Another NFA Example Fall 2004 COMP 335 1
![Another NFA Example Fall 2004 COMP 335 1 Another NFA Example Fall 2004 COMP 335 1](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-1.jpg)
![Language accepted (redundant state) Fall 2004 COMP 335 2 Language accepted (redundant state) Fall 2004 COMP 335 2](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-2.jpg)
![Remarks: • The symbol never appears on the input tape • Simple automata: Fall Remarks: • The symbol never appears on the input tape • Simple automata: Fall](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-3.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/92f4a119e5404190b84b9dc83d38c0a9/image-4.jpg)
![Formal Definition of NFAs Set of states, i. e. Input aphabet, i. e. Transition Formal Definition of NFAs Set of states, i. e. Input aphabet, i. e. Transition](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-5.jpg)
![Transition Function Fall 2004 COMP 335 6 Transition Function Fall 2004 COMP 335 6](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-6.jpg)
![Fall 2004 COMP 335 7 Fall 2004 COMP 335 7](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-7.jpg)
![Fall 2004 COMP 335 8 Fall 2004 COMP 335 8](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-8.jpg)
![Fall 2004 COMP 335 9 Fall 2004 COMP 335 9](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-9.jpg)
![Extended Transition Function Fall 2004 COMP 335 10 Extended Transition Function Fall 2004 COMP 335 10](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-10.jpg)
![Fall 2004 COMP 335 11 Fall 2004 COMP 335 11](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-11.jpg)
![Fall 2004 COMP 335 12 Fall 2004 COMP 335 12](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-12.jpg)
![Formally : there is a walk from with label Fall 2004 COMP 335 to Formally : there is a walk from with label Fall 2004 COMP 335 to](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-13.jpg)
![The Language of an NFA Fall 2004 COMP 335 14 The Language of an NFA Fall 2004 COMP 335 14](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-14.jpg)
![Fall 2004 COMP 335 15 Fall 2004 COMP 335 15](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-15.jpg)
![Fall 2004 COMP 335 16 Fall 2004 COMP 335 16](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-16.jpg)
![Fall 2004 COMP 335 17 Fall 2004 COMP 335 17](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-17.jpg)
![Fall 2004 COMP 335 18 Fall 2004 COMP 335 18](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-18.jpg)
![Formally The language accepted by NFA is: where (final state) and there is some Formally The language accepted by NFA is: where (final state) and there is some](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-19.jpg)
![Fall 2004 COMP 335 20 Fall 2004 COMP 335 20](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-20.jpg)
![NFA accept Regular Languages Fall 2004 COMP 335 21 NFA accept Regular Languages Fall 2004 COMP 335 21](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-21.jpg)
![Equivalence of FA Definition: An FA is equivalent to FA if that is if Equivalence of FA Definition: An FA is equivalent to FA if that is if](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-22.jpg)
![Example of equivalent FA NFA DFA Fall 2004 COMP 335 23 Example of equivalent FA NFA DFA Fall 2004 COMP 335 23](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-23.jpg)
![We will prove: Languages accepted by NFA Regular Languages That is, NFA and DFA We will prove: Languages accepted by NFA Regular Languages That is, NFA and DFA](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-24.jpg)
![Step 1 Languages accepted by NFA Regular Languages Proof: Every DFA is trivially an Step 1 Languages accepted by NFA Regular Languages Proof: Every DFA is trivially an](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-25.jpg)
![Step 2 Languages accepted by NFA Regular Languages Proof: Any NFA can be converted Step 2 Languages accepted by NFA Regular Languages Proof: Any NFA can be converted](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-26.jpg)
![Convert NFA to DFA NFA DFA Fall 2004 COMP 335 27 Convert NFA to DFA NFA DFA Fall 2004 COMP 335 27](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-27.jpg)
