Download presentation
Presentation is loading. Please wait.
Published byBrett Allen Modified over 9 years ago
1
Costas Busch - LSU1 Non-Deterministic Finite Automata
2
Costas Busch - LSU2 Alphabet = Nondeterministic Finite Automaton (NFA)
3
Costas Busch - LSU3 No transition Two choices No transition Alphabet =
4
Costas Busch - LSU4 First Choice
5
Costas Busch - LSU5 First Choice
6
Costas Busch - LSU6 “accept” First Choice All input is consumed
7
Costas Busch - LSU7 Second Choice
8
Costas Busch - LSU8 Second Choice “reject” Input cannot be consumed Automaton Halts
9
Costas Busch - LSU9 An NFA accepts a string: if there is a computation of the NFA that accepts the string i.e., all the input string is processed and the automaton is in an accepting state
10
Costas Busch - LSU10 is accepted by the NFA: “accept” “reject” because this computation accepts this computation is ignored
11
Costas Busch - LSU11 Rejection example
12
Costas Busch - LSU12 First Choice “reject”
13
Costas Busch - LSU13 Second Choice
14
Costas Busch - LSU14 Second Choice “reject”
15
Costas Busch - LSU15 Another Rejection example
16
Costas Busch - LSU16 First Choice
17
Costas Busch - LSU17 “reject” First Choice Input cannot be consumed Automaton halts
18
Costas Busch - LSU18 Second Choice
19
Costas Busch - LSU19 Second Choice “reject” Input cannot be consumed Automaton halts
20
Costas Busch - LSU20 An NFA rejects a string: if there is no computation of the NFA that accepts the string. All the input is consumed and the automaton is in a non accepting state The input cannot be consumed OR For each computation:
21
Costas Busch - LSU21 is rejected by the NFA: “reject” All possible computations lead to rejection
22
Costas Busch - LSU22 is rejected by the NFA: “reject” All possible computations lead to rejection
23
Costas Busch - LSU23 Language accepted:
24
Costas Busch - LSU24 Lambda Transitions
25
Costas Busch - LSU25
26
Costas Busch - LSU26
27
Costas Busch - LSU27 input tape head does not move Automaton changes state
28
Costas Busch - LSU28 “accept” String is accepted all input is consumed
29
Costas Busch - LSU29 Rejection Example
30
Costas Busch - LSU30
31
Costas Busch - LSU31 (read head doesn’t move)
32
Costas Busch - LSU32 “reject” String is rejected Input cannot be consumed Automaton halts
33
Costas Busch - LSU33 Language accepted:
34
Costas Busch - LSU34 Another NFA Example
35
Costas Busch - LSU35
36
Costas Busch - LSU36
37
Costas Busch - LSU37 “accept”
38
Costas Busch - LSU38 Another String
39
Costas Busch - LSU39
40
Costas Busch - LSU40
41
Costas Busch - LSU41
42
Costas Busch - LSU42
43
Costas Busch - LSU43
44
Costas Busch - LSU44 “accept”
45
Costas Busch - LSU45 Language accepted
46
Costas Busch - LSU46 Another NFA Example
47
Costas Busch - LSU47 Language accepted (redundant state)
48
Costas Busch - LSU48 Remarks: The symbol never appears on the input tape Simple automata:
49
Costas Busch - LSU49 NFADFA NFAs are interesting because we can express languages easier than DFAs
50
Costas Busch - LSU50 Formal Definition of NFAs Set of states, i.e. Input aplhabet, i.e. Transition function Initial state Accepting states
51
Costas Busch - LSU51 Transition Function resulting states with following one transition with symbol
52
Costas Busch - LSU52 States reachable from scanning
53
Costas Busch - LSU53 States reachable from scanning
54
Costas Busch - LSU54 States reachable from scanning nothing
55
Costas Busch - LSU55 States reachable from scanning
56
Costas Busch - LSU56 Extended Transition Function Same with but applied on strings
