Deterministic Finite Automata (DFAs)

Algorithms & Models of Computation CS/ECE 374, Fall 27 Deterministic Finite Automata (DFAs) Lecture 3 Tuesday, September 5, 27 Sariel Har-Peled (UIUC) CS374 Fall 27 / 36

Part I DFA Introduction 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 Sariel Har-Peled (UIUC) CS374 3 Fall 27 3 / 36

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 4 Fall 27 4 / 36

Another view Machine has input written on a read-only tape Start in specified start state Start at left, scan symbol, change state and move right Circled states are accepting Machine accepts input string if it is in an accepting state after scanning the last symbol. Sariel Har-Peled (UIUC) CS374 5 Fall 27 5 / 36

Graphical Representation/State Machine q3 q2, q q Directed graph with nodes representing states and edge/arcs representing transitions labeled by symbols in Σ For each state (vertex) q and symbol a Σ there is exactly one outgoing edge labeled by a Initial/start state has a pointer (or labeled as s, q or start ) Some states with double circles labeled as accepting/final states Sariel Har-Peled (UIUC) CS374 6 Fall 27 6 / 36

Graphical Representation q q 3 q q 2, Where does lead?? Figure : DFA M for problem 5 Which strings end up in accepting state? Can you prove it? Every string w has a unique walk that it follows from a given hange the initial state to B? state q by reading one letter of w from left to right. hange the set of final states to be {B} (with initial state A)? Sariel Har-Peled (UIUC) CS374 7 Fall 27 7 / 36

Graphical Representation q q 3 q q 2, Definition Figure : DFA M for problem 5 A DFA M accepts a string w iff the unique walk starting at the start state and spelling out w ends in an accepting state. hange Definition the initial state to B? The language accepted (or recognized) by a DFA M is denote by L(M) and defined as: L(M) = {w M accepts w}. hange the set of final states to be {B} (with initial state A)? Sariel Har-Peled (UIUC) CS374 8 Fall 27 8 / 36

Warning M accepts language L does not mean simply that that M accepts each string in L. It means that M accepts each string in L and no others. Equivalently M accepts each string in L and does not accept/rejects strings in Σ \ L. M recognizes L is a better term but accepts is widely accepted (and recognized) (joke attributed to Lenny Pitt) Sariel Har-Peled (UIUC) CS374 9 Fall 27 9 / 36

Formal Tuple Notation Definition A deterministic finite automata (DFA) M = (Q, Σ, δ, s, A) is a five tuple where Q is a finite set whose elements are called states, Σ is a finite set called the input alphabet, δ : Q Σ Q is the transition function, s Q is the start state, A Q is the set of accepting/final states. Common alternate notation: q for start state, F for final states. Sariel Har-Peled (UIUC) CS374 Fall 27 / 36

DFA Notation M = set of all states transition func ( {}}{{}}{ Q, }{{} Σ, alphabet δ, s }{{} start state, set of all accept states {}}{ A ) Sariel Har-Peled (UIUC) CS374 Fall 27 / 36

Example q q 3 q q 2, Q = {q, q, q, q 3 } Figure : DFA M for problem 5 Σ = {, } δ s = q hange the initial state to B? A = {q } hange the set of final states to be {B} (with initial state A)? Sariel Har-Peled (UIUC) CS374 2 Fall 27 2 / 36

Extending the transition function to strings Given DFA M = (Q, Σ, δ, s, A), δ(q, a) is the state that M goes to from q on reading letter a Useful to have notation to specify the unique state that M will reach from q on reading string w Transition function δ : Q Σ Q defined inductively as follows: δ (q, w) = q if w = ɛ δ (q, w) = δ (δ(q, a), x) if w = ax. Sariel Har-Peled (UIUC) CS374 3 Fall 27 3 / 36

Formal definition of language accepted by M Definition The language L(M) accepted by a DFA M = (Q, Σ, δ, s, A) is {w Σ δ (s, w) A}. Sariel Har-Peled (UIUC) CS374 4 Fall 27 4 / 36

Example q q 3 q q 2, What is: δ (q, ɛ) Figure : DFA M for problem 5 δ (q, ) δ (q, ) hange the initial state to B? δ (q 4, ) hange the set of final states to be {B} (with initial state A)? Sariel Har-Peled (UIUC) CS374 5 Fall 27 5 / 36

Example continued q q 3 q q 2, What is L(M) if start state is changed to q? Figure : DFA M for problem 5 What is L(M) if final/accept states are set to {q 2, q 3 } instead of {q }? hange the initial state to B? hange the set of final states to be {B} (with initial state A)? Sariel Har-Peled (UIUC) CS374 6 Fall 27 6 / 36

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 q, δ (q, uv) = δ (δ (q, u), v). Sariel Har-Peled (UIUC) CS374 7 Fall 27 7 / 36

Part II Constructing DFAs 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) Sariel Har-Peled (UIUC) CS374 9 Fall 27 9 / 36

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 2 Fall 27 2 / 36

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 Sariel Har-Peled (UIUC) CS374 2 Fall 27 2 / 36

Part III Product Construction and Closure Properties Sariel Har-Peled (UIUC) CS374 22 Fall 27 22 / 36

Part IV Complement Sariel Har-Peled (UIUC) CS374 23 Fall 27 23 / 36

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? q3 q2, q q Sariel Har-Peled (UIUC) CS374 24 Fall 27 24 / 36

Complement Example... Just flip the state of the states! q3 q2, q3 q2, q q q q Sariel Har-Peled (UIUC) CS374 25 Fall 27 25 / 36

Complement Theorem Languages accepted by DFAs are closed under complement. 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) CS374 26 Fall 27 26 / 36

Part V Product Construction Sariel Har-Peled (UIUC) CS374 27 Fall 27 27 / 36

Union and Intersection 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) CS374 28 Fall 27 28 / 36

Example M accepts # = odd M 2 accepts # = odd Sariel Har-Peled (UIUC) CS374 29 Fall 27 29 / 36

Example M accepts # = odd M 2 accepts # = odd Cross-product machine Sariel Har-Peled (UIUC) CS374 3 Fall 27 3 / 36

Example II Accept all binary strings of length divisible by 3 and 5 2 3 4,, 2, 3, 4,,, 2, 3, 4, 2, 2, 2 2, 2 3, 2 4, 2 Assume all edges are labeled by,. Sariel Har-Peled (UIUC) CS374 3 Fall 27 3 / 36

Product construction for intersection 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 = {(q, q 2 ) q Q, q 2 Q 2 } s = (s, s 2 ) δ : Q Σ Q where δ((q, q 2 ), a) = (δ (q, a), δ 2 (q 2, a)) A = A A 2 = {(q, q 2 ) q A, q 2 A 2 } Theorem L(M) = L(M ) L(M 2 ). Sariel Har-Peled (UIUC) CS374 32 Fall 27 32 / 36

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 Sariel Har-Peled (UIUC) CS374 33 Fall 27 33 / 36

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 = {(q, q 2 ) q Q, q 2 Q 2 } s = (s, s 2 ) δ : Q Σ Q where δ((q, q 2 ), a) = (δ (q, a), δ 2 (q 2, a)) A = {(q, q 2 ) q A or q 2 A 2 } Theorem L(M) = L(M ) L(M 2 ). Sariel Har-Peled (UIUC) CS374 34 Fall 27 34 / 36

Set Difference 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) CS374 35 Fall 27 35 / 36

Things to know: 2-way DFA Question: Why are DFAs required 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) CS374 36 Fall 27 36 / 36