r 2 ),a) = (δ 1 (r 1,a), δ 2 (r 2,a)). 4. q 0 is the pair (q 1, q 2 ). 5. F is the set of pair in which either members in an accept state of M 1 or M 2. F = (F 1 Q 2 ) (Q 1 F 2 ) F F 1 F 2 Theory of Languages and Automata Prof. Movaghar 17

18 Closure under Concatenation THEREM The class of regular languages is closed under the concatenation operation. To prove this theorem we introduce a new technique called nondeterminism. Theory of Languages and Automata Prof. Movaghar 18

19 Nondeterminism In a nondeterministic machine, several choices may exit for the next state at any point. Nondeterminism is a generalization of determinism, so every deterministic finite automaton is automatically a nondeterministic finite automaton. Theory of Languages and Automata Prof. Movaghar 19

20 Differences between DFA & NFA First, very state of a DFA always has exactly one exiting transition arrow for each symbol in the alphabet. In an NFA a state may have zero, one, or more exiting arrows for each alphabet symbol. Second, in a DFA, labels on the transition arrows are symbols from the alphabet. An NFA may have arrows labeled with members of the alphabet or ε. Zero, one, or many arrows may exit from each state with the label ε. Theory of Languages and Automata Prof. Movaghar 20

21 Deterministic vs. Nondeterministic Computation Theory of Languages and Automata Prof. Movaghar 21

22 Example Consider the computation of N 1 on input Theory of Languages and Automata Prof. Movaghar 22

23 Example (cont.) Theory of Languages and Automata Prof. Movaghar 23

24 Formal Definition A nondeterministic finite automaton is a 5-tuple (Q,Σ,δ,q 0, F), where 1. Q is a finite set of states, 2. Σ is a finite alphabet, 3. δ : Q Σ ε P(Q) is the transition function, 4. q 0 Q is the start state, and 5. F Q is the set of accept states. Theory of Languages and Automata Prof. Movaghar 24

25 Example N 1 = (Q, Σ, δ, q 0, F), where 1. Q = {q 1, q 2, q 3, q 4 }, 2. Σ = {0,1}, 3. δ is given as 0 1 ε q 1 {q 1 } {q 1,q 2 } q 2 {q 3 } {q 4 } q 3 {q 4 } q 4 {q 4 } {q 4 } 1. q 1 is the start state, and 2. F = {q 4 }. Theory of Languages and Automata Prof. Movaghar 25

26 Equivalence of NFAs & DFAs THEREM Every nondeterministic finite automaton has an equivalent deterministic finite automaton. PRF IDEA convert the NFA into an equivalent DFA that simulates the NFA. If k is the number of states of the NFA, so the DFA simulating the NFA will have 2 k states. Theory of Languages and Automata Prof. Movaghar 26

27 Proof Let N = (Q,Σ,δ,q 0, F) be the NFA recognizing A. We construct a DFA M =(Q',Σ',δ',q 0 ', F ) recognizing A. let's first consider the easier case wherein N has no ε arrows. 1. Q' = P(Q) q 0 = {q 0 }. 4. F' = {R Q R contains an accept state of N}. Theory of Languages and Automata Prof. Movaghar 27

28 Proof (cont.) Now we need to consider the ε arrows. for R Q let E(R) = {q q can be reached from R by traveling along 0 or more ε arrows}. 1. Q' = P(Q). 2. δ' (R,a) ={q Q q E(δ(r,a)) for some r R}. 3. q 0 = E({q 0 }). 4. F' = {R Q R contains an accept state of N}. Theory of Languages and Automata Prof. Movaghar 28

29 Corollary A language is regular if and only if some nondeterministic finite automaton recognizes it. Theory of Languages and Automata Prof. Movaghar 29

30 Example D s state set is {,{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}. The start state is E({1}) = {1,3}. The accept states are {{1},{1,2},{1,3},{1,2,3}}. Theory of Languages and Automata Prof. Movaghar 30

31 Example (cont.) After removing unnecessary states Theory of Languages and Automata Prof. Movaghar 31

32 CLSURE UNDER THE REGULAR PERATINS [Using NFA] Theory of Languages and Automata Prof. Movaghar 32

33 Closure Under Union The class of regular languages is closed under the Union operation. Let NFA1 recognize A1 and NFA2 recognize A2. Construct NFA3 to recognize A1 U A2. Theory of Languages and Automata Prof. Movaghar 33

34 Proof (cont.) Theory of Languages and Automata Prof. Movaghar 34

35 Closure Under Concatenation peration The class of regular languages is closed under the concatenation operation. Theory of Languages and Automata Prof. Movaghar 35

36 Proof (cont.) Theory of Languages and Automata Prof. Movaghar 36

37 Closure Under Star operation The class of regular languages is closed under the star operation. We represent another NFA to recognize A*. Theory of Languages and Automata Prof. Movaghar 37

38 Proof (cont.) 1. = The states of N are the states of N1 plus a new start state. 2. The state q 0 is the new start state. 3. = 4. The accept states are the old accept states plus the new start state. 5. Define so that for any : Theory of Languages and Automata Prof. Movaghar 38

39 Regular Expression Say that R is a regular expression if R is: 1. a for some a in the alphabet Σ , where and are regular exp. 5., where and are regular exp. 6., where and are regular exp. Recursive Definition? Theory of Languages and Automata Prof. Movaghar 39

40 Regular Expression Language Let R be a regular expression. L( R ) is the language that is defined by R: 1. if = for Σ then L( R )={a} 2. if = then = { } 3. if = then = 4. if = then = 5. i = then = 6. if = then = Theory of Languages and Automata Prof. Movaghar 40

