Deterministic Finite Automata (DFAs)

Transcription

1 Algorithms & Models of Computation CS/ECE 374, Fall 27 Deterministic Finite Automata (DFAs) Lecture 3 Tuesday, September 5, 27 Part I DFA Introduction Sariel Har-Peled (UIUC) CS374 Fall 27 / 36 Sariel Har-Peled (UIUC) CS374 2 Fall 27 2 / 36 DFAs also called Finite State Machines (FSMs) The simplest model for computers? State machines that are common in practice. Vending machines Elevators Digital watches Simple network protocols Programs with fixed memory A simple program Program to check if a given input string w has odd length int n = While input is not finished read next character c n n + endwhile If (n is odd) output YES Else output NO bit x = While input is not finished read next character c x flip(x) endwhile If (x = ) output YES Else output NO Sariel Har-Peled (UIUC) CS374 3 Fall 27 3 / 36 Sariel Har-Peled (UIUC) CS374 4 Fall 27 4 / 36

2 alphabet, is a finite set called the input alphabet, function, nal states. Another view : Q! Q is the transition function, Graphical Representation/State Machine s 2 Q is the start state, A Q is the set of accepting/final states. 3 2,,s,A), string w = w w 2 w k, where for each i, w i 2, and Definition states 5. For a DFA M =(Q,,,s,A), string w = w w 2 w k, where for each i, w i 2, and states a seuence of states r,r,...r k such that (a) r = p, (b) for p, each 2 Q, i, wesayp w! M if there is a seuence of states r,r,...r k such that (a) r = p, (b) for each i, (r i,w i+ )=r i+, and (c) r k =. p, and string w 2,thereisauniue state such that pproblem w! M. 4. Prove that for any state p, and string w 2,thereisauniue state such that p w! M. Notation. Directed graph with nodes representing states and edge/arcs M (p, w) = where p w! M representing transitions labeled by symbols in Σ Machine has input written on a read-only tape Q,,,s,A). Definition 6. Consider a DFA M =(Q,,,s,A). For each state (vertex) and symbol a Σ there is exactly one Start in specified start state (s, w) 2 A. M accepts string w 2 i outgoing edge labeled by a M (s, w) 2 A. Start at left, scan symbol, change state and move right ized by a DFA M is L(M) ={w 2 Initial/start state has a pointer (or labeled as s, or start ) Circled states are accepting M accepts w}. The language accepted/recognized by a DFA M is L(M) ={w 2 M accepts w}. d/recognized by M i L = L(M). A set L Some states with double circles labeled as accepting/final states Machine accepts input string if it is in an accepting state is said to accepted/recognized by M i L = L(M). after scanning the last symbol. Sariel Har-Peled (UIUC) CS374 5 Fall 27 5 / 36 Sariel Har-Peled (UIUC) CS374 6 Fall 27 6 / 36 Problem 5. Graphical Representation 3. Which of the following is true? B! M B A! M D D! M C A! M2 B Graphical Representation 3 2 Where does lead?? Figure : DFA M for problem 5 Which strings end up in accepting state? 2. What is the following?, M 2 (A, ) = (B,) = (C, ) = Definition Can you prove it? 3. What is L(M)? Every string w has a uniue walk that it follows from a given if we change the initial state to B? 4. What is the language recognized if we change the initial state to B? state by reading one letter of w from left to right. 2 Figure : DFA M for problem 5 A DFA M accepts a string w iff the uniue walk starting at the start state and spelling out w ends in an accepting state. Definition The language accepted (or recognized) by a DFA M is denote by L(M) and defined as: L(M) = {w M accepts w}. if we change the set of final states to be {B} (with initial state 5. A)? What is the language recognized if we change the set of final states to be {B} (with initial state A)?, Sariel Har-Peled (UIUC) CS374 7 Fall 27 7 / 36 Sariel Har-Peled (UIUC) CS374 8 Fall 27 8 / 36

