Finite Automata and Regular Languages
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1 Finite Automata and Regular Languages Topics to be covered in Chapters 1-4 include: deterministic vs. nondeterministic FA, regular expressions, one-way vs. two-way FA, minimization, pumping lemma for regular sets, closure properties, Myhill-Nerode Theorem,... H. Yen (NTUEE) Theory of Computation Fall / 67
2 Definitions and Notation An alphabet is finite set of set of symbols or letters. Eg. A = {a, b, c}, Σ = {0, 1}. A string or word over an alphabet A is a finite sequence of letters from A. Eg. aaba is string over {a, b, c}. Empty string denoted by ɛ. Set of all strings over A denoted by A. What is the size or cardinality of A? Infinite but Countable: Can enumerate in lexicographic order: ɛ, a, b, c, aa, ab,... Operation of concatenation on words: String u followed by string v: written u v or simply uv. Eg. aabb aaa = aabbaaa. H. Yen (NTUEE) Theory of Computation Fall / 67
3 Definitions and Notation A language over an alphabet Σ is a set L of strings over Σ, i.e., L Σ. Eg. for Σ = {a, b, c}: L = {abc, aaba}. L 1 = {ɛ, b, aa, bb, aab, aba, baa, bbb,...}. L 2 = {}. L 3 = {ɛ}. How many languages are there over a given alphabet A? Uncountably infinite Example (Some languages) Chinese, Swedish, English, Spanish, French,... Any programming language, {ɛ} and Σ are languages over any Σ The set of prime natural numbers {1, 3, 5, 7, 11,...} H. Yen (NTUEE) Theory of Computation Fall / 67
4 Some Operations on Languages Union, Intersection,... : As for any set Concatenation: L 1 L 2 = {u v u L 1, v L 2 }. Closure: Eg. {abc, aaba} {ɛ, a, bb} = {abc, aaba, abca, aabaa, abcbb, aababb}. L = n N where L 0 = {ɛ}, L n+1 = L n L. Note: We have = {ɛ} and L = L 0 L 1 L 2... = {ɛ} {x 1...x n n > 0, x i L} L n H. Yen (NTUEE) Theory of Computation Fall / 67
5 Definitions and notation: DFA A Deterministic Finite Automaton (DFA) A is a 5-tuple (Q, Σ, q 0, δ, F ) where Q is a finite set of states, Σ is the input alphabet, q 0 Q is the start state (or initial state), δ : Q Σ Q is the transition function, and F Q is the set of final states (or accepting states). H. Yen (NTUEE) Theory of Computation Fall / 67
6 Example Let the DFA (Q, Σ, δ, q 0, F ) be given by: Q = {q 0, q 1, q 2 } Σ = {0, 1} F = {q 2 } δ : Q Σ Q δ(q 0, 0) = q 1 ; δ(q 1, 0) = q 1 ; δ(q 2, 0) = q 2 ; δ(q 0, 1) = q 0 ; δ(q 1, 1) = q 2 ; δ(q 2, 1) = q 2 What does the DFA do? H. Yen (NTUEE) Theory of Computation Fall / 67
7 How to Represent a DFA? Transition diagram: Transition table: The start state is indicated with. The final states are indicated with H. Yen (NTUEE) Theory of Computation Fall / 67
8 When Does a DFA Accept a Word? When reading the word the automaton moves according to δ. Definition: If after reading the input it stops in a final state, it accepts the word. Example Only the word then is accepted. We have a (non-accepting) stop or dead state q 5. H. Yen (NTUEE) Theory of Computation Fall / 67
9 Example: DFA Let Σ = {0, 1}. We want to accept words in L = {x010y x, y Σ }, i.e., words that contain 010 as a subword. H. Yen (NTUEE) Theory of Computation Fall / 67
