CISC 4090: Theory of Computation Chapter 1 Regular Languages Section 1.1: Finite Automata Xiaolan Zhang, adapted from slides by Prof. Werschulz Fordham University Department of Computer and Information Sciences Spring, 2014 1 / 95 2 / 95 What is a computer? Finite automata Models of computers with extremely limited memory ◮ Not a simple question to answer precisely ◮ Many simple computers have extremely limited memories and ◮ Computers are quite complicated are (in fact) finite state machines. ◮ We start with a computational model ◮ Can you name any? (Hint: several are in this building, but ◮ Different models will have different features, and may match a have nothing specifically to do with our department.) real computer better in some ways, and worse in others ◮ Vending machine ◮ Our first model is the finite state machine or finite state ◮ Elevators ◮ Thermostat automaton ◮ Automatic door at supermarket 3 / 95 4 / 95
Automatic door Automatic door (cont’d) ◮ What is the desired behavior? Describe the actions and then front , rear , both rear , both , neither list the states. ◮ Person approaches, door should open front ◮ Door should stay open while person going through ◮ Door should shut if no one near doorway ◮ States are Open and Closed closed open ◮ More details about automatic door ◮ Components: front pad, door, rear pad ◮ Describe behavior now: neither ◮ Hint: action depends not only on what happens, but also on current state neither front rear both ◮ If you walk through, door should stay open when you’re on closed closed open closed closed rear pad open closed open open open ◮ But if door is closed and someone steps on rear pad, door does not open 5 / 95 6 / 95 More on finite automata A finite automaton M 1 1 0 0 1 q 1 q 2 q 3 ◮ How may bits of data does this FSM store? ◮ 1 bit: open or closed 0,1 ◮ What about state information for elevators, thermostats, vending machines, etc.? Finite automaton M 1 with three states: ◮ FSM used in speech processing, special character recognition, ◮ We see the state diagram compiler construction . . . ◮ Start state q 1 ◮ Have you implemented an FSM? When? ◮ Accept state q 2 (double circle) ◮ Several transitions ◮ Network protocol for the game “Hangman” ◮ A string like 1101 will be accepted if M 1 ends in accept state, and rejects otherwise. What will it do? ◮ Can you describe all strings that M 1 will accept? ◮ All strings ending in 1 , and ◮ All strings having an even number of 0 ’s following the final 1 7 / 95 8 / 95
Formal definition of finite state automata Describe M 1 using formal definition 0 1 0 1 q 1 q 2 q 3 A finite (state) automaton (FA) is a 5-tuple ( Q , Σ , δ, q 0 , F ): 0,1 ◮ Q is a finite set of states ◮ Σ is a finite set, called the alphabet M 1 = ( Q , Σ , δ, q 0 , F ), where ◮ δ : Q × Σ → Q is the transition function ◮ Q = { q 1 , q 2 , q 3 } ◮ q 0 ∈ Q is the start state ◮ Σ = { 0 , 1 } ◮ q 1 is the start state ◮ F ⊆ Q is the set of accepting (or final ) states . ◮ F = { q 2 } (only one accepting state) ◮ Transition function δ given by δ 0 1 q 1 q 1 q 2 q 2 q 3 q 2 q 3 q 2 q 2 9 / 95 10 / 95 The language of an FA What is the language of M 1 ? ◮ We write L ( M 1 ) = A , i.e., M 1 recognizes A . ◮ If A is the set of all strings that a machine M accepts, then A ◮ What is A ? is the language of M . ◮ A = { w ∈ { 0 , 1 } ∗ : . . . } . ◮ Write L ( M ) = A . ◮ We have ◮ Also say that M recognizes or accepts A . ◮ A machine may accept many strings, but only one language. w ∈ { 0 , 1 } ∗ : w contains at least one 1 � A = ◮ Convention: M accepts strings but recognizes a language . � and an even number of 0 ’s follow the last 1 11 / 95 12 / 95
