Announcements. Problem Set Four due Thursday at 7:00PM (right before the midterm).

Transcription

1 Finite Automata

2 Announcements Problem Set Four due Thursday at 7:PM (right before the midterm). Stop by OH with questions! with questions! Review session tonight, 7PM until whenever people are done asking questions, in Covers practice midterm questions. Alternate time tomorrow as well; we'll announce a location once we have a room reserved.

3 Problem Set 2 Scores Mean: 8 Median: 2 Stdev: 6

4 Midterm Details Midterm is Thursday, 7:-:PM in Hewlett 2 (right next door). Alternate: Wednesday, 7: :PM, room TBA Alternate: Thursday, 9:AM Noon, room TBA Open-book, open-note, open-computer, closed-network. Feel free to download lecture slides. Do not Google the answers, your friends, read xkcd, etc. If you can disable your network connection, please do so. Material is up through and including Wednesday's lecture. Don't panic; any question on this week's material will be extremely easy and should not require much thought.

5 What problems can we solve with a computer?

6 What problems can we solve with a computer? What kind of computer?

7 Computers are Messy

8 Computers are Messy

9 Computers are Messy

10 Computers are Messy

11 We need a simpler way of discussing computing machines.

12 An automaton (plural: automata) is a mathematical model of a computing device.

13 Automata make it possible to reason about computability by abstracting away the implementation complexity of real computing systems.

14 Computers are Messy

15 Automata are Clean q q

16 Computers are Messy

17 Automata are Clean q q

18 Computers are Messy

19 Automata are Clean q q

20 Computers are Messy

21 Automata are Clean q q

22 Why Build Models? The models of computation we will explore in this class correspond to different conceptions of what a computer could do. Finite automata (this week) are an abstraction of computers with finite resource constraints. Provide upper bounds for the computing machines that we can actually build. Turing machines (later) are an abstraction of computers with unbounded resources. Provide upper bounds for what we could ever hope to accomplish.

23 What problems can we solve with a computer?

24 What problems can we solve with a computer? What is a problem?

25 Problems with Problems Before we can talk about what problems we can solve, we need a formal definition of a problem. We want a definition that corresponds to the problems we want to solve, captures a large class of problems, and is mathematically simple to reason about. No one definition has all three properties.

26 Decision Problems In this class, we will consider decision problems, problems with yes/no answers. Examples: Does have 3 as a divisor? Is P the shortest path from u to v? If the velociraptors can run at 3m/s, will I survive for more than ten minutes? Do we have to get down on Friday?

27 Decision Problems are Powerful Many questions that are not decision problems can be phrased as decision problems. Example: What is ? can be phrased as a decision problem Is = 77? There are much better ways to do this; we'll see them later on. In this sense, decision problems are at least as hard as any other problem. If we can't solve decision problems, we certainly can't solve these other problems.

28 Strings An alphabet is a finite set Σ of characters. A string is a finite sequence of characters drawn from some alphabet. Example: Σ = {, }, some valid strings include The empty string contains no characters and is denoted ε.

29 Languages A formal language is a set of strings. We say that L is a language over Σ if it is a set of strings formed from the characters of Σ. Example: The language of palindromes over Σ = {,, 2} is the set {ε,,, 2,,, 22,,, 2,,, } The set of all strings composed from letters in Σ is denoted Σ*.

30 Decision Problems and Languages Languages give a compact and flexible way to encode decision problems. Consider these examples: { p p is an encoding of a prime number } { w w is an encoding of (x, y), where x < y } { x x is an encoding of a graph G and a path P, where P is the longest path in G } Any decision problem can be represented by a language of strings encoding inputs to which the answer is yes. All the automata we will discuss in this class will be machines for answering the question is string x in language L?

31 To Summarize An automaton is an idealized mathematical computing machine. A language is a set of strings. A decision problem is a yes/no question (though it can be quite complex). The automata we will study will accept as input a string and (attempt to) output whether that string is contained in a particular language.

32 Finite Automata

33 A finite automaton is a mathematical machine for determining whether a string is contained within some language.

34 Each finite automaton consists of a set of states connected by transitions.

35 A Simple Finite Automaton q q

36 A Simple Finite Automaton q q Each circle represents a state of the automaton.

37 A Simple Finite Automaton q q

38 A Simple Finite Automaton q q One special state is designated as the state.

39 A Simple Finite Automaton q q

40 A Simple Finite Automaton q q

41 A Simple Finite Automaton q q The automaton is run on an input string and answers yes or no.

42 A Simple Finite Automaton q q

43 A Simple Finite Automaton q q

44 A Simple Finite Automaton q q The automaton can be in one state at a time. It begins in the state.

45 A Simple Finite Automaton q q

46 A Simple Finite Automaton q q

47 A Simple Finite Automaton q q The automaton now begins processing characters in the order in which they appear.

48 A Simple Finite Automaton q q

49 A Simple Finite Automaton q q

50 A Simple Finite Automaton q q Each arrow in this diagram represents a transition. The automaton always follows the transition corresponding to the current symbol being read.