3 is a finite set called the input alphabet, Warning : Q! Q is the transition function, s 2 Q is the start state, Formal Tuple Notation M accepts language L does not mean simply that A that Q is the M accepts set of accepting/final states. Definition each string in L. A deterministic finite automata (DFA) M = (Q, Σ, δ, s, A) is a Definition 5. For a DFA M =(Q,,,s,A), string w = w It means that M accepts each string in L and no others. Euivalently five tuple where w 2 w k, where for each i, w i 2, and states p, 2 Q, wesayp w! M if there is a seuence of states r,r,...r k such that (a) r = p, (b) for each i, M accepts each string in L and does not accept/rejects (r i,w i+ )=rstrings i+, and in(c) r k =. Q is a finite set whose elements are called states, Σ \ L. Σ is a finite set called the input alphabet, Problem 4. Prove that for any state p, and string w 2,thereisauniue state such that p w! M. δ : Q Σ Q is the transition function, M recognizes L is a better term but accepts is widely Notation. accepted M (p, w) = where p w! M s Q is the start state, (and recognized) (joke attributed to Lenny Pitt) Definition 6. Consider a DFA M =(Q,,,s,A). A Q is the set of accepting/final states. M accepts string w 2 i M (s, w) 2 A. Common alternate notation: for start state, F for final states. The language accepted/recognized by a DFA M is L(M) ={w 2 M accepts w}. A set L is said to accepted/recognized by M i L = L(M). Sariel Har-Peled (UIUC) CS374 9 Fall 27 9 / 36 Sariel Har-Peled (UIUC) CS374 Fall 27 / 36 DFA Notation M = Problem 5.. Which of the following is true? B! M B A! M D set of all states transition func set ofd all accept! M Cstates ( {}}{{}}{{}}{ ) Q, }{{} Σ, δ, }{{} s, A A! M2 B alphabet start state 2. What is the following? (A, ) = (B,) = (C, ) = Example 3. What is L(M)? δ s = 4. What is the language recognized if we change the initial state to B? A = { } 3 2 Q = {,,, 3 } Figure : DFA M for problem 5 Σ = {, }, 5. What is the language recognized if we change the set of final states to be {B} (with initial state A)? Sariel Har-Peled (UIUC) CS374 Fall 27 / 36 Sariel Har-Peled (UIUC) CS374 2 Fall 27 2 / 36

4 alphabet, is a finite set called the input alphabet, function, Extending the transition function to strings : Q! Q is the transition function, s 2 Q is the start state, Formal definition of language accepted by M nal states. Given DFA M = (Q, Σ, δ, s, A), δ(, a) is the Astate Q is that themset goes of accepting/final states. Definition to from on reading letter a The language L(M) accepted by a DFA M = (Q, Σ, δ, s, A) is,s,a), string w = w w 2 w k, where for each i, w i 2, and Definition states 5. For a DFA M =(Q,,,s,A), string w = w w 2 w k, where for each i, w i 2, and states a seuenceuseful of states to have r,r,...r notation k suchtothat specify (a) r the = p, uniue (b) for p, state each 2 Q, that i, wesayp M will w! reach M if there is a seuence of states r,r,...r k such that {w Σ (a) r δ = p, (b) for each i, (s, w) A}. from on reading string w (r i,w i+ )=r i+, and (c) r k =. p, and string w 2 Transition,thereisauniue function δ state : Q Σ such that pproblem Q defined! M. 4. Prove that for any state p, and string w 2 inductively as,thereisauniue state such that p w! M. follows: δ (, w) = if w = ɛ Notation. M (p, w) = where p w! M Q,,,s,A). δ (, w) = δ Definition 6. Consider a DFA M =(Q,,,s,A). (δ(, a), x) if w = ax. (s, w) 2 A. M accepts string w 2 i M (s, w) 2 A. ized by a DFA M is L(M) ={w 2 M accepts w}. d/recognized by M i L = L(M). The language accepted/recognized by a DFA M is L(M) ={w 2 M accepts w}. A set L is said to accepted/recognized by M i L = L(M). Sariel Har-Peled (UIUC) CS374 3 Fall 27 3 / 36 Sariel Har-Peled (UIUC) CS374 4 Fall 27 4 / 36 Problem 5. Example 3. Which of the following is true? B! M B A! M D D! M C A! M2 B Example continued 3 What is: δ (, ɛ) δ (, ) δ (, ) if we change the initial state to B? δ ( 4, ) 2 Figure : DFA M for problem 5 2. What is the following?, M 2 (A, ) = (B,) = (C, ) = 3. What is L(M)? 4. What is the language recognized if we change the initial state to B? 2 What is L(M) if start state is changed to? Figure : DFA M for problem 5 What is L(M) if final/accept states are set to { 2, 3 } instead of { }?, if we change the set of final states to be {B} (with initial state 5. A)? What is the language recognized if we change the set of final states to be {B} (with initial state A)? Sariel Har-Peled (UIUC) CS374 5 Fall 27 5 / 36 Sariel Har-Peled (UIUC) CS374 6 Fall 27 6 / 36

5 Advantages of formal specification Necessary for proofs Necessary to specify abstractly for class of languages Exercise: Prove by induction that for any two strings u, v, any state, δ (, uv) = δ (δ (, u), v). Part II Constructing DFAs Sariel Har-Peled (UIUC) CS374 7 Fall 27 7 / 36 Sariel Har-Peled (UIUC) CS374 8 Fall 27 8 / 36 DFAs: State = Memory How do we design a DFA M for a given language L? That is L(M) = L. DFA is a like a program that has fixed amount of memory independent of input size. The memory of a DFA is encoded in its states The state/memory must capture enough information from the input seen so far that it is sufficient for the suffix that is yet to be seen (note that DFA cannot go back) DFA Construction: Example Assume Σ = {, } L =, L = Σ, L = {ɛ}, L = {}. L = {w {, } w is divisible by 5} L = {w {, } w ends with } L = {w {, } w contains as substring} L = {w {, } w contains or as substring} L = {w w has a k positions from the end} Sariel Har-Peled (UIUC) CS374 9 Fall 27 9 / 36 Sariel Har-Peled (UIUC) CS374 2 Fall 27 2 / 36

