Theory of Computation Nondeterministic Finite Automaton NFA An
![Theory of Computation Theory of Computation](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-1.jpg)
![Nondeterministic Finite Automaton (NFA) An NFA is a collection of three things 1) Finite Nondeterministic Finite Automaton (NFA) An NFA is a collection of three things 1) Finite](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-2.jpg)
![Observations The string is supposed to be accepted, if there exists at least one Observations The string is supposed to be accepted, if there exists at least one](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-3.jpg)
![Example a 1 - a 2+ 3 a 4 b 5+ The above NFA Example a 1 - a 2+ 3 a 4 b 5+ The above NFA](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-4.jpg)
![Example a, b 1 - a, b a 2 a 3+ The above NFA Example a, b 1 - a, b a 2 a 3+ The above NFA](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-5.jpg)
![Non-determinism This refers to situations in which the next state of a computation is Non-determinism This refers to situations in which the next state of a computation is](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-6.jpg)
![Converting an FA to an equivalent NFA n If a language is accepted by Converting an FA to an equivalent NFA n If a language is accepted by](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-7.jpg)
![Example n Consider the following FA corresponding to (a+b)*b a b b n + Example n Consider the following FA corresponding to (a+b)*b a b b n +](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-8.jpg)
![Example n Consider the following FA b a n a, b b a 1 Example n Consider the following FA b a n a, b b a 1](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-9.jpg)
![Note n n Every FA can be considered to be an NFA, but the Note n n Every FA can be considered to be an NFA, but the](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-10.jpg)
![NFA to FA An NFA can be considered to be a TG, so a NFA to FA An NFA can be considered to be a TG, so a](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-11.jpg)
![2 Example b a 1 - 4+ b a 3 a 1 - 2 2 Example b a 1 - 4+ b a 3 a 1 - 2](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-12.jpg)
![Nondeterministic Finite Automaton (NFA) Cont… Alphabet = 13 Nondeterministic Finite Automaton (NFA) Cont… Alphabet = 13](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-13.jpg)
![Alphabet = Two choices 14 Alphabet = Two choices 14](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-14.jpg)
![Alphabet = Two choices No transition 15 Alphabet = Two choices No transition 15](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-15.jpg)
![First Choice 16 First Choice 16](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-16.jpg)
![First Choice 17 First Choice 17](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-17.jpg)
![First Choice 18 First Choice 18](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-18.jpg)
![First Choice All input is consumed “accept” 19 First Choice All input is consumed “accept” 19](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-19.jpg)
![Second Choice 20 Second Choice 20](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-20.jpg)
![Second Choice 21 Second Choice 21](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-21.jpg)
![Second Choice No transition: the automaton hangs 22 Second Choice No transition: the automaton hangs 22](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-22.jpg)
![Second Choice Input cannot be consumed “reject” 23 Second Choice Input cannot be consumed “reject” 23](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-23.jpg)
![An NFA accepts a string: if there is a computation of the NFA that An NFA accepts a string: if there is a computation of the NFA that](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-24.jpg)
![Example is accepted by the NFA: “accept” because this computation accepts “reject” 25 Example is accepted by the NFA: “accept” because this computation accepts “reject” 25](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-25.jpg)
![Rejection example 26 Rejection example 26](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-26.jpg)
![First Choice 27 First Choice 27](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-27.jpg)
![First Choice “reject” 28 First Choice “reject” 28](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-28.jpg)
![Second Choice 29 Second Choice 29](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-29.jpg)
![Second Choice 30 Second Choice 30](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-30.jpg)
![Second Choice “reject” 31 Second Choice “reject” 31](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-31.jpg)
![Example is rejected by the NFA: “reject” All possible computations lead to rejection 32 Example is rejected by the NFA: “reject” All possible computations lead to rejection 32](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-32.jpg)
![Rejection example 33 Rejection example 33](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-33.jpg)
![First Choice 34 First Choice 34](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-34.jpg)
![First Choice No transition: the automaton hangs 35 First Choice No transition: the automaton hangs 35](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-35.jpg)
![First Choice Input cannot be consumed “reject” 36 First Choice Input cannot be consumed “reject” 36](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-36.jpg)