What is the language of M 2 ? What is the language of M 3 ? 0 1 1 0 1 1 q 1 q 2 q 1 q 2 0 0 ◮ M 3 = {{ q 1 , q 2 } , { 0 , 1 } , δ, q 1 , { q 1 }} is M 2 , but with accept M 2 = {{ q 1 , q 2 } , { 0 , 1 } , δ, q 1 , { q 2 }} where state set { q 1 } instead of { q 2 } . ◮ I leave δ as an exercise. ◮ What is the language of M 3 ? ◮ What is the language of M 2 ? ◮ L ( M 3 ) = { w ∈ { 0 , 1 } ∗ : . . . } . ◮ L ( M 2 ) = { w ∈ { 0 , 1 } ∗ : . . . } . ◮ Guess L ( M 3 ) = { w ∈ { 0 , 1 } ∗ : w ends in a 0 } . ◮ L ( M 2 ) = { w ∈ { 0 , 1 } ∗ : w ends in a 1 } . Not quite right. Why? ◮ L ( M 3 ) = { w ∈ { 0 , 1 } ∗ : w = ε or w ends in a 0 } . 13 / 95 14 / 95 What is the language of M 4 ? Construct M 5 to do modular arithmetic ◮ M 4 is a five-state automaton (Figure 1.12 on page 38). ◮ Let Σ = {� reset � , 0 , 1 , 2 } . ◮ What does M 4 accept? ◮ All strings that start and end with a or start and end with b . ◮ Construct M 5 to accept a string iff the sum of each input ◮ More simply, L ( M 4 ) is all strings starting and ending with the symbol is a multiple of 3, and � reset � sets the sum back to 0. same symbol. ◮ Note that string of length 1 is okay. 15 / 95 16 / 95
Now generalize M 5 Formal definition of acceptance ◮ Generalize M 5 to accept if sum of symbols is a multiple of i instead of 3. Let M = ( Q , Σ , δ, Q 0 , F ) be an FA and let w = w 1 w 2 . . . w n ∈ Σ ∗ . M = {{ q 0 , q 1 , q 2 , . . . , q i − 1 } , { 0 , 1 , 2 , � reset �} , δ i , q 0 , { q 0 }} , We say that M accepts w if there exists a sequence r 0 , r 1 , . . . , r n ∈ Q of states such that where ◮ r 0 = q 0 . ◮ δ i ( q j , 0 ) = q j . ◮ δ i ( q j , 1 ) = q k , where k = j + 1 mod i . ◮ δ ( r i , w i +1 ) = r i +1 for i ∈ { 0 , 1 , . . . , n − 1 } ◮ δ i ( q j , 2 ) = q k , where k = j + 2 mod i . ◮ r n ∈ F . ◮ δ i ( q j , � reset � ) = q 0 . ◮ Note: As long as i is finite, we are okay and only need finite memory (number of states). ◮ Could you generalize to Σ = { 0 , 1 , 2 , . . . , k } ? 17 / 95 18 / 95 Regular languages Designing automata ◮ You will need to design an FA that accept a given language L . ◮ Strategies: A language L is regular if it is recognized by some finite ◮ Determine what you need to remember (The states). automaton. ◮ How many states to determine even/odd number of 1 ’s in an ◮ That is, there is a finite automaton M such that L ( M ) = A , input? i.e., M accepts all of the strings in the language, and rejects ◮ What does each state represent? all strings not in the language. ◮ Set the start and finish states, based on what each state ◮ Why should you expect proofs by construction coming up in represents. ◮ Assign the transitions. your next homework? ◮ Check your solution: it should accept every w ∈ L , and it should not accept any w �∈ L . ◮ Be careful about ε . 19 / 95 20 / 95
You try designing FA Regular operations Let A and B be languages. We define three regular operations : ◮ Design an FA to accept the language of binary strings having ◮ Union : A ∪ B = { x : x ∈ A or x ∈ B } . an odd number of 1 ’s (page 43). ◮ Concatenation : A · B = { xy : x ∈ A and y ∈ B } . ◮ Design an FA to accept all strings containing the ◮ Kleene star : A ∗ = { x 1 x 2 . . . x k : k ≥ 0 and each x i ∈ A } . substring 001 (page 44). ◮ What do you need to remember? ◮ Kleene star is a unary operator on a single language. ◮ A ∗ consists of (possibly empty!) concatenations of strings ◮ Design an FA to accept strings containing the substring abab . from A . 21 / 95 22 / 95 Examples of regular operations Closure Let A = { good , bad } and B = { boy , girl } . What are the ◮ A set of objects is closed under an operation if applying that following? operations to members of that set always results in a member ◮ A ∪ B = { good , bad , boy , girl } . of that set. ◮ A · B = { goodboy , goodgirl , badboy , badgirl } . ◮ The natural numbers N = { 1 , 2 , . . . } are closed under ◮ A ∗ = addition and multiplication, but not subtraction or division. { ε, good , bad , goodgood , goodbad , badgood , badbad , . . . } . 23 / 95 24 / 95
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