57
Costas Busch - LSU57 States reachable from scanning
58
Costas Busch - LSU58 States reachable from scanning
59
Costas Busch - LSU59 Special case: for any state
60
Costas Busch - LSU60 : there is a walk from to with label In general
61
Costas Busch - LSU61 The Language of an NFA The language accepted by is: Where for each and there is some (accepting state)
62
Costas Busch - LSU62
63
Costas Busch - LSU63
64
Costas Busch - LSU64
65
Costas Busch - LSU65
66
Costas Busch - LSU66
67
Costas Busch - LSU67
68
Costas Busch - LSU68 NFAs accept the Regular Languages
69
Costas Busch - LSU69 Equivalence of Machines Definition: Machine is equivalent to machine if
70
Costas Busch - LSU70 NFA DFA Example of equivalent machines
71
Costas Busch - LSU71 Theorem: Languages accepted by NFAs Regular Languages NFAs and DFAs have the same computation power, namely, they accept the same set of languages Languages accepted by DFAs
72
Costas Busch - LSU72 Languages accepted by NFAs Regular Languages accepted by NFAs Regular Languages we need to show Proof: AND
73
Costas Busch - LSU73 Languages accepted by NFAs Regular Languages Proof-Step 1 Every DFA is trivially a NFA Any language accepted by a DFA is also accepted by a NFA
74
Costas Busch - LSU74 Languages accepted by NFAs Regular Languages Proof-Step 2 Any NFA can be converted to an equivalent DFA Any language accepted by a NFA is also accepted by a DFA
75
Costas Busch - LSU75 Conversion of NFA to DFA NFA DFA
76
Costas Busch - LSU76 NFA DFA
77
Costas Busch - LSU77 NFA DFA empty set trap state
78
Costas Busch - LSU78 NFA DFA union
79
Costas Busch - LSU79 NFA DFA union
80
Costas Busch - LSU80 NFA DFA trap state
81
Costas Busch - LSU81 NFA DFA END OF CONSTRUCTION
82
Costas Busch - LSU82 General Conversion Procedure Input: an NFA Output: an equivalent DFA with
83
Costas Busch - LSU83 The NFA has states The DFA has states from the power set
84
Costas Busch - LSU84 1. Initial state of NFA: Initial state of DFA: Conversion Procedure Steps step
85
Costas Busch - LSU85 NFA DFA Example
86
Costas Busch - LSU86 2. For every DFA’s state compute in the NFA add transition to DFA Union step
87
Costas Busch - LSU87 NFA DFA Example
88
Costas Busch - LSU88 3. Repeat Step 2 for every state in DFA and symbols in alphabet until no more states can be added in the DFA step
89
Costas Busch - LSU89 NFA DFA Example
90
Costas Busch - LSU90 4. For any DFA state if some is accepting state in NFA Then, is accepting state in DFA step
91
Costas Busch - LSU91 NFA DFA Example
92
Costas Busch - LSU92 If we convert NFA to DFA then the two automata are equivalent: Lemma: Proof: AND We need to show:
93
Costas Busch - LSU93 First we show: We only need to prove:
94
Costas Busch - LSU94 symbols Consider NFA
95
Costas Busch - LSU95 denotes a possible sub-path like symbol
96
Costas Busch - LSU96 We will show that if then NFA DFA state label state label
97
Costas Busch - LSU97 More generally, we will show that if in : (arbitrary string) then NFA DFA
98
Costas Busch - LSU98 Proof by induction on Induction Basis: is true by construction of NFA DFA
99
Costas Busch - LSU99 Induction hypothesis: NFA DFA Suppose that the following hold
100
Costas Busch - LSU100 Induction Step: NFA DFA Then this is true by construction of
101
Costas Busch - LSU101 Therefore if NFA DFA then
102
Costas Busch - LSU102 We have shown: With a similar proof we can show: END OF PROOF Therefore:
Similar presentations
© 2024 SlidePlayer.com Inc.
All rights reserved.