![Convert NFA to DFA NFA DFA Fall 2004 COMP 335 28 Convert NFA to DFA NFA DFA Fall 2004 COMP 335 28](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-28.jpg)
![Convert NFA to DFA NFA DFA Fall 2004 COMP 335 29 Convert NFA to DFA NFA DFA Fall 2004 COMP 335 29](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-29.jpg)
![Convert NFA to DFA NFA DFA Fall 2004 COMP 335 30 Convert NFA to DFA NFA DFA Fall 2004 COMP 335 30](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-30.jpg)
![Convert NFA to DFA NFA DFA Fall 2004 COMP 335 31 Convert NFA to DFA NFA DFA Fall 2004 COMP 335 31](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-31.jpg)
![Convert NFA to DFA NFA DFA Fall 2004 COMP 335 32 Convert NFA to DFA NFA DFA Fall 2004 COMP 335 32](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-32.jpg)
![Convert NFA to DFA NFA DFA Fall 2004 COMP 335 33 Convert NFA to DFA NFA DFA Fall 2004 COMP 335 33](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-33.jpg)
![NFA to DFA: Remarks We are given an NFA We want to convert it NFA to DFA: Remarks We are given an NFA We want to convert it](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-34.jpg)
![If the NFA has states Then the DFA has states in the powerset Fall If the NFA has states Then the DFA has states in the powerset Fall](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-35.jpg)
![Procedure NFA to DFA 1. Initial state of NFA: Initial state of DFA: Fall Procedure NFA to DFA 1. Initial state of NFA: Initial state of DFA: Fall](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-36.jpg)
![Example NFA DFA Fall 2004 COMP 335 37 Example NFA DFA Fall 2004 COMP 335 37](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-37.jpg)
![Procedure NFA to DFA 2. For every DFA’s state Compute in the NFA Add Procedure NFA to DFA 2. For every DFA’s state Compute in the NFA Add](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-38.jpg)
![Example NFA DFA Fall 2004 COMP 335 39 Example NFA DFA Fall 2004 COMP 335 39](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-39.jpg)
![Procedure NFA to DFA Repeat step 2 for all symbols in the alphabet ∑, Procedure NFA to DFA Repeat step 2 for all symbols in the alphabet ∑,](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-40.jpg)
![Example NFA DFA Fall 2004 COMP 335 41 Example NFA DFA Fall 2004 COMP 335 41](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-41.jpg)
![Procedure NFA to DFA 3. For any DFA state: If some is a final Procedure NFA to DFA 3. For any DFA state: If some is a final](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-42.jpg)
![Example NFA DFA Fall 2004 COMP 335 43 Example NFA DFA Fall 2004 COMP 335 43](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-43.jpg)
![Take NFA Theorem Apply the procedure to obtain DFA Then, Fall 2004 and are Take NFA Theorem Apply the procedure to obtain DFA Then, Fall 2004 and are](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-44.jpg)
![Proof AND Fall 2004 COMP 335 45 Proof AND Fall 2004 COMP 335 45](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-45.jpg)
![First we show: Take arbitrary string : We will prove: Fall 2004 COMP 335 First we show: Take arbitrary string : We will prove: Fall 2004 COMP 335](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-46.jpg)
![Fall 2004 COMP 335 47 Fall 2004 COMP 335 47](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-47.jpg)
![We will show that if Fall 2004 COMP 335 48 We will show that if Fall 2004 COMP 335 48](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-48.jpg)
![More generally, we will show that if in : (arbitrary string) Fall 2004 COMP More generally, we will show that if in : (arbitrary string) Fall 2004 COMP](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-49.jpg)
![Proof by induction on The basis case: Fall 2004 COMP 335 50 Proof by induction on The basis case: Fall 2004 COMP 335 50](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-50.jpg)
![Induction hypothesis: Fall 2004 COMP 335 51 Induction hypothesis: Fall 2004 COMP 335 51](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-51.jpg)
![Induction Step: Fall 2004 COMP 335 52 Induction Step: Fall 2004 COMP 335 52](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-52.jpg)
![Induction Step: Fall 2004 COMP 335 53 Induction Step: Fall 2004 COMP 335 53](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-53.jpg)
