1 CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 3 Mälardalen University 2010

2 Content Finite Automata, FA Deterministic Finite Automata, DFA Nondeterministic Automata NFA NFA  DFA Equivalence

3 Finite Automata FA (Finite State Machines)

4 There is no formal general definition for "automaton". Instead, there are various kinds of automata, each with it's own formal definition. has some form of input has some form of output has internal states, may or may not have some form of storage is hard-wired rather than programmable Generally, an automaton

5 Finite Automaton Input String Output String Finite Automaton

6 Finite Accepter Input “Accept” or “Reject” String Finite Automaton Output

7 Nodes = States Edges = Transitions An edge with several symbols is a short-hand for several edges: FA as Directed Graph

8 Deterministic Finite Automata DFA

9 Deterministic there is no element of choice Finite only a finite number of states and arcs Acceptors produce only a yes/no answer DFA

10 Transition Graph initial state final state “accept” state transition abba -Finite Acceptor Alphabet =

11 Formally For a DFA Language accepted by : alphabet transition function initial state final states

12 Observation Language accepted by Language rejected by

13 Regular Languages All regular languages form a language family A language is regular if there is a DFA such that

14 Formal definitions Deterministic Finite Accepter (DFA) : set of states : input alphabet : transition function : initial state : set of final states

15 Input Aplhabet

16 Set of States

17 Initial State

18 Set of Final States

19 Transition Function

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23 Transition Function

24 Extended Transition Function

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28 Observation: There is a walk from to with label

29 Recursive Definition

30      ,,,,,,* ),,(*,* q bq baq baq baq abq          

31 Languages Accepted by DFAs Take DFA Definition: The language contains all input strings accepted by = { strings that drive to a final state}

32 Example accept Alphabet =

33 Another Example accept Alphabet =

34 Formally For a DFA Language accepted by : alphabet transition function initial state final states

35 Observation Language accepted by Language rejected by

36 More Examples accept trap state Alphabet =

37 = { all strings with prefix } accept Alphabet =

38 = { all strings without substring } Alphabet =

39 Regular Languages All regular languages form a language family A language is regular if there is a DFA such that

40 Example is regular The language Alphabet =

41 Nondeterministic Automata NFA

42 Nondeterministic there is an element of choice: in a given state NFA can act on a given string in different ways. Several start/final states are allowed. -transitions are allowed. Finite only a finite number of states and arcs Acceptors produce only a yes/no answer NFA

43 Two choices Alphabet = Nondeterministic Finite Accepter (NFA)

44 First Choice

45 First Choice

46 First Choice

47 “accept” First Choice

48 Second Choice

49 Second Choice

50 Second Choice No transition: the automaton hangs

51 Second Choice “reject”

52 Observation An NFA accepts a string if there is a computation of the NFA that accepts the string

53 Example is accepted by the NFA:

54 Lambda Transitions

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57 (read head doesn’t move)

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59 “accept” String is accepted

60 Language accepted:

61 Another NFA Example Alphabet =

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65 “accept”

66 Another String Alphabet =

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73 “accept”

74 Language accepted Alphabet =

75 Another NFA Example Alphabet =

76 Language accepted

77 Formal Definition of NFA Set of states, i.e. Input alphabet, i.e. Transition function Initial state Final states

78 Transition Function

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82 Extended Transition Function (Utvidgad övergångsfunktion)

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85 Formally if and only if there is a walk from to with label

86 The Language of an NFA Alphabet =

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91 Formally The language accepted by NFA (final state) where and there is some (at least one) is:

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93 NFA  DFA Equivalence

94 Equivalence of NFAs and DFAs Accept the same languages? YES! NFAs  DFAs ? Same power?

95 We will prove: Languages accepted by NFAs Languages accepted by DFAs NFAs and DFAs have the same computation power!

96 Languages accepted by NFAs Languages accepted by DFAs Step 1 Proof Every DFA is also an NFA A language accepted by a DFA is also accepted by an NFA

97 Languages accepted by NFAs Languages accepted by DFAs Step 2 Proof Any NFA can be converted to an equivalent DFA A language accepted by an NFA is also accepted by a DFA

98 Procedure NFA to DFA 1. Initial state of NFA: Initial state of DFA:

99 Example NFA DFA Step 1

100 Procedure NFA to DFA 2. For every DFA’s state Compute in the NFA Add transition

101 Example NFA DFA Step 2

102 Procedure NFA to DFA Repeat Step 2 for all letters in alphabet, until no more transitions can be added.

103 Example NFA DFA Step 3

104 Procedure NFA to DFA 3. For any DFA state If some is a final state in the NFA Then is a final state in the DFA

105 Example NFA DFA Step 4

106 Theorem Take NFA Apply procedure to obtain DFA Then and are equivalent :

107 Languages accepted by NFAs Languages accepted by DFAs We have proven (proof by construction): Regular Languages END OF PROOF

108 Nondeterministic vs. Deterministic Automata

109 Formal Definition of NFA Set of states, i.e. Input alphabet, i.e. Transition function Initial state Final (accepting) states NFA is a mathematical model defined as a quintuple:

110 Deterministic Finite Automata A deterministic finite automaton (DFA) is a special case of a nondeterministic finite automaton in which 1. no state has an -transition, i.e. a transition on input, and 2. for each state q and input symbol a, there is at most one edge labeled a leaving q.