![Second Choice 37 Second Choice 37](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-37.jpg)
![Second Choice 38 Second Choice 38](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-38.jpg)
![Second Choice No transition: the automaton hangs 39 Second Choice No transition: the automaton hangs 39](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-39.jpg)
![Second Choice Input cannot be consumed “reject” 40 Second Choice Input cannot be consumed “reject” 40](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-40.jpg)
![is rejected by the NFA: “reject” All possible computations lead to rejection 41 is rejected by the NFA: “reject” All possible computations lead to rejection 41](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-41.jpg)
![Language accepted: 42 Language accepted: 42](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-42.jpg)
![Lambda Transitions 43 Lambda Transitions 43](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-43.jpg)
![44 44](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-44.jpg)
![45 45](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-45.jpg)
![(read head does not move) 46 (read head does not move) 46](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-46.jpg)
![47 47](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-47.jpg)
![all input is consumed “accept” String is accepted 48 all input is consumed “accept” String is accepted 48](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-48.jpg)
![Rejection Example 49 Rejection Example 49](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-49.jpg)
![50 50](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-50.jpg)
![(read head doesn’t move) 51 (read head doesn’t move) 51](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-51.jpg)
![No transition: the automaton hangs 52 No transition: the automaton hangs 52](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-52.jpg)
![Input cannot be consumed “reject” String is rejected 53 Input cannot be consumed “reject” String is rejected 53](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-53.jpg)
![Language accepted: 54 Language accepted: 54](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-54.jpg)
![Another NFA Example 55 Another NFA Example 55](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-55.jpg)
![56 56](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-56.jpg)
![57 57](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-57.jpg)
![58 58](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-58.jpg)
![“accept” 59 “accept” 59](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-59.jpg)
![Another String 60 Another String 60](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-60.jpg)
![61 61](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-61.jpg)
![62 62](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-62.jpg)
![63 63](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-63.jpg)
![64 64](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-64.jpg)
![65 65](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-65.jpg)
![66 66](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-66.jpg)
![“accept” 67 “accept” 67](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-67.jpg)
![Language accepted 68 Language accepted 68](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-68.jpg)
![Another NFA Example 69 Another NFA Example 69](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-69.jpg)
![Language accepted (redundant state) 70 Language accepted (redundant state) 70](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-70.jpg)
![Remarks: • The symbol never appears on the input state • Simple automata: 71 Remarks: • The symbol never appears on the input state • Simple automata: 71](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-71.jpg)
![• NFAs are interesting because we can express languages easier than DFAs NFA • NFAs are interesting because we can express languages easier than DFAs NFA](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-72.jpg)
![Formal Definition of NFAs n Set of states, i. e. Input aplhabet, i. e. Formal Definition of NFAs n Set of states, i. e. Input aplhabet, i. e.](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-73.jpg)
![Transition Function 74 Transition Function 74](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-74.jpg)
![75 75](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-75.jpg)
![76 76](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-76.jpg)
![77 77](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-77.jpg)
![Extended Transition Function 78 Extended Transition Function 78](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-78.jpg)
![79 79](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-79.jpg)
![80 80](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-80.jpg)
![We will prove: Languages accepted by NFAs Regular Languages accepted by DFAs 81 We will prove: Languages accepted by NFAs Regular Languages accepted by DFAs 81](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-81.jpg)
![NFA Convert NFA to DFA 82 NFA Convert NFA to DFA 82](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-82.jpg)
![Convert NFA to DFA NFA DFA 83 Convert NFA to DFA NFA DFA 83](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-83.jpg)
![Convert NFA to DFA NFA n DFA 84 Convert NFA to DFA NFA n DFA 84](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-84.jpg)