41 Examples(cont.) = Σ 1Σ = { contains at least one 1} = { contains a single 1} 4. Σ 001Σ = { contains 001 as a substring} = {01,10} 6. (ΣΣ) = { is a string of even length} 7. (ΣΣΣ) = { the lentgh of w is a multiple of 3} 8(ΣΣΣΣ) = { the lentgh of w is a multiple 4} = (1 ) = {, 0,1,01} = 12. = { } Theory of Languages and Automata Prof. Movaghar 41

42 Equivalence of DFA and Regular Expression A language is regular if and only if some regular expression describes it. Lemma: If a language is described by a regular expression, then it is regular. If a language is regular, then it is described by a regular expression. Theory of Languages and Automata Prof. Movaghar 42

43 Building an NFA from the Regular Expression We consider the six cases in the formal definition of regular expressions Theory of Languages and Automata Prof. Movaghar 43

44 Examples Theory of Languages and Automata Prof. Movaghar 44

45 ther direction of the proof We need to show that, if a language A is regular, a regular expression describes it! First we show how to convert DFAs into GNFAs, and then GNFAs into regular expressions. We can easily convert a DFA into a GNFA in the special form. Theory of Languages and Automata Prof. Movaghar 45

46 Formal Definition A generalized nondeterministic finite automaton is a 5-tuple, a 5-tuple (Q,Σ,δ,q start, q accept ), where 1. Q is a finite set called states, 2. Σ is a the input alphabet, 3. δ : ( { }) (Q { } is the transition function, 4. q start is the start state, and 5. q accept is the accept state. Theory of Languages and Automata Prof. Movaghar 46

47 Assumptions For convenience we require that GNFAs always have a special form that meets the following conditions: 1. The start state has transition arrows going to every other state but no arrows coming in from any other state. 2. There is only a single accept state, and it has arrows coming in from every other state but no arrows going to any other state. Furthermore, the accept state is not the same as the start state. 3. Except for the start and accept states, one arrow goes from every state to every other state and also from each state to itself. Theory of Languages and Automata Prof. Movaghar 47

48 Acceptance of Languages for GNFA A GNFA accepts a string w in Σ* if w = w 1 w 2 w k, where each w i is in Σ* is in Σ* and a sequence of q 0, q 1,, q k exists such that 1. q 0 = q start is the start state, 2. q k = q accept is the accept state, and 3. For each i, we have w i L(R i ) where R i = δ(q i-1, q i ); in other words R i is the expression on the arrow from q i-1 to q i.

49 How to Eliminate a State? Theory of Languages and Automata Prof. Movaghar 49

50 Example Theory of Languages and Automata Prof. Movaghar 50

51 Example Theory of Languages and Automata Prof. Movaghar 51

52 Grammar A grammar G is a 4-tuple G = (V, Σ, R, S) where: 1. V is a finite set of variables, 2. Σ is a finite, disjoint from V, of terminals, 3. R is a finite set of rules, 4. S is the start variable. Theory of Languages and Automata Prof. Movaghar 52

53 Rule A rule is of the form where ( Σ) + and ( Σ) The rules are applied in the following manner: given a string w of the form w = uxv, We say that the rule x y is applicable to this string, and we may use it to replace x with y, thereby obtaining a new string z = uyv, This is written as. Theory of Languages and Automata Prof. Movaghar 53

54 If Derivation we say that W 1 derives W n and write Thus, we always have Theory of Languages and Automata Prof. Movaghar 54

55 Language of a Grammar Let G = (V, Σ, R, S) be a grammar. Then, the set = { Σ : } is the language generated by G. Theory of Languages and Automata Prof. Movaghar 55

56 Example Consider the grammar G = ({S}, {a,b}, P, S} with P given by S asb S ε Then So we can write Then, L(G) = {a n b n : n 0} Theory of Languages and Automata Prof. Movaghar 56

57 A Notation for Grammars Consider the grammar G = ({S}, {a,b}, P, S} with P given by S asb S ε The above grammar is usually written as: G: S asb ε Theory of Languages and Automata Prof. Movaghar 57

58 Regular Grammar A grammar G = (V, Σ, R, S) is said to be right-linear if all rules are of the form A xb A x Where A, B V, and X Σ*. A grammar is said to be leftlinear if all rules are of the form A Bx A x A regular grammar is one that is either right-linear or left-linear. Theory of Languages and Automata Prof. Movaghar 58

59 Theorem Let G = (V, Σ, R, S) be a right-linear grammar. Then: L(G) is a regular language. Theory of Languages and Automata Prof. Movaghar 59

60 Example Construct a NFA that accepts the language generated by the grammar V 0 av 1 V 1 abv 0 b V 0 a V 1 b V f b a V 2 Theory of Languages and Automata Prof. Movaghar 60

61 Theorem Let L be a regular language on the alphabet Σ. Then: There exists a right-linear grammar G = (V, Σ, R, S) Such that L = L(G). Theory of Languages and Automata Prof. Movaghar 61

62 Theorem Theorem A language is regular if and only if there exists a left-linear grammar G such that L = L(G). utline of the proof: Given any left-linear grammar with rules of the form A Bx A x We can construct a right-linear Ĝ by replacing every such rule of G with A x R B A x R We have L(G) = L(Ĝ) R. Theory of Languages and Automata Prof. Movaghar 62

63 Theorem A language L is regular if and only if there exists a regular grammar G such that L = L(G). : = ; Theory of Languages and Automata Prof. Movaghar 63