111 STATE INPUT SYMBOL ab {0, 1} - {0} {2} {3} Transition table for the finite automaton above A nondeterministic finite automaton b 0 start 1 a 2 bb 3 a Example

112 NFA accepting aa* + bb* 0 start 1 a 2 a 3 b 4 b Example

113 NFA accepting (a+b)*abb 0 start 1 a 2 bb b aa a b 3 a Example

114 NFA recognizing three different patterns. (a) NFA for a, abb, and a*b +. (b) Combined NFA. Example 4 1 start a 2 3 a 65 b b 7 b 8 b a 4 1 a 2 3 a 65 bb 7 b 8 b a 0

115 Ways to think of nondeterminism always make the correct guess “backtracking” (systematically try all possibilities) For a particular string, imagine a tree of possible state transitions: q0q0 q3q3 q0q0 q4q4 q2q2 q1q1 a a a a b a

116 Advantages of nondeterminism an NFA can be smaller, easier to construct and easier to understand than a DFA that accepts the same language useful for proving some theorems good introduction to nondeterminism in more powerful computational models, where nondeterminism plays an important role

117 Space and time taken to recognize regular expressions: - NFA more compact but take time to backtrack all choices - DFA take place, but save time AUTOMATONSPACETIME NFA DFA O(|r|) O(2 |r| ) O(|r|  |x|) O(|x|) Determinism vs. nondeterminism (Where r is regular expression, and x is input string)

118 Equivalent automata Two finite automata M 1 and M 2 are equivalent if L(M 1 ) = L(M 2 ) that is, if they both accept the same language.

119 Equivalence of NFAs and DFAs To show that NFAs and DFAs accept the same class of languages, we show two things: –Any language accepted by a DFA can also be accepted by some NFA (As DFA is a special case of NFA) –Any language accepted by a NFA can also be accepted by some (corresponding, specially constructed) DFA

120 Proof strategy To show that any language accepted by a NFA is also accepted by some DFA, we describe an algorithm that takes any NFA and converts it into a DFA that accepts the same language. The algorithm is called the “subset construction algorithm”. We can use mathematical induction (on the length of a string accepted by the automaton) to prove the DFA that is constructed accepts the same language as the NFA.

121 Converting NFA to DFA Subset Construction

122 Subset construction Given a NFA constructs a DFA that accepts the same language The equivalent DFA simulates the NFA by keeping track of the possible states it could be in. Each state of the DFA is a subset of the set of states of the NFA -hence, the name of the algorithm. If the NFA has n states, the DFA can have as many as 2 n states, although it usually has many less.

123 Steps of subset construction The initial state of the DFA is the set of all states the NFA can be in without reading any input. For any state {q i,q j,…,q k } of the DFA and any input a, the next state of the DFA is the set of all states of the NFA that can result as next states if the NFA is in any of the states q i,q j,…,q k when it reads a. This includes states that can be reached by reading a, followed by any number of -moves. Use this rule to keep adding new states and transitions until it is no longer possible to do so. The accepting states of the DFA are those states that contain an accepting state of the NFA.

124 Example Here is a NFA that we want to convert to an equivalent DFA

125 {0,1} The start state of the DFA is the set of states the NFA can be in before reading any input. This includes the start state of the NFA and any states that can be reached by a -transition. NFA DFA

126 {0,1} a b {2} For start state {0,1}, make transitions for each possible input, here a and b. Reading b from start {0,1}, we reach state {2}. Means from either {0}, or {1} we reach {2}. NFA DFA

127 For state {2}, we create a transition for each possible input, a and b. From {2}, with b we are either back to {2} (loop) or we reach {1}- see the little framed original NFA. So from {2}, with b we end in state {1, 2}. Reading a leads us from {2} to {0} in the original NFA, which means state {0, 1} in the new DFA. {0,1} {1,2} {2} NFA DFA

128 For state {1, 2}, we make again transition for each possible input, a and b. From {2} a leads us to {0}. From {1} with a we are back to {1}. So, we reach {0, 1} with a from {1,2}. With b we are back to {1,2}. At this point, a transition is defined for every state-input pair. {0,1} {1,2} {2} DFA NFA

129 The last step is to mark the final states of the DFA. As {1} was the accepting state in NFA, all states containing {1} in DFA will be accepting states: ({0, 1} and {1, 2}). {0,1} {1,2} {2} DFA NFA

130 Subset Construction Algorithm

131 Subset Construction States of nondeterministic M´ will correspond to sets of states of deterministic M Where q 0 is start state of M, use {q 0 } as start state of M´. Accepting states of M´ will be those state-sets containing at least one accepting state of M.

132 Subset Construction (cont.) For each state-set S and for each s in alphabet of M, we draw an arc labeled s from state S to that state-set consisting of all and only the s- successors of members of S. Eliminate any state-set, as well as all arcs incident upon it, such that there is no path leading to it from {q 0 }.

133 The power set of a finite set, Q, consists of 2 |Q| elements The DFA corresponding to a given NFA with Q states have a finite number of states, 2 |Q|. If |Q| = 0 then Q is the empty set, | P(Q)| = 1 = 2 0. If |Q| = N and N  1, we construct subset of a given set so that for each element of the initial set there are two alternatives, either is the element member of a subset or not. So we have 2 · 2 · 2 · 2 · 2 · 2 · 2…. ·2 = 2 N N times

134 From an NFA to a DFA Subset Construction Operation Description - closure(s) - closure(T) Move(T,a) Set of NFA states reachable from an NFA state s on -transitions along Set of NFA states reachable from some NFA state s in T on - transitions along Set of NFA states reachable from some NFA state set with a transition on input symbol a

135 From an NFA to a DFA Subset Construction Initially, -closure (s 0 ) is the only states in D and it is unmarked while there is an unmarked state T in D do mark T; for each input symbol a do U:= e-closure(move(T,a)); if U is not in D then add U as an unmarked state to D Dtran[T,a]:=U; end(for) end(while)