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Chapter 2

Chapter 2. Finite Automata (part a). Hokkaido, Japan. Outline. (part a --- in this PPT) 2.0 Introduction 2.1 An Informal Picture of Finite Automata 2.2 Deterministic Finite automata 2.3 Nondeterministic Finite Automata (part b --- in another PPT) 2.4 An Application: Text Search

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Chapter 2

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  1. Chapter 2 Finite Automata (part a) Hokkaido, Japan

  2. Outline (part a --- in this PPT) • 2.0 Introduction • 2.1 An Informal Picture of Finite Automata • 2.2 Deterministic Finite automata • 2.3 Nondeterministic Finite Automata (part b --- in another PPT) • 2.4 An Application: Text Search • 2.5 Finite Automata with Epsilon-Transitions

  3. 2.0 Introduction • Two types of finite automata (FA): • Deterministic FA (DFA) • Nondeterministic FA (NFA) • Both types are of the same power, • The former is easier for hardware and software implementations. • The latter is more efficient for descriptions of applications of FA.

  4. 2.1 An Informal Picture of Finite Automata • A complete application example of finite automata for protocol design and verification • Read by yourself! • Involving a concept of “product of two automata”

  5. $10 $10 $5 $10 $5 $0 $5 $5 $5 Start $20 $10 $10 $20 $15 transition diagram 2.2 Deterministic Finite Automata • Recall the example of a vending machine selling 20-dolllar food in Chapter 0

  6. 2.2 Deterministic Finite Automata • 2.2.1 Definition of DFA • A DFA A consists of 5-tuples (1/2): • a finite (nonempty) set of states Q; • a finite (nonempty) set of input symbols S; • a (state) transition function dsuch that d(q, a) = p means “automaton A, in state q, takes input a • and enters state p;”

  7. 2.2 Deterministic Finite Automata • 2.2.1 Definition of DFA • A DFA A consists of 5-tuples (2/2): • a start state q0; • a set of (nonempty) final or accepting states F. • A may be written as a “5-tuple” A = (Q, S, d, q0, F).

  8. tape tape reader finite control 2.2 Deterministic Finite Automata • 2.2.1 Definition of DFA • graphic model for a DFA

  9. 2.2 Deterministic Finite Automata • 2.2.2 How a DFA processes strings • Given an input string x = a1a2…an, if d(q0, a1) = q1, d(q1, a2) = q2, …, d(qi1, ai) = qi, ..., d(qn1, an) = qn, and qnF, then x is “accepted”; otherwise, “rejected.” • Every transition is deterministic.

  10. 2.2 Deterministic Finite automata • Example 2.1 • Design an FA A to accept the language L = {x01y | x and y are any strings of 0’s and 1’s}. • Examples of strings in L: 01, 11010, 100011…

  11. 2.2 Deterministic Finite automata • Example 2.1 • Transitions: • d(q0, 1) = q0, d(q0, 0) = q2, d(q2, 1) = q1, • d(q2, 0) = q2, d(q1, 0) = q1, d(q1, 1) = q1 • 5-Tuple: A = ({q0, q1, q2}, {0, 1}, d, q0, {q1})

  12. 1 0 1 0 q1 q0 Start 0, 1 q2 2.2 Deterministic Finite automata • 2.2.3 Simpler Notations for DFA’s • Transition diagram ---

  13. 2.2 Deterministic Finite automata • 2.2.3 Simpler Notations for DFA’s • Transition table --- • : initial state; *: final state

  14. 2.2 Deterministic Finite automata • 2.2.4 Extending transition function to strings • Extended transition function If x = a1a2…an and d is such that d(p, a1) = q1, d(q1, a2) = q2, …, d(qi1, ai) = qi, ..., d(qn1, an) = q, then we define to be (p, x) = q.

  15. 2.2 Deterministic Finite automata • 2.2.4 Extending Transition Function to Strings • Also may be defined recursively as in the textbook. • Recursive definition for Basis: (q, e) = q. Induction: if w = xa (a is the last symbol of w), then (q, w) = d( (q, x), a).