![Therefore if Fall 2004 COMP 335 54 Therefore if Fall 2004 COMP 335 54](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-54.jpg)
![We have shown: We also need to show: (proof is similar) Fall 2004 COMP We have shown: We also need to show: (proof is similar) Fall 2004 COMP](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-55.jpg)
- Slides: 55
![Another NFA Example Fall 2004 COMP 335 1 Another NFA Example Fall 2004 COMP 335 1](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-1.jpg)
Another NFA Example Fall 2004 COMP 335 1
![Language accepted redundant state Fall 2004 COMP 335 2 Language accepted (redundant state) Fall 2004 COMP 335 2](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-2.jpg)
Language accepted (redundant state) Fall 2004 COMP 335 2
![Remarks The symbol never appears on the input tape Simple automata Fall Remarks: • The symbol never appears on the input tape • Simple automata: Fall](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-3.jpg)
Remarks: • The symbol never appears on the input tape • Simple automata: Fall 2004 COMP 335 3
![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/92f4a119e5404190b84b9dc83d38c0a9/image-4.jpg)
• NFAs are interesting because we can express languages easier than DFAs NFA Fall 2004 DFA COMP 335 4
![Formal Definition of NFAs Set of states i e Input aphabet i e Transition Formal Definition of NFAs Set of states, i. e. Input aphabet, i. e. Transition](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-5.jpg)
Formal Definition of NFAs Set of states, i. e. Input aphabet, i. e. Transition function Initial state Final states Fall 2004 COMP 335 5
![Transition Function Fall 2004 COMP 335 6 Transition Function Fall 2004 COMP 335 6](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-6.jpg)
Transition Function Fall 2004 COMP 335 6
![Fall 2004 COMP 335 7 Fall 2004 COMP 335 7](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-7.jpg)
Fall 2004 COMP 335 7
![Fall 2004 COMP 335 8 Fall 2004 COMP 335 8](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-8.jpg)
Fall 2004 COMP 335 8
![Fall 2004 COMP 335 9 Fall 2004 COMP 335 9](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-9.jpg)
Fall 2004 COMP 335 9
![Extended Transition Function Fall 2004 COMP 335 10 Extended Transition Function Fall 2004 COMP 335 10](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-10.jpg)
Extended Transition Function Fall 2004 COMP 335 10
![Fall 2004 COMP 335 11 Fall 2004 COMP 335 11](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-11.jpg)
Fall 2004 COMP 335 11
![Fall 2004 COMP 335 12 Fall 2004 COMP 335 12](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-12.jpg)
Fall 2004 COMP 335 12
![Formally there is a walk from with label Fall 2004 COMP 335 to Formally : there is a walk from with label Fall 2004 COMP 335 to](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-13.jpg)
Formally : there is a walk from with label Fall 2004 COMP 335 to 13
![The Language of an NFA Fall 2004 COMP 335 14 The Language of an NFA Fall 2004 COMP 335 14](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-14.jpg)
The Language of an NFA Fall 2004 COMP 335 14
![Fall 2004 COMP 335 15 Fall 2004 COMP 335 15](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-15.jpg)
Fall 2004 COMP 335 15
![Fall 2004 COMP 335 16 Fall 2004 COMP 335 16](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-16.jpg)
Fall 2004 COMP 335 16
![Fall 2004 COMP 335 17 Fall 2004 COMP 335 17](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-17.jpg)
Fall 2004 COMP 335 17
![Fall 2004 COMP 335 18 Fall 2004 COMP 335 18](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-18.jpg)
Fall 2004 COMP 335 18
![Formally The language accepted by NFA is where final state and there is some Formally The language accepted by NFA is: where (final state) and there is some](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-19.jpg)
Formally The language accepted by NFA is: where (final state) and there is some Fall 2004 COMP 335 19
![Fall 2004 COMP 335 20 Fall 2004 COMP 335 20](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-20.jpg)
Fall 2004 COMP 335 20
![NFA accept Regular Languages Fall 2004 COMP 335 21 NFA accept Regular Languages Fall 2004 COMP 335 21](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-21.jpg)
