| [0.05s -> 14.45s] Hello everyone. In this lecture, we will be studying about minimization of DFA. We have already studied about DFA. We have also studied about NFA. And we have also studied how to convert NFA to DFA. |
| [14.45s -> 26.61s] And in this lecture, we will be studying about minimization of DFA. But what is minimization and why is it required? Minimization of DFA is required to |
| [26.61s -> 38.32s] to obtain the minimal version of any DFA which consists of the minimum number of states possible. Alright, so what does this mean? |
| [38.32s -> 52.67s] example to explain this definition. Suppose you are given the task to design a DFA. Any DFA. And then you design this DFA using 5 states. Alright. And your friend |
| [52.67s -> 56.37s] Design the same DFA using 4 states. |
| [56.78s -> 71.09s] Both the DFS are correct. Both the DFS perform the exact same task. But one of them is designed using 5 states and another is designed using 4 states. Both of them are correct. |
| [71.09s -> 84.21s] Here we see that the same DFA can be designed using a lesser number of states. Now which one do you think is more efficient? 5 state DFA or the 4 state DFA? Obviously it will be the one with the |
| [84.21s -> 96.78s] lesser number of states. So, we want to design the DFA using the minimum number of states possible. That is known as the minimal version of any DFA. |
| [96.78s -> 110.16s] If you try to design a DFA directly in such a way to get the minimal version, it may be difficult for you. It is not impossible but it may be difficult. It is possible only after you |
| [110.16s -> 121.65s] practice and practice but there is a way of minimizing a given DFA. Given a DFA you can apply some technique and minimize it and make it |
| [121.65s -> 127.09s] to the minimal version. And that is what we are going to study in this lecture. |
| [127.34s -> 141.04s] So, how can we minimize DFA? So, let's say that you use these 5 states 1, 2, 3, 4, 5. You have these 5 states and you want to minimize this DFA. That means you want to reduce the number of |
| [141.04s -> 155.12s] states but keep the DFA performing the same thing. So how can you do this? What you can do is you can combine two states. Let's say these two states you combine them together and you make this a single state. |
| [155.15s -> 169.04s] And then now you have 1, 2, 3, 4 states. So, that is how you can minimize it. But how can you simply combine two states? You cannot just simply combine two states. There is a condition when you can combine two states. |
| [169.04s -> 183.28s] And what is that condition? Two states can be combined only when these two states are equivalent. Now when are two states said to be equivalent? |
| [183.28s -> 191.70s] Equivalent. What is the meaning of equivalence? Two states A and B are said to be equivalent if |
| [192.02s -> 206.46s] A on getting a particular input string x. Here x is any input string. So, if the state A on seeing the input string x goes to a final state. |
| [206.46s -> 220.90s] And at the same time if state B also on getting that same input string goes to any of the final states then A and B are said to be equivalent. |
| [220.90s -> 223.57s] or if |
| [224.05s -> 237.71s] a on getting an input string x does not go to the final state and also b on getting the particular input string x does not go to any of the final states then |
| [237.71s -> 249.49s] also a and b are said to be equivalent okay and this will become more clear to you when we take some examples and now what we have to study is |
| [249.65s -> 261.55s] Types of equivalents. There are some different kinds of equivalents like 0 equivalents, 1 equivalents, 2 equivalents and so on. So, next we will be seeing what is that. |
| [262.35s -> 272.90s] So here we see that if modulo x equal to 0 this means that if the length of the string x here we have taken |
| [272.90s -> 286.67s] x as any input string. If the length of that string x is 0, then a and b are said to be 0 equivalent. Alright. And if the length of x is equal to 1, |
| [286.67s -> 300.91s] then a and b are said to be 1 equivalent. And if the length of string x is equal to 2, then a and b are said to be 2 equivalent. So, in general we can write that if the |
| [300.91s -> 314.14s] length of the string x is equal to n then a and b are said to be n equivalent all right so these are the type of equivalences that we have and we already studied |
| [314.14s -> 327.63s] When are two states a and b set to be equivalent? It is with these conditions. When on seeing a particular input string x, if both a and b either goes to the final state or |
| [327.63s -> 339.97s] It does not go to any final state. Then they are said to be equivalent. And why do we need this equivalent property? We need it in order to combine the states, in order to reduce the number of states to get the |
| [339.97s -> 354.24s] minimum number of states possible in order to design the minimal version of any DFA. So, this was the theoretical explanation and in the next lecture we will be seeing an example which will make it. |
| [354.24s -> 357.97s] very clear to you. So, see you in the next one with an example. |
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