FINITE AUTOMATA. Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 1

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1 FINITE AUTOMATA Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 1

2 Protocol for e-commerce using e-money Allowed events: P The customer can pay the store (=send the money- File to the store) C The customer can cancel the money (like putting a stop on a check) S The store can ship the goods to the customer R The store can redeem the money (=cash the check) T The bank can transfer the money to the store Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 2

3 The protocol for each participant Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 3

4 Some actions Customer may try to copy the money file, use it to pay several times, or both pay and cancel the money, thus getting the goods for free. Store should ship the goods but should be careful as well. It should not ship goods until it is sure it has been given valid money for the goods. The bank must behave responsibly, or it cannot be a bank. It must not allow money to be both cancelled and redeemed. Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 4

5 Completed Protocols Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 5

6 State (2,c) is accessible and corresponds to a situation where the bank received a cancel message before redeem message where as store shipped the good. Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 6

7 Deterministic Finite Automata (DFA) Formal Definition of Finite Automaton 1. Finite set of states, typically Q. 2. Alphabet of input symbols, typically Σ. 3. One state is the start/initial state, typically q 0 Q. 4. Zero or more final/accepting states; the set is typically F Q. 5. δ is a transition function (q; a) p. This function: Takes a state and input symbol as arguments. Returns a state. One rule" of δ would be written δ(q; a) = p, where q and p are states, and a is an input symbol. Intuitively: if the DFA is in state q, and input a is received, then the DFA goes to state p (note: q = p OK). A DFA is represented as the 5-tuple: A = (Q; Σ ; δ ; q 0 ; F). Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 7

8 DFA Example An automaton A that accepts L = {x01y : x; y {0; 1}} The Automaton as a transition diagram: The automaton A = ({q 0 ; q 1 ; q 2 }; {0; 1}; δ; q 0 ; {q 1 }) Transition table for the DFA 0 1 q 0 q 2 q 0 *q 1 q 1 q 1 q 2 q 2 q 1 Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 8

9 Extending the Transition Function to Strings An DFA accepts a string ω = a 1 a 2 a n if there is a path in the transition diagram that 1. Begins at a start state 2. Ends at an accepting state 3. Has sequence of labels a 1 a 2 a n Example: The following DFA accepts e.g. the string Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 9

10 Extending the transition functions to strings δ The transition function can be extended to that operates on states and strings (as opposed to states and symbols) ˆ( q, ε ) = q δ δ ( q, a ) 1 = p1 Basis: and ˆ δˆ Induction: Assume: ˆ( q, δ x) = p Then ˆ( q, δ xa) = δ ( ˆ( q, δ x), a) = δ ( p, a) Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 10

11 DFA for language L Now we can formally define a DFA A = (Q; Σ ; δ ; q 0 ; F). This language is denoted L(A), and is defined by L( A) = { w ˆ( δ q0, w) F} If Language L is L(A) for some DFA A, then we say L is a regular language. Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 11

12 Design a DFA to accept the language DFA accepting all and only strings with an even number of 0's and an even number of 1's Both even 0 s even 1 s even Both odd 0 1 * q 0 q 2 q 1 q 1 q 3 q 0 q 2 q 0 q 3 q 3 q 1 q 2 The automaton A = ({q 0, q 1, q 2, q 3 }, {0, 1}, δ, q 0, {q 0 }) Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 12

13 Marble-rolling toy from p. 53 of textbook A marble is dropped at A or B. Levers x1, x2 or x3 cause the to fall either to the left or to the right. Whenever a marble encounters a lever, it causes the lever to reverse after the marble passes, so the next marble will take the opposite branch. Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 13

14 A state is represented as sequence of three its followed by r or a (previous input rejected or accepted) A B 000r 100r 011r *000a 100r 011r *001a 101r 000a 010r 110r 001a *010a 110r 001a 011r 111r 010a 100r 010r 111r *100a 010r 111r 101r 011r 100a *101a 011r 100a 110r 000a 101a *110a 000a 101a 111r 001a 110a Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 14

15 Nondeterministic Finite Automata (NFA) A NFA can be in several states at once, or, viewed another way, it can guess" which state to go to next. This ability is expressed as an ability to guess something about its input. Like DFA, NFA has a finite set of states, a finite set of input symbols, one start state and a set of accepting states. It has also transition function δ. The difference is in the type of δ. NFA δ function returns a set of zero, one, or more states rather than returning exactly one state, as the DFA must. Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 15

16 Example: An automaton that accepts all and only strings ending in 01 Here is what happens when the NFA processes the input Input is not finished Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 16

17 Definition of Nondeterministic Finite Automata 1. Finite set of states, typically Q. 2. Alphabet of input symbols, typically Σ. 3. One state is the start/initial state, typically q 0 Q. 4. Zero or more final/accepting states; the set is typically F Q. δ is a transition function from Q x Σ to the powerset of Q A NFA is represented as the 5-tuple: A = (Q; Σ ; δ ; q 0 ; F). Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 17