  16. q1 0, 1 a x (q, x) … (q, w=xa) d 2.2 Deterministic Finite automata • 2.2.4 Extending Transition Function to Strings A graphic diagram for the following concept: Induction: if w = xa (a is the last symbol of w), then (q, w) = d( (q, x), a).

  17. 2.2 Deterministic Finite automata • 2.2.5 The Language of a DFA • The language of a DFA Ais defined as L(A) = {w | (q0, w) is in F}. • If L is L(A) for some DFA A, then we say L is a regular language.

  18. 1 0 1 0 q1 q0 Start 0, 1 q2 $20 $20 0, 1 2.3 Nondeterministic Finite Automata • 2.3.1 An informal view of NFA’s • Review of a previous example of DFA (of Example 2.1) • An NFA version of above DFA (more intuitive!) 1 0 q1 q0 Start 0, 1 q2

  19. $20 0, 1 1 0 q1 q0 Start 0, 1 q2 2.3 Nondeterministic Finite Automata • Some properties of NFA’s • Some transitions may “die,” like d(q2, 0). • Some transitions have multiple choices, like d(q0, 0) = q0 and q2.

  20. 0, 1 1 0 q2 q0 Start q1 $20 2.3 Nondeterministic Finite Automata • Example 2.6 • Design an NFA accepting the following language L = {w | w{0, 1}* and ends in 01}. • Nondeterminism creates many transition paths, but if there is one path leading to a final state, then the input is accepted.

  21. 0, 1 1 0 q2 q0 Start q1 $20 q0 q0 q0 q0 q0 q0 q0 q0 q1 q1 q1 q1 q2 q2 q2 2.3 Nondeterministic Finite Automata • Example 2.6 (cont’d) • When input x = 00101, the NFA processes x in the following way: 0 0 1 0 1 q0 Stuck! Stuck! Accept! Fig. 2.10

  22. 2.3 Nondeterministic Finite Automata • 2.3.2 Definition of NFA • An NFA Ais a 5-tuple A = (Q, S, d, q0, F) where • Q = a finite (nonempty) set of states; • S = a finite (nonempty) set of input symbols; • q0 = a start state; • F = a set of (nonempty) final or accepting states;

  23. 2.3 Nondeterministic Finite Automata • 2.3.2 Definition of NFA • An NFA Ais a 5-tuple A = (Q, S, d, q0, F) where • d = a (state) transition functionsuch that d(q, a) = {p1, p2, …, pm} which means • “automaton A, in state q, takes input a and enters • one of the states p1, p2, …, pm.”

  24. 0, 1 1 0 q2 q0 Start q1 $20 2.3 Nondeterministic Finite Automata • Example 2.7 • Transition table of NFA of the last example ---

  25. 2.3 Nondeterministic Finite Automata • 2.3.3 Extended Transition Function • Recursive definition of (with string as input) • Basis: (q, e) = {q}. • Induction: if w = xa (a is the last symbol of w), (q, x) = {p1, p2, …, pk}, and • d(pi, a) = {r1, r2, …, rm} • then • (q, w) = {r1, r2, …, rm}.

  26. (q, w=xa) = {r1 r2 . . . rm} 2.3 Nondeterministic Finite Automata (supplemental) • 2.3.3 Extended Transition Function • A graphic diagram for the following concept: Induction: if w = xa (a is the last symbol of w), (q, x) = {p1, p2, …, pk}, and d(pi, a) = {r1, r2, …, rm} then (q, w) = {r1, r2, …, rm}. (q, x)= {p1 p2 . . . pk} a a x … a d

  27. 0, 1 1 0 q2 q0 Start q1 $20 2.3 Nondeterministic Finite Automata • Example 2.8 --- • For the input w = 00101, we have • (q0, e) = {q0}. 2. (q0, e 0) = d(q0, 0) = {q0, q1}. 3. (q0, 00) = d(q0, 0)∪d(q1, 0)= {q0, q1}∪f ={q0, q1} …

  28. 2.3 Nondeterministic Finite Automata • 2.3.4 The Language of an NFA • Definition --- If A = (Q, S, d, q0, F) is an NFA, then the language accepted by A is L(A) = {w | (q0, w)∩Ff}. • That is, as long as a final state is reached, which is among possibly many, the input is accepted.