NFA accept Regular Languages Fall 2004 COMP 335 21
![Equivalence of FA Definition An FA is equivalent to FA if that is if Equivalence of FA Definition: An FA is equivalent to FA if that is if](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-22.jpg)
Equivalence of FA Definition: An FA is equivalent to FA if that is if both accept the same language. Fall 2004 COMP 335 22
![Example of equivalent FA NFA DFA Fall 2004 COMP 335 23 Example of equivalent FA NFA DFA Fall 2004 COMP 335 23](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-23.jpg)
Example of equivalent FA NFA DFA Fall 2004 COMP 335 23
![We will prove Languages accepted by NFA Regular Languages That is NFA and DFA We will prove: Languages accepted by NFA Regular Languages That is, NFA and DFA](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-24.jpg)
We will prove: Languages accepted by NFA Regular Languages That is, NFA and DFA have the same computation power Fall 2004 COMP 335 Languages accepted by DFA 24
![Step 1 Languages accepted by NFA Regular Languages Proof Every DFA is trivially an Step 1 Languages accepted by NFA Regular Languages Proof: Every DFA is trivially an](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-25.jpg)
Step 1 Languages accepted by NFA Regular Languages Proof: Every DFA is trivially an NFA Any language accepted by a DFA is also accepted by an NFA Fall 2004 COMP 335 25
![Step 2 Languages accepted by NFA Regular Languages Proof Any NFA can be converted Step 2 Languages accepted by NFA Regular Languages Proof: Any NFA can be converted](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-26.jpg)
Step 2 Languages accepted by NFA Regular Languages Proof: Any NFA can be converted into an equivalent DFA Fall 2004 Any language accepted by an NFA is also accepted by a DFA COMP 335 26
![Convert NFA to DFA NFA DFA Fall 2004 COMP 335 27 Convert NFA to DFA NFA DFA Fall 2004 COMP 335 27](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-27.jpg)
Convert NFA to DFA NFA DFA Fall 2004 COMP 335 27
![Convert NFA to DFA NFA DFA Fall 2004 COMP 335 28 Convert NFA to DFA NFA DFA Fall 2004 COMP 335 28](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-28.jpg)
Convert NFA to DFA NFA DFA Fall 2004 COMP 335 28
![Convert NFA to DFA NFA DFA Fall 2004 COMP 335 29 Convert NFA to DFA NFA DFA Fall 2004 COMP 335 29](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-29.jpg)
Convert NFA to DFA NFA DFA Fall 2004 COMP 335 29
![Convert NFA to DFA NFA DFA Fall 2004 COMP 335 30 Convert NFA to DFA NFA DFA Fall 2004 COMP 335 30](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-30.jpg)
Convert NFA to DFA NFA DFA Fall 2004 COMP 335 30
![Convert NFA to DFA NFA DFA Fall 2004 COMP 335 31 Convert NFA to DFA NFA DFA Fall 2004 COMP 335 31](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-31.jpg)
Convert NFA to DFA NFA DFA Fall 2004 COMP 335 31
![Convert NFA to DFA NFA DFA Fall 2004 COMP 335 32 Convert NFA to DFA NFA DFA Fall 2004 COMP 335 32](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-32.jpg)
Convert NFA to DFA NFA DFA Fall 2004 COMP 335 32
![Convert NFA to DFA NFA DFA Fall 2004 COMP 335 33 Convert NFA to DFA NFA DFA Fall 2004 COMP 335 33](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-33.jpg)
Convert NFA to DFA NFA DFA Fall 2004 COMP 335 33
![NFA to DFA Remarks We are given an NFA We want to convert it NFA to DFA: Remarks We are given an NFA We want to convert it](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-34.jpg)
NFA to DFA: Remarks We are given an NFA We want to convert it into an equivalent DFA That is, Fall 2004 COMP 335 34
![If the NFA has states Then the DFA has states in the powerset Fall If the NFA has states Then the DFA has states in the powerset Fall](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-35.jpg)
If the NFA has states Then the DFA has states in the powerset Fall 2004 COMP 335 35
![Procedure NFA to DFA 1 Initial state of NFA Initial state of DFA Fall Procedure NFA to DFA 1. Initial state of NFA: Initial state of DFA: Fall](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-36.jpg)
Procedure NFA to DFA 1. Initial state of NFA: Initial state of DFA: Fall 2004 COMP 335 36