51 A Simple Finite Automaton q q

52 A Simple Finite Automaton q q

53 A Simple Finite Automaton q q

54 A Simple Finite Automaton q q

55 A Simple Finite Automaton q q After transitioning, the automaton considers the next symbol in the input.

56 A Simple Finite Automaton q q

57 A Simple Finite Automaton q q

58 A Simple Finite Automaton q q

59 A Simple Finite Automaton q q

60 A Simple Finite Automaton q q

61 A Simple Finite Automaton q q

62 A Simple Finite Automaton q q

63 A Simple Finite Automaton q q

64 A Simple Finite Automaton q q

65 A Simple Finite Automaton q q

66 A Simple Finite Automaton q q

67 A Simple Finite Automaton q q

68 A Simple Finite Automaton q q

69 A Simple Finite Automaton q q

70 A Simple Finite Automaton q q

71 A Simple Finite Automaton q q

72 A Simple Finite Automaton q q

73 A Simple Finite Automaton q q

74 A Simple Finite Automaton q q

75 A Simple Finite Automaton q q

76 A Simple Finite Automaton q q

77 A Simple Finite Automaton q q Now that the automaton has looked at all this input, it can decide whether to say yes or no.

78 A Simple Finite Automaton q q The double circle Now that the automaton has looked at all this input, it can decide whether to say yes or no. indicates that this state is an accepting state, so the automaton outputs yes.

79 A Simple Finite Automaton q q The double circle Now that the automaton has looked at all this input, it can decide whether to say yes or no. indicates that this state is an accepting state, so the automaton outputs yes.

80 A Simple Finite Automaton q q The double circle Now that the automaton has looked at all this input, it can decide whether to say yes or no. indicates that this state is an accepting state, so the automaton outputs yes.

81 A Simple Finite Automaton q q

82 A Simple Finite Automaton q q

83 A Simple Finite Automaton q q

84 A Simple Finite Automaton q q

85 A Simple Finite Automaton q q

86 A Simple Finite Automaton q q

87 A Simple Finite Automaton q q

88 A Simple Finite Automaton q q

89 A Simple Finite Automaton q q

90 A Simple Finite Automaton q q

91 A Simple Finite Automaton q q

92 A Simple Finite Automaton q q

93 A Simple Finite Automaton q q

94 A Simple Finite Automaton q q

95 A Simple Finite Automaton q q

96 A Simple Finite Automaton q q

97 A Simple Finite Automaton q q

98 A Simple Finite Automaton q q

99 A Simple Finite Automaton q q

100 A Simple Finite Automaton q q

101 A Simple Finite Automaton q q

102 A Simple Finite Automaton q q

103 A Simple Finite Automaton q q

104 A Simple Finite Automaton q q

105 A Simple Finite Automaton q q

106 A Simple Finite Automaton q q

107 A Simple Finite Automaton q q

108 A Simple Finite Automaton q q

109 A Simple Finite Automaton This state is not q q an accepting state, so the automaton says no.

110 A Simple Finite Automaton This state is not q q an accepting state, so the automaton says no.

111 A Simple Finite Automaton This state is not q q an accepting state, so the automaton says no.

112 A Simple Finite Automaton q q

113 A Simple Finite Automaton q q Try it yourself! Does the automaton accept (say yes) or reject (say no)?

114 The Story So Far A finite automaton is a collection of states joined by transitions. Some state is designated as the state. Some states are designated as accepting states. The automaton processing a string by beginning in the state and following the indicated transitions. If the automaton ends in an accepting state, it accepts the input. Otherwise, the automaton rejects the input.

115 Accepting States, Revisited q q q 5

116 Accepting States, Revisited q q q 5

117 Accepting States, Revisited q q q 5

118 Accepting States, Revisited q q q 5

119 Accepting States, Revisited q q q 5

120 Accepting States, Revisited q q q 5

121 Accepting States, Revisited q q q 5

122 Accepting States, Revisited q q q 5

123 Accepting States, Revisited q q q 5

124 Accepting States, Revisited q q q 5

125 Accepting States, Revisited q q q 5

126 Accepting States, Revisited q q q 5

127 Accepting States, Revisited q q q 5

128 Accepting States, Revisited q q q 5

129 Accepting States, Revisited q q q 5

130 A finite automaton does not accept as soon as the input enters an accepting state. A finite automaton accepts if the it ends in an accepting state.