  29. 2.3 Nondeterministic Finite Automata • 2.3.5 Equivalence of DFA and NFA • NFA’s are more intuitive to construct for many languages. • DFA’s are easier for use in software and hardware implementations.

  30. 2.3 Nondeterministic Finite Automata • 2.3.5 Equivalence of DFA and NFA • Every NFA N has an equivalent DFA D, i.e., L(D) = L(N). (Definition: “Two models are said to be equivalent if they have identical languages.”) • The proof needs “subset construction.”

  31. 2.3 Nondeterministic Finite Automata • 2.3.5 Equivalence of DFA and NFA • Subset construction: each state of DFA is a subset of NFA • Given an NFA N = (QN, S, dN, q0, FN), we construct a DFA D = (QD, S, dD, q0', FD) such that: • The two alphabets are identical; • QDis the power set of QN (i.e., set of all subsets of QN); • Inaccessible states in QD should be deleted; (continued in the next page)

  32. 2.3 Nondeterministic Finite Automata • 2.3.5 Equivalence of DFA and NFA • (cont’d) • q0' = {q0}; • Each subset S of QN is in final state set FD if S includes at least one accepting state in FN (i.e., S∩FNf). • For each subset S of QN and for each input symbol a, define dD(S, a) = dN(p, a)

  33. 0, 1 1 0 q2 q0 Start q1 $20 2.3 Nondeterministic Finite Automata • Example 2.10 --- • For the NFA of Example 2.6 below: • The power set of QN = {q0, q1, q2} is QD={f, {q0}, {q1}, {q2}, {q0, q1}, {q0, q2}, {q1, q2}, {q0, q1, q2}}.

  34. 0, 1 1 0 q2 q0 Start q1 $20 2.3 Nondeterministic Finite Automata • Example 2.10 (cont’d) • The transition table is

  35. 2.3 Nondeterministic Finite Automata • Example 2.10 (cont’d) • Change names of the states, we get

  36. 2.3 Nondeterministic Finite Automata • Example 2.10 (cont’d) • Checking accessible states from the start state as red ones; the others are inaccessible. Figure 2.13

  37. 0, 1 1 0 q2 q0 Start q1 $20 2.3 Nondeterministic Finite Automata • Example 2.10 (cont’d) (supplemental) • solved by “Lazy evaluation” using table: starting from the start state, only accessible states are evaluated, in order to avoid generation of inaccessible states. • The transition table is

  38. 1 0 1 0 Start {q0, q2} {q0} {q0, q1} 2.3 Nondeterministic Finite Automata • Example 2.10 (cont’d) --- • solved by “Lazy evaluation” using diagram: starting from the start state, only accessible states are evaluated, in order to avoid generation of inaccessible states. 0 Figure 2.14 1

  39. 2.3 Nondeterministic Finite Automata • Theorem 2.11 If DFA D = (QD, S, dD, {q0}, FD) is constructed from NFA N = (QN, S, dN, q0, FN) by subset construction, then L(D) = L(N). • Proof. See the textbook.

  40. 2.3 Nondeterministic Finite Automata • Theorem 2.12 (equivalence of DFA and NFA): A language L is accepted by some DFA if and only ifL is accepted by some NFA. • Proof. See the textbook.

  41. 2.3 Nondeterministic Finite Automata • 2.3.6 A Bad Case of Subset Construction • The DFA constructed from an NFA usually has the same number of accessible states of the NFA. • But the worse case is m = 2n where • m is the number of states in the DFA. • n is the number of states in the NFA.

  42. e t h n t th start the then 2.3 Nondeterministic Finite Automata • 2.3.7a Regarding NFA’s as DFA’s • In each state of a DFA, for each input there mustbe a next state. • In an NFA, for a certain input there mightbeno next state. So the following is an NFA.

  43. e t h n t th start the then 2.3 Nondeterministic Finite Automata • 2.3.7a Regarding NFA’s as DFA’s • But it is deterministic in nature --- either getting into a next state or becoming “dead” (getting into a “dead” state). • Such a kind of NFA has at most one transition out of any state on any symbol, and may be regarded as a DFA.

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