![Convert NFA to DFA NFA n DFA 85 Convert NFA to DFA NFA n DFA 85](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-85.jpg)
![Convert NFA to DFA NFA n DFA 86 Convert NFA to DFA NFA n DFA 86](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-86.jpg)
![Convert NFA to DFA NFA n DFA 87 Convert NFA to DFA NFA n DFA 87](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-87.jpg)
![Convert NFA to DFA NFA n DFA 88 Convert NFA to DFA NFA n DFA 88](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-88.jpg)
![Thank You! 89 Thank You! 89](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-89.jpg)
- Slides: 89
![Theory of Computation Theory of Computation](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-1.jpg)
Theory of Computation
![Nondeterministic Finite Automaton NFA An NFA is a collection of three things 1 Finite Nondeterministic Finite Automaton (NFA) An NFA is a collection of three things 1) Finite](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-2.jpg)
Nondeterministic Finite Automaton (NFA) An NFA is a collection of three things 1) Finite states with one initial and some final states 2) Finite set of input letters (Σ) from which input strings are formed 3) Finite set of transitions, showing where to move if a letter is input at certain state, there may be more than one transition for certain letters and there may not be any transition for certain letters. 2
![Observations The string is supposed to be accepted if there exists at least one Observations The string is supposed to be accepted, if there exists at least one](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-3.jpg)
Observations The string is supposed to be accepted, if there exists at least one successful path, otherwise rejected. NFA can be considered to be an intermediate structure between FA and TG. 3
![Example a 1 a 2 3 a 4 b 5 The above NFA Example a 1 - a 2+ 3 a 4 b 5+ The above NFA](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-4.jpg)
Example a 1 - a 2+ 3 a 4 b 5+ The above NFA accepts the language consisting of a and ab. 4
![Example a b 1 a b a 2 a 3 The above NFA Example a, b 1 - a, b a 2 a 3+ The above NFA](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-5.jpg)
Example a, b 1 - a, b a 2 a 3+ The above NFA accepts the language of strings, defined over Σ = {a, b}, containing aa. 5
![Nondeterminism This refers to situations in which the next state of a computation is Non-determinism This refers to situations in which the next state of a computation is](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-6.jpg)
Non-determinism This refers to situations in which the next state of a computation is not uniquely determined by the current state. n In these situations we know only a range of possibilities n 6
![Converting an FA to an equivalent NFA n If a language is accepted by Converting an FA to an equivalent NFA n If a language is accepted by](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-7.jpg)
Converting an FA to an equivalent NFA n If a language is accepted by an FA, then there exists a TG accepting that language(Kleene’s theorem). n Since, an NFA is also a TG, there exists an NFA accepting the language accepted by the given FA. In this case these FA and NFA are said to be equivalent to each others. 7
![Example n Consider the following FA corresponding to abb a b b n Example n Consider the following FA corresponding to (a+b)*b a b b n +](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-8.jpg)
Example n Consider the following FA corresponding to (a+b)*b a b b n + a The above FA may be equivalent to the following NFA a, b - b + 8
![Example n Consider the following FA b a n a b b a 1 Example n Consider the following FA b a n a, b b a 1](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-9.jpg)
Example n Consider the following FA b a n a, b b a 1 + The above FA may be equivalent to the following NFA a, b - a 1 a a, b + 9
![Note n n Every FA can be considered to be an NFA but the Note n n Every FA can be considered to be an NFA, but the](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-10.jpg)
Note n n Every FA can be considered to be an NFA, but the converse may not true. Every NFA can be considered to be a TG, but the converse may not true. 10
![NFA to FA An NFA can be considered to be a TG so a NFA to FA An NFA can be considered to be a TG, so a](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-11.jpg)
NFA to FA An NFA can be considered to be a TG, so a RE corresponding to the given NFA can be determined Again using Kleene’s theorem, an FA can be built corresponding to that RE. 11
![2 Example b a 1 4 b a 3 a 1 2 2 Example b a 1 - 4+ b a 3 a 1 - 2](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-12.jpg)
2 Example b a 1 - 4+ b a 3 a 1 - 2 b b a 3 4+ b a a, b 12
![Nondeterministic Finite Automaton NFA Cont Alphabet 13 Nondeterministic Finite Automaton (NFA) Cont… Alphabet = 13](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-13.jpg)
Nondeterministic Finite Automaton (NFA) Cont… Alphabet = 13
![Alphabet Two choices 14 Alphabet = Two choices 14](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-14.jpg)
Alphabet = Two choices 14
![Alphabet Two choices No transition 15 Alphabet = Two choices No transition 15](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-15.jpg)
Alphabet = Two choices No transition 15
![First Choice 16 First Choice 16](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-16.jpg)
First Choice 16