18 Example: An automaton that accepts all and only strings ending in 01 The NFA can be specified fromally as A = ({q 0, q 1, q 2 }, {0, 1}, δ, q 0, {q 2 }). The transition table is as follows: 0 1 q 0 {q 0, q 1 } { q 0 } q 1 Ø { q 2 } *q 2 Ø Ø Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 18

19 The extended transition function Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 19

20 Now we can formally define a NFA A = (Q; Σ ; δ ; q0; F). This language is denoted L(A), and is defined by L( A) = { w ˆ( δ q0, w) I F 0} / Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 20

21 Equivalence of DFA and NFA NFA s are usually easier to program in. DFA in practice has about as many states as the NFA, although it is often has more transitions. In the worst case, however, the smallest DFA can have 2 n states while the smallest NFA for the same language has only n states. Surprisingly, for any NFA N there is a DFA D, such that L(D) = L(N), and vice versa. This involves the subset construction, an important example how an automaton B can be generically constructed from another automaton A. Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 21

22 Equivalence of DFA and NFA Given an NFA N = (Q N, Σ, δ N, q 0,F N ) We will construct DFA D = (Q D, Σ, δ D, {q 0 },F D ) Such that L(D) = L(N). First of all alphabets are same. Q D is the set of subsets of Q N. i.e., Q D is the power set of Q N. If Q N has n states then Q D will have 2 n states. Often not all these states are accessible from the start state of Q D. Q D = {S S Q N } F D = {S Q N S F N Ø} For every S Q N and for each input symbol a in Σ Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 22

23 Equivalence of DFA and NFA example N = ({q 0, q 1, q 2 }, {0, 1}, δ, q 0, {q 2 }). Look all the states p in S, and find the union of the transitions. Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 23

24 0 1 q 0 q 2 q 0 *q 1 q 2 q 0 q 2 q 2 q 1 Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 24

25 Equivalence of DFA and NFA Given an NFA N = (Q N, Σ, δ N, q 0,F N ) We will construct DFA D = (Q D, Σ, δ D, {q 0 },F D ) Such that L(D) = L(N). First of all alphabets are same. Q D is the set of subsets of Q N. i.e., Q D is the power set of Q N. If Q N has n states then Q D will have 2 n states. Often not all these states are accessible from the start state of Q D. Q D = {S S Q N } F D = {S Q N S F N Ø} For every S Q N and for each input symbol a in Σ Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 25

26 Theorem If D = (Q D, Σ, δ D, {q 0 },F D ) is DFA constructed from NFA N = (Q N, Σ, δ N, q 0,F N ) by the subset construction, then L(D) = L(N). Proof: We actually prove first, by induction on ω that Basis: ω = є the claim follows from the definition. Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 26

27 Theorem cont. Induction Hypothesis Construction of subsets Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 27

28 Theorem result We demonstrated that Now, why it follows that L(D)=L(N)? When we observe that both D and N accept ω if and only if extended transition functions, respectively, contain an accepting state F N, we have a complete proof that L(D)=L(N). Theorem: A language L is accepted by some DFA if and only if L is accepted by some NFA. Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 28

29 Exponential blow up It is possible that NFA N with n+1 states that has no equivalent DFA with fewer than 2 n states Suppose an equivalent DFA D with fewer than 2 n states exists. D must remember the last n symbols it has read. There are 2 n bit sequences a 1 a 2 a n Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 29

30 Since any 2 n subsets of the last n symbols could have been 1, if D has fewer than 2 n states, then there would be some state q such that D can be in state q after reading two different sequences of n bits, say a i and b i. Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 30

31 Case 1: q has to be both an accepting and a nonaccepting state. Case 2: Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 31

32 Finite Automata with ε transition The new feature is that we allow transition on ε, the empty string. NFA is allowed to make a transition spontaneously, without receiving an input symbol. Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 32

33 ε transition An ε-nfa accepting decimal numbers consisting of: 1. An optional + or - sign 2. A string of digits 3. a decimal point 4. another string of digits 5. One of the strings (2) are (4) are optional Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 33

34 Definition of ε-nfa 1. Finite set of states, typically Q. 2. Alphabet of input symbols, typically Σ. 3. One state is the start/initial state, typically q 0 Q. 4. Zero or more final/accepting states; the set is typically F Q. δ is a transition function from Q x Σ U {ε} to the powerset of Q A ε-nfa is represented as the 5-tuple: A = (Q; Σ ; δ ; q 0 ; F). Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 34

35 example E = ({q 0, q 1, q 5 }, {.,+,-,1,2,.9},δ,q 0,{q 5 }) Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 35

36 Enclose Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 36

37 Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 37

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39 Details of construction Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 39

40 Example Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 40

41 Theorem A language L is accepted by some ε-nfa if and only if L is accepted by some DFA. Mart 2006 Ankara Üniversitesi Bilgisayar Mühendisliği 41