10 Extended Transition Function Definition of ˆδ We extend δ : Q Σ Q to ˆδ : Q Σ Q inductively by ˆδ(q, w) = ˆδ(q, ɛ) = q w = ɛ ˆδ(q, ax) = ˆδ(δ(q, a), x) w = ax, a Σ, x Σ Note: ˆδ(q, a) = δ(q, a) since the string a = aɛ. ˆδ(q, a) = ˆδ(q, aɛ) = ˆδ(δ(q, a), ɛ) = δ(q, a) Another definition ˆδ ˆδ (q, w) = ˆδ (q, ɛ) = q ˆδ (q, xa) = δ(ˆδ (q, x), a) w = ɛ w = xa, a Σ, x Σ H. Yen (NTUEE) Theory of Computation Fall / 67
11 Some Properties Proposition For any words x and y, and for any state q we have that ˆδ(q, xy) = ˆδ(ˆδ(q, x), y). Proof. We prove the result by induction on x. Basis case: ˆδ(q, ɛy) = ˆδ(q, y) = ˆδ(ˆδ(q, ɛ), y). Inductive step: Our IH is that ˆδ(q, xy) = ˆδ(ˆδ(q, x), y) for any word y and any state q. We should prove that ˆδ(q, (ax)y) = ˆδ(ˆδ(q, ax), y). ˆδ(q, (ax)y) = ˆδ(q, a(xy)) by def of concat = ˆδ(δ(q, a), xy) by def of ˆδ = ˆδ(ˆδ(δ(q, a), x), y) by IH = ˆδ(ˆδ(q, ax), y) by def of ˆδ H. Yen (NTUEE) Theory of Computation Fall / 67
12 Language Accepted by a DFA Definition The language accepted by the DFA M = (Q, Σ, δ, q 0, F ), denoted by L(M), is the set {x x Σ, ˆδ(q 0, x) F }. Example is accepted but is not. Definition A language L Σ is called regular if there is a DFA M over Σ such that L(M) = L. H. Yen (NTUEE) Theory of Computation Fall / 67
13 Configuration A configuration of a finite automaton A = (Q, Σ, δ, q 0, F ) is given by a state of its control unit and the content of its tape that was not read yet. The set of all configurations (Conf) is Q Σ For an input word w Σ, (q 0, w) is the initial configuration, and (q f, ɛ) is an final configuration where q f F. Definition (q, w) (q, w ) iff w = aw and q = δ(q, a) for some a Σ. Definition A computation of an automaton is a sequence of configurations C 0, C 1, C 2,..., C k where C i are configurations, C 0 is an initial configuration, C k is a final configuration, and for all i, we have C i C i+1. H. Yen (NTUEE) Theory of Computation Fall / 67
14 Relation Definition The relation is the reflexive and transitive closure of the relation, i.e., it is the smallest reflexive and transitive relation containing the relation. We also write q w q to denote that ˆδ(q, w) = q, i.e., starting in state q goes to state q by reading word w. NOTE: (q, w) (q, ɛ) iff ˆδ(q, w) = q iff q w q, for DFA. Definition The language accepted by the DFA M = (Q, Σ, δ, q 0, F ) is the set L(M) = {x Σ ˆδ(q 0, x) F } = {x Σ (q 0, x) (q f, ɛ), q f F } = {x Σ q 0 x qf, q f F } H. Yen (NTUEE) Theory of Computation Fall / 67
15 Accepting a Word H. Yen (NTUEE) Theory of Computation Fall / 67
16 Product of Automata Definition Given two DFA D 1 = (Q 1, Σ, δ 1, q 1, F 1 ) and D 2 = (Q 2, Σ, δ 2, q 2, F 2 ) with the same alphabet Σ, we can define the product D = (Q, Σ, δ, q 0, F ), also denoted by D 1 D 2, as follows: Q = Q 1 Q 2 δ((r 1, r 2 ), a) = (δ 1 (r 1, a), δ 2 (r 2, a)) q 0 = (q 1, q 2 ) F = F 1 F 2 Proposition ˆδ(r 1, r 2 ), x) = (ˆδ 1 (r 1, x), ˆδ 2 (r 2, x)). Proof by Induction. H. Yen (NTUEE) Theory of Computation Fall / 67
17 An Example Example H. Yen (NTUEE) Theory of Computation Fall / 67
18 Language Accepted by a Product Automaton Theorem Given two DFA D 1 and D 2, then L(D 1 D 2 ) = L(D 1 ) L(D 2 ). Proof. ˆδ(q 0, x) = (ˆδ 1 (q 1, x), ˆδ 2 (q 2, x)) F iff ˆδ 1 (q 1, x) F 1 and ˆδ 2 (q 2, x) F 2, that is, x L(D 1 ) and x L(D 2 ), so x L(D 1 ) L(D 2 ). H. Yen (NTUEE) Theory of Computation Fall / 67