![First Choice 17 First Choice 17](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-17.jpg)
First Choice 17
![First Choice 18 First Choice 18](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-18.jpg)
First Choice 18
![First Choice All input is consumed accept 19 First Choice All input is consumed “accept” 19](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-19.jpg)
First Choice All input is consumed “accept” 19
![Second Choice 20 Second Choice 20](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-20.jpg)
Second Choice 20
![Second Choice 21 Second Choice 21](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-21.jpg)
Second Choice 21
![Second Choice No transition the automaton hangs 22 Second Choice No transition: the automaton hangs 22](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-22.jpg)
Second Choice No transition: the automaton hangs 22
![Second Choice Input cannot be consumed reject 23 Second Choice Input cannot be consumed “reject” 23](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-23.jpg)
Second Choice Input cannot be consumed “reject” 23
![An NFA accepts a string if there is a computation of the NFA that An NFA accepts a string: if there is a computation of the NFA that](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-24.jpg)
An NFA accepts a string: if there is a computation of the NFA that accepts the string i. e. , all the input is consumed and the automaton is in an accepting state 24
![Example is accepted by the NFA accept because this computation accepts reject 25 Example is accepted by the NFA: “accept” because this computation accepts “reject” 25](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-25.jpg)
Example is accepted by the NFA: “accept” because this computation accepts “reject” 25
![Rejection example 26 Rejection example 26](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-26.jpg)
Rejection example 26
![First Choice 27 First Choice 27](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-27.jpg)
First Choice 27
![First Choice reject 28 First Choice “reject” 28](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-28.jpg)
First Choice “reject” 28
![Second Choice 29 Second Choice 29](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-29.jpg)
Second Choice 29
![Second Choice 30 Second Choice 30](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-30.jpg)
Second Choice 30
![Second Choice reject 31 Second Choice “reject” 31](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-31.jpg)
Second Choice “reject” 31
![Example is rejected by the NFA reject All possible computations lead to rejection 32 Example is rejected by the NFA: “reject” All possible computations lead to rejection 32](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-32.jpg)
Example is rejected by the NFA: “reject” All possible computations lead to rejection 32
![Rejection example 33 Rejection example 33](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-33.jpg)
Rejection example 33
![First Choice 34 First Choice 34](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-34.jpg)
First Choice 34
![First Choice No transition the automaton hangs 35 First Choice No transition: the automaton hangs 35](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-35.jpg)
First Choice No transition: the automaton hangs 35
![First Choice Input cannot be consumed reject 36 First Choice Input cannot be consumed “reject” 36](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-36.jpg)
First Choice Input cannot be consumed “reject” 36
![Second Choice 37 Second Choice 37](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-37.jpg)
Second Choice 37
![Second Choice 38 Second Choice 38](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-38.jpg)
Second Choice 38
![Second Choice No transition the automaton hangs 39 Second Choice No transition: the automaton hangs 39](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-39.jpg)
Second Choice No transition: the automaton hangs 39
![Second Choice Input cannot be consumed reject 40 Second Choice Input cannot be consumed “reject” 40](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-40.jpg)
Second Choice Input cannot be consumed “reject” 40
![is rejected by the NFA reject All possible computations lead to rejection 41 is rejected by the NFA: “reject” All possible computations lead to rejection 41](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-41.jpg)
is rejected by the NFA: “reject” All possible computations lead to rejection 41
![Language accepted 42 Language accepted: 42](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-42.jpg)
Language accepted: 42
![Lambda Transitions 43 Lambda Transitions 43](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-43.jpg)
Lambda Transitions 43
![44 44](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-44.jpg)
44
![45 45](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-45.jpg)
45
![read head does not move 46 (read head does not move) 46](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-46.jpg)
(read head does not move) 46
![47 47](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-47.jpg)
47
![all input is consumed accept String is accepted 48 all input is consumed “accept” String is accepted 48](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-48.jpg)
all input is consumed “accept” String is accepted 48
![Rejection Example 49 Rejection Example 49](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-49.jpg)
Rejection Example 49