131 What Does This Accept? q q q 5

132 What Does This Accept? q q q 5

133 What Does This Accept? q q q 5

134 What Does This Accept? q q q 5

135 What Does This Accept? q q q 5

136 What Does This Accept? q q q 5

137 What Does This Accept? q q q 5

138 What Does This Accept? q No matter where we q in the automaton, after seeing two 's, we end up in accepting state. q 5

139 What Does This Accept? q q q 5

140 What Does This Accept? q q q 5

141 What Does This Accept? q q q 5

142 What Does This Accept? q q q 5

143 What Does This Accept? q q q 5

144 What Does This Accept? q q q 5

145 What Does This Accept? q No matter where we q in the automaton, after seeing two 's, we end up in accepting state q 5. q 5

146 What Does This Accept? q q q 5

147 What Does This Accept? q q This automaton accepts any string ending in or. q 5

148 The language of an automaton is the set of strings that it accepts. We denote this L(A)

149 A language L is called a regular language if there is some finite automaton A for which L(A) = L

150 A Small Problem q q

151 A Small Problem q q

152 A Small Problem q q

153 A Small Problem q q

154 A Small Problem q q

155 A Small Problem q q

156 A Small Problem q q

157 A Small Problem q q

158 A Small Problem q q Problem?

159 Another Small Problem, q, q,

160 Another Small Problem, q, q,

161 Another Small Problem, q, q,

162 Another Small Problem, q, q,

163 Another Small Problem, q, q,

164 Another Small Problem, q, q,

165 Another Small Problem, q, q,

166 Another Small Problem, q, q, Problem?

167 The Need for Formalism In order to reason about the limits of what finite automata can and cannot do, we need to formally specify their behavior in all cases. All of the following need to be defined or disallowed: What happens if there is no transition out of a state on some input? What happens if there are multiple transitions out of a state on some input?

168 DFAs A DFA is a Deterministic Finite Automaton DFAs are the simplest type of automaton that we will see in this course.

169 DFAs, Informally A DFA is defined relative to some alphabet Σ. For each state in the DFA, there must be exactly one transition defined for each symbol in the alphabet. This is the deterministic part of DFA. There is a unique state. There may be multiple accepting states.

170 Is this a DFA?

171 Is this a DFA? q q

172 Is this a DFA?

173 Is this a DFA? q q q 5

174 Is this a DFA?

175 Is this a DFA? q q

176 Is this a DFA? q q

177 Is this a DFA?

178 Is this a DFA? q, q,, q,

179 Is this a DFA?

180 Is this a DFA?, q, q,

181 Is this a DFA?, q, q,

182 Is this a DFA?

183 Is this a DFA?

184 Is this a DFA? Deadly Futuristic Assassin

185 Recognizing Languages with DFAs L = { w w contains as a substring }

186 Recognizing Languages with DFAs L = { w w contains as a substring } q

187 Recognizing Languages with DFAs L = { w w contains as a substring } q q

188 Recognizing Languages with DFAs L = { w w contains as a substring } q q

189 Recognizing Languages with DFAs L = { w w contains as a substring } q q

190 Recognizing Languages with DFAs L = { w w contains as a substring } q q

191 Recognizing Languages with DFAs L = { w w contains as a substring } q q

192 Recognizing Languages with DFAs L = { w w contains as a substring } q q,

193 Recognizing Languages with DFAs L = { w every even character of w is a }

194 Recognizing Languages with DFAs L = { w every even character of w is a } YES NO

195 Recognizing Languages with DFAs L = { w every even character of w is a }

196 Recognizing Languages with DFAs L = { w every even character of w is a } q

197 Recognizing Languages with DFAs L = { w every even character of w is a } q q

198 Recognizing Languages with DFAs L = { w every even character of w is a } q q

199 Recognizing Languages with DFAs L = { w every even character of w is a } q q,

200 Recognizing Languages with DFAs L = { w every even character of w is a } q q,,

201 Recognizing Languages with DFAs L = { w every even character of w is a } q q, This state is sometimes called a dead state.,

202 Recognizing Languages with DFAs L = { w every even character of w is a } q q,,

203 More Elaborate DFAs L = { w w is a legal address }

204 More Elaborate DFAs L = { w w is a legal @,. q 8 q a q a a q 4. q 5 a q 6 a a a

205 More Elaborate DFAs L = { w w is a C-style comment }

206 More Elaborate DFAs L = { w w is a C-style comment } a, *, / a, * q 5 a, *, / /, a q / q * * a / q 4 a, / *

207 More Elaborate DFAs L = { w w is a C-style comment } a, *, / a, * q 5 a, *, / /, a q / q * * a / q 4 a, / *

208 More Elaborate DFAs L = { w w is a C-style comment } Σ a, * q 5 Σ /, a q / q * * a / q 4 a, / *

209 More Elaborate DFAs L = { w w is a C-style comment } Σ a, * q 5 Σ /, a q / q * * a / q 4 a, / *