19 Variation of the Product Definition We define D 1 D 2 similarly to D 1 D 2 but with a different notion of accepting state: a state (r 1, r 2 ) is accepting iff r 1 F 1 or r 2 F 2 Theorem Given two DFA D 1 and D 2, then L(D 1 D 2 ) = L(D 1 ) L(D 2 ). H. Yen (NTUEE) Theory of Computation Fall / 67
20 Complement Definition Given the automaton D = (Q, Σ, δ, q 0, F ) we define the complement D of D as the automaton D = (Q, Σ, δ, q 0, Q F ). Theorem Given a DFA D we have that L( D) = Σ L(D). Proof. L( D) Σ L(D) w L( D) ˆδ(q 0, w) (Q F ) ˆδ(q 0, w) F w L(D) w Σ L(D) L( D) Σ L(D) H. Yen (NTUEE) Theory of Computation Fall / 67
21 Closure properties Theorem The class of regular languages is closed under 1 complement, 2 intersection, 3 union, 4 concatenation, 5 Kleene iteration. (4) and (5) will be shown later. H. Yen (NTUEE) Theory of Computation Fall / 67
22 Nondeterministic Finite Automata A nondeterministic finite automaton (NFA) M is a 5-tuple (Q, Σ, q 0, δ, F ), where Example Q is a finite set of states, Σ is the input alphabet, q 0 Q is the start state (or initial state), δ : Q Σ 2 Q is the transition relation, and F Q is the set of final states (or accepting states). Given a state and the next symbol, the automata can move to many states. H. Yen (NTUEE) Theory of Computation Fall / 67
23 Extending the Transition Function to Strings Definition ˆδ : Q Σ 2 Q ˆδ(q, ɛ) = {q} ˆδ(q, ax) = p δ(q,a) ˆδ(p, x) That is, if δ(q, a) = {p 1,..., p n } then ˆδ(q, ax) = ˆδ(p 1, x)... ˆδ(p n, x) Definition The language accepted by the NFA N = (Q, Σ, δ, q 0, F ) is the set L(N) = {x Σ ˆδ(q 0, x) F }. That is, a word x is accepted if ˆδ(q 0, x) contains at least one accepting state. H. Yen (NTUEE) Theory of Computation Fall / 67
24 DFA and NFA A DFA can be turned into an NFA that accepts the same language. If δ(q, a) = p, let the NFA have δ(q, a) = {p}. Then the NFA is always in a set containing exactly one state - the state the DFA is in after reading the same input. Surprisingly, for any NFA there is a DFA that accepts the same language. Proof is the subset construction. The number of states of the DFA can be exponential in the number of states of the NFA. Thus, NFA! s accept exactly the regular languages. H. Yen (NTUEE) Theory of Computation Fall / 67
25 Subset Construction Subset construction Given an NFA N = (Q N, Σ, δ N, q 0, F N ), we will construct a DFA D = (Q D, Σ, δ D, {q 0 }, F D ) such that L(D) = L(N) Q D = {S : S Q N } F D = {S : S Q N, and S F N } S Q N, a Σ, δ D (S, a) = p S δ N (p, a) The DFA states have names that are sets of NFA states. But as a DFA state, an expression like {p, q} must be read as a single symbol, not as a set. Analogy: a class of objects whose values are sets of objects of another class. H. Yen (NTUEE) Theory of Computation Fall / 67
26 Subset Construction (cont d) Example: H. Yen (NTUEE) Theory of Computation Fall / 67
27 Subset Construction (cont d) Example (cont d): H. Yen (NTUEE) Theory of Computation Fall / 67
28 Subset Construction (cont d) Example (cont d): H. Yen (NTUEE) Theory of Computation Fall / 67
29 Subset Construction (cont d) Example (cont d): H. Yen (NTUEE) Theory of Computation Fall / 67
30 Subset Construction (cont d) Example (cont d): H. Yen (NTUEE) Theory of Computation Fall / 67
31 Subset Construction (cont d) Example (cont d): H. Yen (NTUEE) Theory of Computation Fall / 67