6 DFA Construction: Example L = {Binary numbers congruent to mod 5} Example: = 7 = 2 mod 5, = = mod 5 Key observation: w mod 5 = a implies w mod 5 = 2a mod 5 and w mod 5 = (2a + ) mod 5 Part III Product Construction and Closure Properties Sariel Har-Peled (UIUC) CS374 2 Fall 27 2 / 36 Sariel Har-Peled (UIUC) CS Fall / 36 Part IV Complement Question: If M is a DFA, is there a DFA M such that L(M ) = Σ \ L(M)? That is, are languages recognized by DFAs closed under complement? 3 2, Complement Sariel Har-Peled (UIUC) CS Fall / 36 Sariel Har-Peled (UIUC) CS Fall / 36

7 Complement Complement Example... Theorem Languages accepted by DFAs are closed under complement. Just flip the state of the states! 3 2, 3 2, Proof. Let M = (Q, Σ, δ, s, A) such that L = L(M). Let M = (Q, Σ, δ, s, Q \ A). Claim: L(M ) = L. Why? δ M = δ M. Thus, for every string w, δ M (s, w) = δ M (s, w). δ M (s, w) A δ M (s, w) Q \ A. δ M (s, w) A δ M (s, w) Q \ A. Sariel Har-Peled (UIUC) CS Fall / 36 Sariel Har-Peled (UIUC) CS Fall / 36 Union and Intersection Part V Product Construction Question: Are languages accepted by DFAs closed under union? That is, given DFAs M and M 2 is there a DFA that accepts L(M ) L(M 2 )? How about intersection L(M ) L(M 2 )? Idea from programming: on input string w Simulate M on w Simulate M 2 on w If both accept than w L(M ) L(M 2 ). If at least one accepts then w L(M ) L(M 2 ). Catch: We want a single DFA M that can only read w once. Solution: Simulate M and M 2 in parallel by keeping track of states of both machines Sariel Har-Peled (UIUC) CS Fall / 36 Sariel Har-Peled (UIUC) CS Fall / 36

8 Example Example M accepts # = odd M accepts # = odd M 2 accepts # = odd M 2 accepts # = odd Cross-product machine Sariel Har-Peled (UIUC) CS Fall / 36 Sariel Har-Peled (UIUC) CS374 3 Fall 27 3 / 36 Example II Accept all binary strings of length divisible by 3 and 5 Product construction for intersection M = (Q, Σ, δ, s, A ) and M 2 = (Q, Σ, δ 2, s 2, A 2 ) 2 3 4,, 2, 3, 4,,, 2, 3, 4, Create M = (Q, Σ, δ, s, A) where Q = Q Q 2 = {(, 2 ) Q, 2 Q 2 } s = (s, s 2 ) δ : Q Σ Q where δ((, 2 ), a) = (δ (, a), δ 2 ( 2, a)) 2, 2, 2 2, 2 3, 2 4, 2 Assume all edges are labeled by,. A = A A 2 = {(, 2 ) A, 2 A 2 } Theorem L(M) = L(M ) L(M 2 ). Sariel Har-Peled (UIUC) CS374 3 Fall 27 3 / 36 Sariel Har-Peled (UIUC) CS Fall / 36

9 Correctness of construction Lemma For each string w, δ (s, w) = (δ (s, w), δ 2 (s 2, w)). Exercise: Assuming lemma prove the theorem in previous slide. Proof of lemma by induction on w Product construction for union M = (Q, Σ, δ, s, A ) and M 2 = (Q, Σ, δ 2, s 2, A 2 ) Create M = (Q, Σ, δ, s, A) where Q = Q Q 2 = {(, 2 ) Q, 2 Q 2 } s = (s, s 2 ) δ : Q Σ Q where δ((, 2 ), a) = (δ (, a), δ 2 ( 2, a)) A = {(, 2 ) A or 2 A 2 } Theorem L(M) = L(M ) L(M 2 ). Sariel Har-Peled (UIUC) CS Fall / 36 Sariel Har-Peled (UIUC) CS Fall / 36 Set Difference Things to know: 2-way DFA Theorem M, M 2 DFAs. There is a DFA M such that L(M) = L(M ) \ L(M 2 ). Exercise: Prove the above using two methods. Using a direct product construction Using closure under complement and intersection and union Sariel Har-Peled (UIUC) CS Fall / 36 Question: Why are DFAs reuired to only move right? Can we allow DFA to scan back and forth? Caveat: Tape is read-only so only memory is in machine s state. Can define a formal notion of a 2-way DFA Can show that any language recognized by a 2-way DFA can be recognized by a regular (-way) DFA Proof is tricky simulation via NFAs Sariel Har-Peled (UIUC) CS Fall / 36