![50 50](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-50.jpg)
50
![read head doesnt move 51 (read head doesn’t move) 51](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-51.jpg)
(read head doesn’t move) 51
![No transition the automaton hangs 52 No transition: the automaton hangs 52](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-52.jpg)
No transition: the automaton hangs 52
![Input cannot be consumed reject String is rejected 53 Input cannot be consumed “reject” String is rejected 53](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-53.jpg)
Input cannot be consumed “reject” String is rejected 53
![Language accepted 54 Language accepted: 54](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-54.jpg)
Language accepted: 54
![Another NFA Example 55 Another NFA Example 55](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-55.jpg)
Another NFA Example 55
![56 56](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-56.jpg)
56
![57 57](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-57.jpg)
57
![58 58](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-58.jpg)
58
![accept 59 “accept” 59](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-59.jpg)
“accept” 59
![Another String 60 Another String 60](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-60.jpg)
Another String 60
![61 61](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-61.jpg)
61
![62 62](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-62.jpg)
62
![63 63](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-63.jpg)
63
![64 64](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-64.jpg)
64
![65 65](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-65.jpg)
65
![66 66](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-66.jpg)
66
![accept 67 “accept” 67](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-67.jpg)
“accept” 67
![Language accepted 68 Language accepted 68](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-68.jpg)
Language accepted 68
![Another NFA Example 69 Another NFA Example 69](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-69.jpg)
Another NFA Example 69
![Language accepted redundant state 70 Language accepted (redundant state) 70](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-70.jpg)
Language accepted (redundant state) 70
![Remarks The symbol never appears on the input state Simple automata 71 Remarks: • The symbol never appears on the input state • Simple automata: 71](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-71.jpg)
Remarks: • The symbol never appears on the input state • Simple automata: 71
![NFAs are interesting because we can express languages easier than DFAs NFA • NFAs are interesting because we can express languages easier than DFAs NFA](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-72.jpg)
• NFAs are interesting because we can express languages easier than DFAs NFA DFA 72
![Formal Definition of NFAs n Set of states i e Input aplhabet i e Formal Definition of NFAs n Set of states, i. e. Input aplhabet, i. e.](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-73.jpg)
Formal Definition of NFAs n Set of states, i. e. Input aplhabet, i. e. Transition function Initial state Accepting states 73
![Transition Function 74 Transition Function 74](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-74.jpg)
Transition Function 74
![75 75](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-75.jpg)
75
![76 76](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-76.jpg)
76
![77 77](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-77.jpg)
77
![Extended Transition Function 78 Extended Transition Function 78](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-78.jpg)
Extended Transition Function 78
![79 79](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-79.jpg)
79
![80 80](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-80.jpg)
80
![We will prove Languages accepted by NFAs Regular Languages accepted by DFAs 81 We will prove: Languages accepted by NFAs Regular Languages accepted by DFAs 81](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-81.jpg)
We will prove: Languages accepted by NFAs Regular Languages accepted by DFAs 81
![NFA Convert NFA to DFA 82 NFA Convert NFA to DFA 82](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-82.jpg)
NFA Convert NFA to DFA 82
![Convert NFA to DFA NFA DFA 83 Convert NFA to DFA NFA DFA 83](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-83.jpg)
Convert NFA to DFA NFA DFA 83
![Convert NFA to DFA NFA n DFA 84 Convert NFA to DFA NFA n DFA 84](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-84.jpg)
Convert NFA to DFA NFA n DFA 84
![Convert NFA to DFA NFA n DFA 85 Convert NFA to DFA NFA n DFA 85](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-85.jpg)
Convert NFA to DFA NFA n DFA 85
![Convert NFA to DFA NFA n DFA 86 Convert NFA to DFA NFA n DFA 86](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-86.jpg)
Convert NFA to DFA NFA n DFA 86
![Convert NFA to DFA NFA n DFA 87 Convert NFA to DFA NFA n DFA 87](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-87.jpg)
Convert NFA to DFA NFA n DFA 87
![Convert NFA to DFA NFA n DFA 88 Convert NFA to DFA NFA n DFA 88](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-88.jpg)
Convert NFA to DFA NFA n DFA 88
![Thank You 89 Thank You! 89](https://slidetodoc.com/presentation_image/6235168d421dca0bc72344cb71f6af6c/image-89.jpg)
Thank You! 89
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