32 Subset Cconstruction (cont d) Example (cont d): H. Yen (NTUEE) Theory of Computation Fall / 67
33 Subset Construction (cont d) Example (cont d): H. Yen (NTUEE) Theory of Computation Fall / 67
34 Proof of Equivalence By Induction on w that δ N (q 0, w) = δ D ({q 0 }, w) Basis: w = ɛ: δ N (q 0, ɛ) = δ D ({q 0 }, ɛ) = {q 0 }. Ind. Hypothesis: Assume IH for strings shorter than w. Ind. Step: Let w = xa; IH holds for x. H. Yen (NTUEE) Theory of Computation Fall / 67
35 Exponential Blow-Up There is an NFA N with n + 1 states that has no equivalent DFA with fewer than 2 n states. L(N) = {x 1 c 2 c 3 c n : x {0, 1}, c i {0, 1}}. 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 bitsequences a 1 a 2 a n. q, a 1 a 2 a n, b 1 b 2 b n : q ˆδ N (q 0, a 1 a 2 a n ), q ˆδ N (q 0, b 1 b 2 b n ), a 1 a 2 a n b 1 b 2 b n H. Yen (NTUEE) Theory of Computation Fall / 67
36 Exponential Blow-Up (Cont d) a 1 a i1 1a i+1 a n b 1 b i1 0b i+1 b n Now And ˆδ N (q 0, a 1 a i 1 1a i+1 a n 0 i 1 ) = ˆδ N (q 0, b 1 b i 1 0b i+1 b n 0 i 1 ) ˆδ N (q 0, a 1 a i 1 1a i+1 a n 0 i 1 ) F D A contradiction! ˆδ N (q 0, b 1 b i 1 0b i+1 b n 0 i 1 ) F D H. Yen (NTUEE) Theory of Computation Fall / 67
37 NFA with ɛ-transitions We can allow state-to-state transitions on ɛ input. These transitions are done spontaneously, without looking at the input string. Example A convenience at times, but still only regular languages are accepted. H. Yen (NTUEE) Theory of Computation Fall / 67
38 Closure of States CL(q) = set of states you can reach from state q following only arcs labeled ɛ. CL(A) = {A}; CL(E) = {B, C, D, E}. Closure of a set of states = union of the closure of each state. H. Yen (NTUEE) Theory of Computation Fall / 67
39 Extended Delta ˆδ Basis: ˆδ(q, ɛ) = CL(q) Induction: ˆδ(q, xa) = CL(p) Example p δ(ˆδ(q,x),a) ˆδ(A, ɛ) = CL(A) = {A}. ˆδ(A, 0) = CL({E}) = {B, C, D, E}. ˆδ(A, 01) = CL({C, D}) = {C, D}. H. Yen (NTUEE) Theory of Computation Fall / 67
40 Equivalence of ɛ-nfa and DFA Given an ɛ-nfa E = (Q E, Σ, δ E, q 0, F E ), we will construct a DFA D = (Q D, Σ, δ D, q D, F D ) such that L(D) = L(E) Details of the construction: Q D = {S : S Q E, and S = CL(S)} q D = CL(q 0 ) F D = {S : S Q D, and S F E } δ D (S, a) = {CL(p) : p δ(t, a), for some t S} H. Yen (NTUEE) Theory of Computation Fall / 67
41 Regular Expression Regular expressions are an algebraic way to denote languages. Syntax of regular expressions Syntax of regular expresions over an alphabet Σ: r ::= a r + r r r r, where a Σ. Semantics of regular expressions Semantics: associate a language L(r) Σ with regexp r. L( ) = {} L(a) = {a} L(r + r ) = L(r) L(r ) L(r r ) = L(r) L(r ) L(r ) = L(r). H. Yen (NTUEE) Theory of Computation Fall / 67
42 Algebraic Laws for Regular Expressions The following equalities hold for any RE R, S and T: H. Yen (NTUEE) Theory of Computation Fall / 67
43 Kleene s Theorem: RE = DFA Class of languages defined by regular expressions coincides with regular languages. Proof. RE DFA: Use closure properties of regular languages. DFA RE: H. Yen (NTUEE) Theory of Computation Fall / 67
44 DFA RE: Kleene s Construction H. Yen (NTUEE) Theory of Computation Fall / 67
45 DFA RE: Kleene s Construction (cont d) H. Yen (NTUEE) Theory of Computation Fall / 67
46 DFA RE: Kleene s Construction (cont d) H. Yen (NTUEE) Theory of Computation Fall / 67