![Example NFA DFA Fall 2004 COMP 335 37 Example NFA DFA Fall 2004 COMP 335 37](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-37.jpg)
Example NFA DFA Fall 2004 COMP 335 37
![Procedure NFA to DFA 2 For every DFAs state Compute in the NFA Add Procedure NFA to DFA 2. For every DFA’s state Compute in the NFA Add](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-38.jpg)
Procedure NFA to DFA 2. For every DFA’s state Compute in the NFA Add the following transition to the DFA Fall 2004 COMP 335 38
![Example NFA DFA Fall 2004 COMP 335 39 Example NFA DFA Fall 2004 COMP 335 39](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-39.jpg)
Example NFA DFA Fall 2004 COMP 335 39
![Procedure NFA to DFA Repeat step 2 for all symbols in the alphabet Procedure NFA to DFA Repeat step 2 for all symbols in the alphabet ∑,](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-40.jpg)
Procedure NFA to DFA Repeat step 2 for all symbols in the alphabet ∑, until no more transitions can be added. Fall 2004 COMP 335 40
![Example NFA DFA Fall 2004 COMP 335 41 Example NFA DFA Fall 2004 COMP 335 41](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-41.jpg)
Example NFA DFA Fall 2004 COMP 335 41
![Procedure NFA to DFA 3 For any DFA state If some is a final Procedure NFA to DFA 3. For any DFA state: If some is a final](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-42.jpg)
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 Fall 2004 COMP 335 42
![Example NFA DFA Fall 2004 COMP 335 43 Example NFA DFA Fall 2004 COMP 335 43](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-43.jpg)
Example NFA DFA Fall 2004 COMP 335 43
![Take NFA Theorem Apply the procedure to obtain DFA Then Fall 2004 and are Take NFA Theorem Apply the procedure to obtain DFA Then, Fall 2004 and are](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-44.jpg)
Take NFA Theorem Apply the procedure to obtain DFA Then, Fall 2004 and are equivalent: COMP 335 44
![Proof AND Fall 2004 COMP 335 45 Proof AND Fall 2004 COMP 335 45](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-45.jpg)
Proof AND Fall 2004 COMP 335 45
![First we show Take arbitrary string We will prove Fall 2004 COMP 335 First we show: Take arbitrary string : We will prove: Fall 2004 COMP 335](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-46.jpg)
First we show: Take arbitrary string : We will prove: Fall 2004 COMP 335 46
![Fall 2004 COMP 335 47 Fall 2004 COMP 335 47](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-47.jpg)
Fall 2004 COMP 335 47
![We will show that if Fall 2004 COMP 335 48 We will show that if Fall 2004 COMP 335 48](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-48.jpg)
We will show that if Fall 2004 COMP 335 48
![More generally we will show that if in arbitrary string Fall 2004 COMP More generally, we will show that if in : (arbitrary string) Fall 2004 COMP](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-49.jpg)
More generally, we will show that if in : (arbitrary string) Fall 2004 COMP 335 49
![Proof by induction on The basis case Fall 2004 COMP 335 50 Proof by induction on The basis case: Fall 2004 COMP 335 50](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-50.jpg)
Proof by induction on The basis case: Fall 2004 COMP 335 50
![Induction hypothesis Fall 2004 COMP 335 51 Induction hypothesis: Fall 2004 COMP 335 51](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-51.jpg)
Induction hypothesis: Fall 2004 COMP 335 51
![Induction Step Fall 2004 COMP 335 52 Induction Step: Fall 2004 COMP 335 52](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-52.jpg)
Induction Step: Fall 2004 COMP 335 52
![Induction Step Fall 2004 COMP 335 53 Induction Step: Fall 2004 COMP 335 53](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-53.jpg)
Induction Step: Fall 2004 COMP 335 53
![Therefore if Fall 2004 COMP 335 54 Therefore if Fall 2004 COMP 335 54](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-54.jpg)
Therefore if Fall 2004 COMP 335 54
![We have shown We also need to show proof is similar Fall 2004 COMP We have shown: We also need to show: (proof is similar) Fall 2004 COMP](https://slidetodoc.com/presentation_image/92f4a119e5404190b84b9dc83d38c0a9/image-55.jpg)
We have shown: We also need to show: (proof is similar) Fall 2004 COMP 335 55
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