47 DFA RE: Using State Elimination H. Yen (NTUEE) Theory of Computation Fall / 67
48 DFA RE: Using State Elimination (cont d) For each q F we ll be left with an A q that looks like E q = (R + SU T ) SU E q = R The final expression is q F E q. H. Yen (NTUEE) Theory of Computation Fall / 67
49 DFA RE: Using System of Equations H. Yen (NTUEE) Theory of Computation Fall / 67
50 DFA RE: Using System of Equations (cont d) H. Yen (NTUEE) Theory of Computation Fall / 67
51 DFA RE: Using System of Equations (cont d) L q s are a solution to the system of equations In general there could be many solutions to equations. Consider x = A x. In this case, L q s can be seen to the least solution to the equations. H. Yen (NTUEE) Theory of Computation Fall / 67
52 DFA RE: Using System of Equations (cont d) H. Yen (NTUEE) Theory of Computation Fall / 67
53 RE ɛ-nfa Theorem For every regex R we can construct and ɛ-nfa A, s.t. L(A) = L(R). H. Yen (NTUEE) Theory of Computation Fall / 67
54 Induction Proof H. Yen (NTUEE) Theory of Computation Fall / 67
55 An Example We convert (0 + 1) 1(0 + 1) H. Yen (NTUEE) Theory of Computation Fall / 67
56 Equivalence Relation Definition A binary relation R on a set S is a subset of S S. An equivalence relation on a set satisfies 1 Reflexivity: For all x in S, xrx 2 Symmetry: For x, y S xry yrx 3 Transitivity: For x, y, z S xry yrz xrz Every equivalence relation on S partitions S into equivalence classes. The number of equivalence classes is called the index of the relation. H. Yen (NTUEE) Theory of Computation Fall / 67
57 Right Invariant Definition An equivalence relation on Σ is said to be right invariant with respect to concatenation if x, y Σ and a Σ, xry implies that xarya. H. Yen (NTUEE) Theory of Computation Fall / 67
58 Equivalence relations induced by DFA Let M = (Q, Σ, δ, q 0, F ) be a DFA. Define a relation R M as follows: For x, y Σ, xr M y δ(q 0, x) = δ(q 0, y). Is this an equivalence relation? If so, how many equivalence classes does it have? That is, what is its index? H. Yen (NTUEE) Theory of Computation Fall / 67
59 An Example Example H. Yen (NTUEE) Theory of Computation Fall / 67
60 Refinement Definition An equivalence relation R 1 is a refinement of R 2 if R 1 R 2, i.e. (x, y) R 1 (x, y) R 2 H. Yen (NTUEE) Theory of Computation Fall / 67
61 The Myhill-Nerode Theorem Theorem Let L Σ. The following statements are equivalent: 1 L is recognized by a DFA 2 L is the union of some of the equivalence classes of a right invariant equivalence relation of finite index. 3 Define an equivalence relation R L as follows. For x, y Σ, (x, y) R L z Σ, xz L whenever yz L. Then R L has finite index. To prove this, we need to prove that H. Yen (NTUEE) Theory of Computation Fall / 67
62 Proof: 1 2 (1) L is recognized by a DFA (2) L is the union of some of the equivalence classes of a right invariant equivalence relation of finite index. Let M = (Q, Σ, δ, q 0, F ) be the DFA that recognizes L. The relation R M defined earlier as follows: For x, y Σ, xr M y δ(q 0, x) = δ(q 0, y) is an equivalence relation which is of finite index and is also right invariant as M is a DFA. L is just the union of the equivalence classes corresponding to the final states of M H. Yen (NTUEE) Theory of Computation Fall / 67
63 Proof: 2 3 (2) L is the union of some of the equivalence classes of a right invariant equivalence relation of finite index. (3) Define an equivalence relation R L as follows. For x, y Σ, (x, y) R L z Σ, xz L whenever yz L. Then R L has finite index. Let E be the equivalence relation in (2). We show that E is a refinement of R L. Hence since E has finite index, so does R L. We show that if (x, y) E then (x, y) R L, i.e. E R L and hence E is a refinement of R L. (x, y) E z Σ, (xz, yz) E by repeated use of right invariance. Hence xz is in an equivalence class in L iff yz is, thereby implying that (x, z) R L. Thus R L has finite index. H. Yen (NTUEE) Theory of Computation Fall / 67
64 Proof: 3 1 (3) Define an equivalence relation R L as follows. For x, y Σ, (x, y) R L z Σ, xz L whenever yz L. Then R L has finite index. (1) L is recognized by a DFA Construct a DFA M L = (Q L, Σ, δ L, q 0L, F L ). We first show that R L is right invariant. To see this assume that (x, y) R L. Assume R L is not right invariant. Then there exists a Σ such that (xa, ya) R L. Let z be a distinguishing string for the pair (xa, ya). Then az is a distinguishing string for (x, y) contradicting the assumption that (x, y) R L. Therefore R L is right invariant. H. Yen (NTUEE) Theory of Computation Fall / 67
65 Proof: 3 1 (cont d) Let [x] denote the equivalence class containing x. The state set Q L = {[x] : x Σ }. The transition function δ L is defined as δ L ([x], a) = [xa]. The definition is consistent because had we chosen some other representative [y] of the equivalence class, [xa] = [ya] by right invariance of R L so the transition is to the same state of M L. We need to prove that x L δ L (q 0L, x) F L i.e. x is accepted by M L This is easy as δ L (q 0L, x) = [x] and x is accepted by M L iff [x] F L x L. H. Yen (NTUEE) Theory of Computation Fall / 67
66 Applications of the Myhill-Nerode Theorem The MN theorem can be used to show that a particular language is regular without actually constructing the automaton or to show conclusively that a language is not regular. Example. Is the following language regular 1 L 1 = {xy : x = y, x, y Σ }? 2 Example. What about the language L 2 = {xy : x = y, x, y Σ and y ends with a 1 }? 3 Example. What about the language L 3 = {xy : x = y, x, y Σ and y contains a 1}? H. Yen (NTUEE) Theory of Computation Fall / 67
67 Applications of the Myhill-Nerode Theorem (cont d) 1 For the language L 1 there are two equivalence classes of R L1. The first C 1 contains all strings of even length and the second C 2 all strings of odd length. 2 For L 2 we have the additional constraint that y ends with a 1. Class C 2 remains the same as that for L 1. Class C 1 is refined into classes C 1 which contains all strings of even length that end in a 1 and C 1 which contains all strings of even length which end in a 0. Thus L 1 and L 2 are both regular. 3 For L 3 we have to distinguish for example, between the even length strings in the sequence 01, 0001, ,..., as 00 distinguishes the first string from all the others after it in the sequence (concatenation of 00 to 01 gives a string not in L 3 but concatenation of 00 to all the others gives a string in L 3 ), 0000 distinguishes the second from all the others... H. Yen (NTUEE) Theory of Computation Fall / 67
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